Numerical Solution of Partial Differential Equations with Fuzzy Transform. By Huda Salim El-Zerii

Transcription

1 Al-Azhar University-Gaza (AUG) Deanship of postgraduate Studies Faculty of science Department of Mathematics Numerical Solution of Partial Differential Equations with Fuzzy Transform By Huda Salim El-Zerii A Thesis Submitted In Partial fulfillment of The Requirements for The Master Degree In Mathematics 2015

2 Al-Azhar University-Gaza (AUG) Deanship of postgraduate Studies Faculty of science Department of Mathematics Numerical Solution of Partial Differential Equations with Fuzzy Transform By Huda Salim El-Zerii Supervised by Dr. Awni M. Abu-Saman Associate Professor of applied Mathematics Al-Azhar University-Gaza (AUG) Gaza, Palestine 2015

3 Al-Azhar University-Gaza (AUG) Deanship of postgraduate Studies Department of Mathematics DECLARATION SHEET DATE: / /2014 I, hereby, certify that this thesis submitted for the Master degree in Mathematics is the result of my own research, except where otherwise acknowledged, and that this thesis (or any part of the same) has not been submitted for a higher degree to any university or institution. Huda Salim El-Zerii Signature :-

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5 Dedication To Spirits of my parents To My brothers To My sisters To all my glorious teachers I

6 Acknowledgement First of all, I would like to express my sincere thanks to Allah the facilitator and specified in all cases. I want to express my great gratitude to my supervisor A/Prof. Awni Abu-Saman for his supervision, guidance, encouragement, and moral support while developing my thesis. Also, I would like to thank all members of the mathematics department at Al-Azhar University, who gave me great support and cooperation during my whole scientific life. I would like to thank all my colleagues and friends everywhere for their encouragement. Finally, words can never express how I am grateful to my family for their unconditional love and endless support in all my life. II

7 Abstract This thesis is devoted for introducing some background about Fuzzy set theory as an introduction to the concept of Fuzzy transform for functions with one variable and the extension of Fuzzy Transform for functions with two variables as a new tool for solving partial differential equations with integer and fractional orders, we will investigate and implement the Fuzzy transform on different types of Partial differential equations. For each type we will give a numerical algorithm, the algorithms will be given will be implemented for different partitions and different basic functions, they will be simulated as a user-subroutine for the mathematical code MATLAB with double precision calculations. Numerical examples will be given to show the efficiency of the algorithms and computed results will be compared with the exact solution. III

8 ملخص ف هزه انشسانت سنقىو بتىض ح األسس ان بن عه ها انتحى م انضباب نهذوال راث ان تغ ش وانذوال راث ان تغ ش ن كأداة جذ ذة نحم ان عادالث انتفاضه ت انجضئ ت سىاء راث انشتب انصح حت أو انكسش ت خالل انذساست سنقىو بتطب ق انتحى م انضباب عهى أنىاع يختهفت ين ان عادالث انتفاضه ت انجضئ ت ك ا س تى وضع خىاسصي ت نحم كم نىع ينها و س تى تطب ق هزه انخىاسصي ت باستخذاو دوال أساس ت يختهفت وق ى يختهفت نهتجضئت كم خىاسصي ت س تى بشيجتها باستخذاو بشنايج Matlab بذقت حساباث يضدوجت ف نها ت كم خىاسصي ت سنقىو بحم يثال عذدي نهتأكذ ين فعان ت انخىاسصي ت ك ا وس تى يقاسنت اننتائج يع انحهىل انحق ق ت. IV

9 Contents Dedication Acknowledgements Abstract Contents List of Figures List of tables Chapter 1 Introduction I II III V VII XI 1 Chapter 2 Fuzzy Set theory 2.1 Introduction Basic definitions Basic operations Fuzzy relations and its operations Fuzzy modeling 14 Chapter 3 F-Transform for functions with one variable 3.1 Introduction Basic concepts F-Transform for functions with one variable F-Transform inversion for functions with one variable 35 Chapter 4 F-Transform for functions with two variables 4.1 Introduction Basic concepts F-Transform for functions with two variables F-Transform inversion for functions with two variables 56 Chapter 5 Numerical solution of Differential Equations using F-transform 5.1 Introduction Second Order Ordinary Differential Equations Fractional order differential equations 73 V

10 Chapter 6 Numerical solution of PDEs using F-Transform 6.1 Introduction Parabolic equations Hyperbolic equations Elliptic equations Partial fractional differential equations Summary 127 Chapter 7 Conclusions 128 References 130 Appendices 1. Matlab subroutine for Second Order Ordinary Differential 134 Equation 2. Matlab subroutine for fractional order differential equation Matlab subroutine for Parabolic equation with non-uniform 136 partition 4. Matlab subroutine for Parabolic equation with uniform 138 partition 5. Matlab subroutine for Hyperbolic equation Matlab subroutine for Elliptic equation Matlab subroutine for time dependent fractional diffusion 144 equation VI

11 List of Figures Figure Page Graphical presentation of convex and non-convex set Graphical presentation of ( ) Graphical presentation of ( ) Graphical presentation of ( ) Basic structure of fuzzy model Triangular shaped basic functions with uniform fuzzy partition Triangular shaped basic functions with non-uniform fuzzy partition Sinusoidal shaped basic functions with uniform fuzzy partition Sinusoidal shaped basic functions with non-uniform fuzzy partition The computed ( ) using 10 sinusoidal shaped basic functions The computed ( ) using 20 sinusoidal shaped basic functions The computed ( ) using 30 sinusoidal shaped basic functions The computed ( ) using 10 triangular shaped basic functions The computed ( ) using 20 triangular shaped basic functions The computed ( ) using 30 triangular shaped basic functions ( ) ( ) ( ) The computed ( ) using 10 sinusoidal shaped basic functions on both axes The computed ( ) using 10 triangular shaped basic functions on both axes The computed ( ) using 80 sinusoidal shaped basic functions on both axes The computed of ( ) using 80 triangular shaped basic functions on both axes The computed solution with triangular shaped basic functions for n= The computed solution with sinusoidal shaped basic functions for The computed solution with triangular shaped basic 69 VII

12 functions for n= The computed solution with sinusoidal shaped basic functions The computed solution with sinusoidal shaped basic functions for n= The computed solution with triangular shaped basic functions for n= The computed solution with triangular shaped basic functions for n= The computed solution with triangular shaped basic functions for n= The computed solution with triangular shaped basic functions for n= The computed solution at with Sinusoidal shaped basic functions for The computed solution at with Sinusoidal shaped basic functions for The computed solution at with triangular shaped basic functions for The computed solution at with triangular shaped basic functions for The computed solution at with triangular shaped basic functions for The computed solution at with triangular shaped basic functions for The exact solution of ( ) The computed solution of resulting from a non-uniform fuzzy partition with triangular shaped basic functions for The computed solution of resulting from a non-uniform fuzzy partition with sinusoidal shaped basic functions on -axis and triangular shaped basic functions on -axis for The computed solution of resulting from non-uniform fuzzy partition with sinusoidal shaped basic functions for The computed solution of resulting from non-uniform fuzzy partition with triangular shaped basic functions for The computed solution of resulting from non-uniform fuzzy partition with sinusoidal shaped basic functions on -axis and triangular shaped basic functions on -axis for 92 VIII

13 6.2.7 The computed solution of resulting from non-uniform fuzzy partition with sinusoidal shaped basic functions for The computed solution of resulting from uniform fuzzy partition with triangular shaped basic functions for The computed solution of resulting from uniform fuzzy partition with sinusoidal shaped basic functions on -axis and triangular shaped basic functions on -axis for The computed solution of resulting from uniform fuzzy partition with sinusoidal shaped basic functions for The computed solution of resulting from uniform fuzzy partition with triangular shaped basic functions for The computed solution of resulting from uniform fuzzy partition with sinusoidal shaped basic functions on -axis and triangular shaped basic functions on -axis for The computed solution of resulting from uniform fuzzy partition with sinusoidal shaped basic functions for The exact solution of ( ) The computed solution of resulting from uniform fuzzy partition with sinusoidal shaped basic functions for The computed solution of resulting from uniform fuzzy partition with triangular shaped basic functions for The computed solution of resulting from uniform fuzzy partition with triangular shaped basic functions on -axis and triangular shaped basic functions on -axis for The computed solution of resulting from uniform fuzzy partition with sinusoidal shaped basic functions for The computed solution of resulting from uniform fuzzy partition with triangular shaped basic functions for The computed solution of resulting from uniform fuzzy 106 IX

14 partition with triangular shaped basic functions on -axis and triangular shaped basic functions on -axis for The exact solution ( ) The computed solution ( ) for with triangular shaped functions The computed solution ( ) for with sinusoidal shaped functions The computed solution ( ) for with triangular shaped functions The computed solution ( ) for with sinusoidal shaped functions The exact solution of ( ) for The computed solution ( ) for with triangular shaped functions The computed solution ( ) for with sinusoidal shaped functions The computed solution ( ) for with triangular shaped functions The computed solution ( ) for with sinusoidal shaped functions The exact solution of ( ) for The computed solution ( ) for with triangular shaped functions The computed solution ( ) for with sinusoidal shaped functions The computed solution ( ) for with triangular shaped functions The computed solution ( ) for with sinusoidal shaped functions The exact solution ( ) for The computed solution ( ) for with triangular shaped functions The computed solution ( ) for with sinusoidal shaped functions The computed solution ( ) for with triangular shaped functions The computed solution ( ) for with sinusoidal shaped functions 127 X

15 List of tables Table The exact and computed solutions with sinusoidal shaped basic function The exact and computed solutions with triangular shaped basic function The exact and computed solutions using triangular shaped basic function The exact and computed solutions using triangular shaped basic function The exact and computed solutions of example with deferent basic functions for The exact and computed solutions of example with deferent basic functions for The exact and computed solutions of example with deferent basic functions for The exact and computed solutions of example with deferent basic functions for The exact and computed solutions of example with deferent basic functions and partitions at The exact and computed solutions of example with deferent basic functions and partitions at The exact and computed solutions of example with deferent basic functions for non-uniform partitions The exact and computed solutions of example with deferent basic functions for non-uniform partitions The exact and computed solutions of example with deferent basic functions for uniform partitions The exact and computed solutions of example with deferent basic functions for uniform partitions The exact and computed solutions of example with deferent basic functions for The exact and computed solutions of example with deferent basic functions for XI Page

16 The exact and computed solutions of example with deferent basic functions for The exact and computed solutions of example with deferent basic for The exact and computed solutions of example with deferent basic for at The exact and computed solutions of example with deferent basic for at The exact and computed solutions of example with deferent basic for at The exact and computed solutions of example with deferent basic for at The exact and computed solutions of example with deferent basic for at The exact and computed solutions of example with deferent basic for at XII

17 Chapter one Introduction Partial differential equations are used for modeling various physical phenomena. Unfortunately, many problems are dynamical and too complicated, developing an accurate differential equation model for such problems require complex and time consuming algorithms hardly implementable in practice[13]. For a long time, scientists goal was to develop constructive and effective methods that reliably compute the partial differential equation with more accuracy as possible. In classical mathematics, various kinds of transforms (Fourier, Laplace, integral, wavelet) are used as powerful methods for construction of approximation models and for solution of differential or integral-differential equations [18]. In 2001 I. Perfilieva introduced F-transform, F-transform proved its ability in many fields data compression and reconstruction, image processing and fusion, removing noise, control systems, [17:26, 30, 31]. In 2004 Stepnicka and Valasek [31] used F-transform to solve wave equation numerically, the results was encouraging, it was interesting to get continuous approximation solution the of wave equation. In 2006 Dankova and Vaslek [5] considered F-transform as an alternative approach to the solution of image fusion problem instead of wavelet transform. F-transform depending basically on Fuzzy set theory, which accepts the partial membership. Fuzzy set Theory deals with all elements in the problem under study, every element is either totally in, totally out, or partially in. In 1965 Lotfi Zadeh introduced his seminal paper about the concept of Fuzzy set theory. [40] Fuzzy set theory is composed of an organized body of mathematical tools particularly well-suited for handling incomplete information, the unsharpness of classes of objects or situations, or the gradualness of preference profiles, in a flexible way. It offers a unifying framework for modeling various types of information ranging from precise numerical, interval-valued 1

18 data, to symbolic and linguistic knowledge, with a stress on semantics rather than syntax. [41] Achieving high levels of precision is a very important subject in all science fields, getting a satisfied precision depending basically on the way we deal with elements in the problem Universe. For many years we were depended on crisp set theory "classical set theory to deal with elements and sets which belong to the problem Universe, but in real world there are many application problems which can't be described nor handled by the crisp set theory.[12] Scientists considered Fuzzy set theory quickly and used it in their researches; actually it proved its ability in solving many problems, which encourage scientists to use it in different fields. Fragments of the Fuzzy set theory have been used to establish a new field called fuzzy approximation, focusing on approximation properties of fuzzy models. Fuzzy transform "F-transform" is belonging to these models. F-transform introduces an approximate representation of continuous functions defined on closed interval this technique actually like other transforms, consists of two transforms direct F-transform and inverse F- transform". In this thesis we will F-transform to different types of Partial differential equations, for the first time we will apply F-transform to fractional order differential equations, also we will use uniform and non-uniform partitions and compare the results between them. We will solve numerical examples for each type using different basic functions and compare the efficiency of each basic function. The structure of this thesis will be as follows: In chapter two we will give short introduction on Fuzzy set theory which F- Transform depend on. In chapter three we will introduce F-Transform technique for functions with one variable and related concepts, which will be used in chapter five to solve ordinary Differential Equations. In chapter four we will introduce F-Transform technique for functions with two variables, and related concepts, which will be used in chapter six to solve Partial Differential Equations. In chapter five F-Transform will be applied on 2

19 Ordinary Differential Equations, a numerical algorithm will be implemented as a user-subroutine to the mathematical code MATLAB with double precision calculations. Numerical examples will present and compare with the exact solution. In chapter Six F-Transform will be applied on Partial Differential Equations, a numerical algorithm will be implemented as a usersubroutine to the mathematical code MATLAB with double precision calculations. Numerical examples will present and compare with the exact solution. Finally Chapter Seven will be devoted for conclusions. 3

20 Chapter Two Fuzzy set Theory 2.1 Introduction In this chapter we will introduce shortly basic concepts of Fuzzy set theory, more details about Fuzzy set theory can be found in [4, 9, 10, 12, 28, 39:41]. 2.2 Basic Definitions In crisp set theory each element in the Universe is either in or out of where characteristic function is used to characterize any crisp set as follows ( ) { but in fuzzy set theory where the partial membership is available, we need to generalize the characteristic function to describe the membership grade of each element in, as we will see in the next definition. Definition 2.2.1[4] If is a collection of objects denoted generically by then a Fuzzy set in is a set of ordered pairs: {( ( )) + ( ) Where ( ) is the membership function of in which maps to, -. Example [10] Let * +, in crisp set the set can be represented as 4

21 * + but in fuzzy set theory it is represented as *( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )+ Remarks each element of can be represented with exactly one ordered pair where the first element from and the second element form the interval, - the value zero is used to represent complete non-membership, the value one is used represent complete membership, and values in between are used to represent intermediate degree of membership. To consider those elements with specific membership degree in a Fuzzy set we will need the next definitions. Definition [12] The set of elements that belong to Fuzzy set where its membership degree is at least is called the" level set": * ( ) + ( ) Definition [12] The set of elements that belong to Fuzzy set where its membership degree is greater than is called the "strong level set": * ( ) + ( ) 5

22 Definition [28] Let then the support of ( ) is a crisp set of all elements such that ( ). ( ) * ( ) + ( ) Definition [28] Let then the kernel of ( ) is a crisp set of all elements such that ( ) ( ) * ( ) + ( ) Definition [40] A Fuzzy set is said to be empty iff its membership function is identically zero on ( ) ( ) Definition [40] Let then iff ( ) ( ) ( ) Example 2.2.2[10] Let and as in example (2.2.1), then * + 6

23 * + ( ) * + ( ) * + Convexity is an interesting property. To define it in Fuzzy set theory, we will take the membership function as a reference. Definition [9] A Fuzzy set is convex if ( ( ) ) ( ( ) ( )) ( ) where, - Fig a Convex Fuzzy set Fig b Noncovex Fuzzy set Fig Graphical presentation of convex and nonconvex set 7

24 Definition [39] Let a finite Fuzzy set then the cardinality of is defined as ( ) ( ) and the relative cardinality of is defined as ( ) 2.3 Basic operations Since the membership function is a crucial component of the Fuzzy set theory, it is normal to define Fuzzy set operations via their membership functions. Definition [40] Let the Fuzzy sets then the intersection is defined by ( ) * ( ) ( )+ ( ) 8

25 Fig Graphical presentation of ( ) Definition 2.3.2[39] Let the Fuzzy sets then the union is defined by ( ) * ( ) ( )+ ( ) Fig.(2.3.2): Graphical presentation of ( ) 9

26 Definition [28] Let the Fuzzy set, then the membership function of complement of, is defined by ( ) ( ) ( ) Fig Graphical presentation of ( ) Example [10] Let the Fuzzy sets, where as in example (1.2.1), and *( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )+ then *( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )+ *( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )+ 11

27 *( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )+ The operations on Fuzzy sets which are listed as equations (2.3.1) to (2.3.3) are called the standard fuzzy operations. These operations are the same as those for classical sets. When the range of membership values is restricted to the unit interval, these standard fuzzy operations are not the only operations that can be applied to Fuzzy sets. For each one of the standard operations, there exists a broad class of functions whose members can be considered a fuzzy generalization of the standard operations. Functions that qualify as fuzzy intersections and fuzzy unions referred to as t-norms and t- conorms respectively, as we will see later. Definition 2.3.4[28] A mapping, -, -, - is called t-norm iff, - satisfies the properties ( ) ( ) ( ( )) ( ( ) ) ( ) ( ) ( ) Definition [28] A mapping, -, -, - is called t-conorm iff, - satisfies the properties 11

28 ( ) ( ) ( ( )) ( ( ) ) ( ) ( ) ( ) Definition [12] Let then the fuzzy Cartesian product is a Fuzzy set with the following membership function ( )( ) ( ( ) ( )) 2.4 Fuzzy relations and its operations A Fuzzy relation is a mapping from the Cartesian space to the interval, -, where the strength of the mapping is expressed by the membership function of the relation for ordered pairs from the two universes, or ( ). Definition [41] Let and be Fuzzy relations on the Cartesian space, then the following operations apply for the membership values set operations. Union ( ) ( ( ) ( )) ( ) 12

29 Intersection ( ) ( ( ) ( )) ( ) Complement ( ) ( ) ( ) Containment ( ) ( ) ( ) Definition [41] Let the Fuzzy relations composition of and, defined by, then sup-min ( )( ) * ( ) ( )+ ( ) The next definition will generalize the Fuzzy relations composition by using any t-norm operation, which is called sup-t composition. Definition [41] Let be a t-norm and let the fuzzy relations, then the of and, defined by ( )( ) * ( ) ( )+ ( ) 13

30 2.5 Fuzzy modeling Fuzzy models or Fuzzy systems are rule based originating from the concepts of Fuzzy sets, Fuzzy rules, and Fuzzy reasoning. A typical Fuzzy system basically consists of four components: Fuzzy-rule base, inference engine, fuzzification interface, and defuzzification interface. Fig.(2.5.1) shows the block diagram of the structure of Fuzzy model. The inference engine is the central component of Fuzzy model. It is a reasoning mechanism which performs the inference procedure upon the Fuzzy rules and given conditions to derive conclusions. The fuzzification interface is a mechanism to transform a real-valued variable to a Fuzzy set. The defuzzification interface is a mechanism to transform a Fuzzy set over an output universe of discourse to a real-valued variable. There are many types of Fuzzy model. F-transform is belonging to Takagi-Sugeno models of the 0- th order. It is a fuzzy approximation method based on two transforms: a direct one "fuzzification" and an inverse one "defuzzification". It deals with a fuzzy partition of the domain given by Fuzzy sets called basic functions, fulfilling several conditions, as we will see later. For simplicity, we will write ( ) instead of ( ) to denote basic function. Basic functions will be defined in next chapter, but first we have to define fuzzy partition. 14

31 Definition [12] Let for, then the n-tuple ( ) of Fuzzy sets is called a fuzzy partition of, iff ( ) 15

32 Chapter Three F-transform for functions with one variable 3.1 Introduction This chapter is devoted for introducing the basic concepts and properties of F-transform. The definitions will depend mainly on the work of Irina Perfilieva [18:26], who introduced F-transform for the first time in 2001, and Matrin Stepnicka [32,33]. 3.2 Basic Concepts First let us consider, - as a common domain for all one variable functions in this thesis. This domain should be partitioned into subintervals to define membership functions which introduce Fuzzy sets, as we will see in the next definition. Definition 3.2.1[18] Let be fixed nodes within such that and. we say that Fuzzy sets are basic functions and form a fuzzy partition of if the next 5 conditions hold true for each 1., -, - ( ) ; 2. ( ) if ( ); 3. is continuous on ; 4. strictly increases on, - and strictly decreases on, -; 5. ( ). 16

33 Remarks If the nodes are not equidistant, then hence, we can see that and If the nodes are equidistant, then and ( ) hence, the following properties will hold for : 6. ( ) ( ), - 7. ( ) ( ), -. Definition 3.2.2[33] Let be fixed nodes within where, and let Fuzzy sets basic functions. Then, the basic functions determine a fuzzy partition with a symmetry iff ( ) ( ) ( ), - Remarks It is easy to prove that every uniform fuzzy partition is a fuzzy partition with symmetry. ( ) ( ) Condition (7) 17

34 ( ) since ( ) " condition(6) " In case of the partition with symmetry since ( ) ( ), - then, by integrating both sides with respect to ( ) ( ) by changing the integration to be with respect to we get ( ) ( ) ( ) The most two popular basic functions are triangular shaped and sinusoidal shaped the triangular shaped basic functions can be defined as: ( ) ( ),( ), - ( ) ( ),( ), - and for ( ) ( ) ( ) ( ) ( ) {, -, - and the sinusoidal shaped basic functions can be defined as 18

35 ( ) { ( ( ( ) ( ) ) ), - ( ) { ( ( ( ) ( ) ) ), - and for ( ) { ( ( ( ) ( ) ) ), - ( ( ( ) ( ) ) ), - Fig Triangular shaped basic functions with uniform fuzzy partition 19

36 Fig Triangular shaped basic functions with non-uniform fuzzy partition. Fig Sinusoidal shaped basic functions with uniform fuzzy partition. 21

37 Fig Sinusoidal shaped basic functions with non-uniform fuzzy partition. Lemma 3.2.1[18] Let a uniform fuzzy partition of. Then be given by basic functions ( ) ( ) ( ) ( ) and for ( ) ( ) Proof Since ( ) ( ), and ( ), from basic function definition then for, - we get by integrating both sides, we get ( ) ( ) 21

38 ( ) ( ) ( ) ( ) since the fuzzy partition is uniform, then the symmetry property holds, thus we can use equation (3.2.2) to get ( ) ( ) ( ) so ( ) ( ) by the same way, for, - since ( ) ( ), - we can obtain ( ) ( ) Now, since ( ) ( ), then using eqn. (3.2.6) 22

39 ( ) ( ) similarly, since ( ) ( ), then using equation (3.2.7) ( ) ( ) finally for, since ( ) ( ), then ( ) ( ) ( ) ( ) and using equations (3.2.6) and (3.2.7) we get ( ) Lemma [33] Let a fuzzy partition with a symmetry of functions Then, for be given by basic ( ) ( ) 23

40 F-transform technique is applied on a continuous function defined on a closed interval and transforms it to another one which is usually simpler to deal with. Then transform it back to the original space. This technique, actually like other transforms, consists of two transforms: The first transform is called "direct F-transform" which transform the continuous function to a finite number of vectors obtained on the basis of the well-established fuzzy partition of the given function domain, and this will be covered in section 3.3. The second transform is called inverse F-transform" which converts this vectors back to the original space as a continuous function which approximates the original one, and this will be covered in section F-transform for functions with one variable Definition [20] Let be basic functions which form a fuzzy partition of and ( ). We say that the -tuple of real numbers, - given by ( ) ( ) ( ) ( ) is the direct(integral) F-transform of, - with respect to the given fuzzy partition., - are called F-transform components of. Remark If the fuzzy partition is uniform, then by Lemma and basic function definition the direct F-transform of can be written as: 24

41 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) so ( ) ( ) ( ) by the same way ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) so ( ) ( ) ( ) and for ( ) ( ) ( ) ( ) ( ) so ( ) ( ) ( ) for the fuzzy partition with symmetry we can use lemma and basic function definition to write F-transform as 25

42 ( ) ( ) ( ) ( ) for If is known only at some nodes, say define discrete F-transform. then we will need to Definition [19] Let a fuzzy partition of be given by basic functions and let be a function known at nodes, the -tuple of real numbers, - given by ( ) ( ) ( ) ( ) for is said to be the discrete direct F-transform of with respect to the given fuzzy partition. Next lemma will prove F-transform linearity which is important when we apply F-transform on differential equations. Lemma [18] Let a fuzzy partition of be given by basic functions ( ) ( ) and let and be continuous functions on such that where are real numbers. Then the following equality holds 26

43 , -, -, - ( ) where, -, -, - are the F-transform components of and with respect to the given fuzzy partition respectively. Proof For ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ( )) ( ) ( ) ( ) ( ) ( ) since is arbitrary, then lemma holds. Before using F-transform in solving differential equations, we must be sure that the direct F-transform components are close to the values of a given function at nodes. Lemma (3.3.2) will prove it for the uniform partition, and lemma (3.3.3) will prove it for any fuzzy partition with symmetry. Lemma 3.3.2[18] let a uniform fuzzy partition of be given by basic functions ( ) ( ), and let ( ) be a twice continuously differentiable function on ( ) Then, for. ( ) (( )) ( ) Proof 27

44 For, by using equation (3.3.4) and the trapezoidal rule ( ) ( ) ( ( ) ( ) ( ) ( ) ) [( ( ( ) ( ) ( ) ( )) ( )) ( ( ( ) ( ) ( ) ( )) ( ))] since are nodes in the fuzzy partition, so by definition (3.2.1) ( ) ( ) ( ) then ( ( ) ( ) ( )) ( ( ) ( )) ( ) ( ) and hence similarly at ( ) ( ) by using equation (3.3.2) and trapezoidal rule ( ) ( ) [ ( ( ) ( ) ( ) ( )) ( )] since are nodes, then by definition(3.2.1) ( ) ( ) so 28

45 ( ) ( ) ( ) ( ) finally at by using equation (3.3.3) and trapezoidal rule ( ) ( ) [ ( ( ) ( ) ( ) ( )) ( )] since are nodes, then by definition (3.2.1) ( ) ( ) so then lemma (3.3.2) holds ( ) ( ) ( ) ( ) Lemma 3.3.3[33] Let a fuzzy partition with a symmetry of be given by basic functions ( ) ( ) and let ( ) be a twice continuously differentiable function on ( ). Then for ( ) ( ( )) ( ) Lemma [33] 29

46 Let a fuzzy partition with a symmetry of be given by basic functions and let ( ) be a twice continuously differentiable function on ( ). Then for each. ( ) moreover ( ) ( ( )) ( ) ( ) ( ) ( ) Proof Let, - for. Then from definition 3.2.1, we know that ( ) ( ) ( ) ( ) ( ) from basic function definition we know that so ( ) ( ) ( ), - ( ) multiply both sides with ( ) 31

47 ( ) ( ) ( ) by integrating both sides ( ) ( ) ( ) ( ) ( ) now we will simplify each term of the previous summation we get the first term will be simplified as ( ) ( ) ( ) ( ) ( ) ( ) since ( ) ( ) so ( ) ( ) ( ) ( ) the last term will be simplified as ( ) ( ) ( ) ( ) ( ) ( ) since ( ) ( ) so ( ) ( ) ( ) ( ) for the other terms where 31

48 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and since ( ) ( ) so ( ) ( ) ( ) ( ) and hence ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) using equation (3.3.5) then ( ) ( ) ( ) using the trapezoidal rule with the term but ( ), and ( ) ( ( ) ( ) ( ) ( ) ) ( ) ( ) ( ) 32

49 using equation (3.3.9). ( ( ))/ ( ) ( ) ( ( )) ( ) after simplifying the first part of the Lemma will be hold ( ) ( ) ( ( )) to prove the second part, by the same way, From definition (3.2.1) ( ) multiply both sides with ( ) we get ( ) ( ) ( ) by integrating both sides ( ) ( ) ( ) ( ) ( ) after simplifying each term of the summation as we did with the first part of the lemma, we get 33

50 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) finally by using equation (3.3.5), we get ( ) ( ) and hence the second part of lemma holds. Lemma 3.3.5[18] Let a uniform fuzzy partition of be given by basic functions and let ( ) be a twice continuously differentiable function on ( ). Then for each ( ) ( ) ( ) ( ) moreover ( ) ( ) ( ) Proof Since the uniform fuzzy partition is a fuzzy partition with symmetry, then using equation (3.3.10). 34

51 ( ) ( ) ( ( )) but then ( ) ( ) ( ) similarly using equation (3.3.11), and ( ) ( ) ( ) ( ) ( ) ( ) ( ) so lemma holds. 3.4 F-transform inversion for functions with one variable After transforming the original function using direct F-transform and solving it in the fuzzy space, we have to transform it back to the original space by the inverse F-transform. 35

52 Definition 3.4.1[19] Let, - be the direct F-transform of with respect to the basic functions. Then, the function ( ) ( ) ( ) is called the inverse F-transform of. If we fix a fuzzy partition, we will be able to deal with the inverse as a mapping from an -tuples of real numbers to the space * + of linear combinations of the basic functions. The next lemma will prove that the space * + is in a one-to-one correspondence with the set of -tuples of reals, that is, with the set Lemma 3.4.1[22] Let a fuzzy partition of be given by basic functions. Then each function * + ( ) ( ) is uniquely determined by the -tuple, -. Proof Let us assume that ( ) ( ) ( ) where such that there exits for fixed since ( ) and ( ) for "from definition (2.2.1)", after substituting in equation (3.4.2) we get 36

53 since is arbitrary then which contradicts with the hypothesis, then lemma holds. Finally it s important to insure that F-transform converges to the original function, so we will introduce corollary 3.4.1, corollary 3.4.2, and theorem Corollary will show that a sequence of the inverse F- transform converges uniformly to the original function; theorem and corollary are used to prove this. Theorem Let ( ). Then for any there exists and a fuzzy partition such that for all ( ) ( ) ( ) where is the inverse F-transform of with respect to the fuzzy partition Proof. Since is continuous, then such that ( ) ( ) let { } be a uniform fuzzy partition of where now let, -, - so using inequality properties now for, and using equation (3.3.4) 37

54 ( ) ( ) ( ) ( ) ( ) ( ) ( ) since is continuous, and, then ( ) ( ) Using equation (3.2.5), then so ( ) ( ) by the same way For,using equation (3.2.3) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) so analogously for ( ) ( ) ( ) ( ) from equations (3.4.5), (3.4.6), and (3.4.7), we see that ( ) ( ) therefore, 38

55 ( ) ( ) ( ) ( ) by inequality properties ( ) ( ) ( ) ( ) using definition (3.2.1) ( ) then ( ) ( ) Since is arbitrary, then the theorem holds. Corollary Let ( ) and * + be a sequence of uniform partition of, one for each. Let * + be the sequence of the inverse F-transforms, each with respect to a given fuzzy partition. Then, for any such that for each and ( ) ( ) ( ) Corollary Let the assumptions of the Corollary (3.4.1) be fulfilled. Then the sequence of inverse F-transforms * + uniformly converges to. After proving theorem 3.4.1, it is easy to prove both Corollary and Corollary using direct prove and definitions of convergence and continuous. Perfiliva proved them in[18]. In this thesis, we will introduce the next example to illustrate the fact of uniform convergence. The following example will demonstrate visually the uniform convergence property as the number of partition is increased. 39

56 Example Consider, -, where ( ) ( ) "the normalized function". F-transform is applied to ( ), it is simulated as a user-subroutine on the mathematical code MATLAB with double precision calculation, F- transform components and inverse F-transform of ( ) are computed from equations (3.3.1) and (3.4.1) respectively for different partitions and different basic functions. The obtained results in tables (3.4.1) and (3.4.2), and figs. (3.4.1) to (3.4.6) indicates that the inverse F-transform of ( ) is uniformly converges to ( ) as the number of partition increase. 41

57 Table 3.4.1: The exact and computed solutions with sinusoidal shaped basic function x Computed solution Exact solution absolute absolute absolute n=10 n=20 n=30 error error error e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-03 Table 3.4.2: The exact and computed solutions with triangular shaped basic function x Computed solution Exact solution absolute absolute absolute n=10 n=20 n=30 error error error e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-03 41

58 Fig The computed ( ) using 10 sinusoidal shaped basic functions Fig The computed ( ) using 20 sinusoidal shaped basic functions 42

59 Fig The computed ( ) using 30 sinusoidal shaped basic functions Fig The computed ( ) using 10 triangular shaped basic functions 43

60 Fig The computed ( ) using 20 triangular shaped basic functions Fig The computed ( ) using 30 triangular shaped basic functions 44

61 Chapter Four F-transform for functions with two variables 4.1 Introduction This chapter is devoted for introducing the F-transform for functions with two variables. Stepnicka introduced the F-transform technique for functions with two variables for the first time in 2003[ 33], and introduced an extension of F-transform technique for functions with an arbitrary finite number of variables in 2007[33]. Thus next concepts and lemmas will depend mainly on his work. 4.2 Basic Concepts At the beginning we have to refer to that the rectangle, -, - will be used as a common domain of all valued functions in this study. We will also use the following notation: the nodes of the fuzzy partition of, - will be denoted by and on the other hand the nodes of the fuzzy partition of, - will be denoted by and which leads to. In case of uniform fuzzy partition then 45

62 and Definition [33] Let a fuzzy partition of, - be given by basic functions, - and let a fuzzy partition of, - be given by basic functions, -. Then the fuzzy partition of is given by the fuzzy Cartesian product * + * + with respect to the product T-norm of these two fuzzy partition. Remarks If fuzzy partitions on both axes are uniform, then the overall fuzzy partition is called uniform partition. A fuzzy partition of is called a fuzzy partition with symmetry, if the fuzzy partition of each axes is a fuzzy partition with symmetry. The partition on -axis and the partition on -axis are independent, so it possible to make a uniform partition on either axes, or nonuniform partition on both axes, or make uniform partition on one axis and non-uniform partition on the other. The basic functions used on axes are also independent, so it is possible to use the same basic functions on both axes or use different basic functions. Lemma 4.2.1[33] Let a fuzzy partition with a symmetry of * + * +, then be given by ( ) ( ) ( )( ) ( ) 46

63 Proof Using properties of double definite integration and lemma 3.2.2, we get ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) Lemma 4.2.2[33] Let a uniform fuzzy partition of * + * +.then be given by ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and for and ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and 47

64 ( ) ( ) ( ) Proof Since and since the partition is uniform then then using lemma (4.2.1) we get ( ) ( ) ( )( ) ( )( ) by the same way ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( )( ) 48

65 ( ) ( ) ( )( ) ( )( ) and for and ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( )( ) and finally 49

66 ( ) ( ) ( )( ) ( )( ) 4.3 F-transform for functions with two variables Similarly to one-dimension direct F-transform, the two-dimensions direct F-transform is introduced as a mapping from the space of continuous functions ( ) to the space of real matrices of type. ( ), - Definition 4.3.1[22] Let a fuzzy partition of be given by * + * + and let ( ). We say that a real matrix, - [ ] given by ( ) ( ) ( ) ( ) ( ) ( ) is the F-transform of with respect to the given fuzzy partition. The real number are called the components of the F-transform of. In case of a fuzzy partition with symmetric then using equation (4.2.1) we get For ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 51

67 ( )( ) ( ) ( ) ( ) ( ) If the partition is uniform then, using equations (4.2.2), (4.2.3), and (4.2.4) the F-transform components can be written as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and for ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 51

68 Analogous to Lemma (3.3.2), next lemma will show that the components of the F-transforms of ( ) are equal to precise values of at the nodes of the symmetric fuzzy partition up to a certain accuracy. Lemma [33] Let a fuzzy partition with a symmetry of be given by * + * + and let ( ) be a twice continuously differentiable function on ( ) ( ). Then for each Proof ( )..( ) ( ) ( ) ( ) // ( ) Using equation (4.3.2) and basic functions definition ( ). / ( ) ( ) ( ) ( ). / ( ) ( ) ( ) using properties of definite integration, we get ( ). / ( ) ( ( ) ( ) ( ) ( ) ) by applying trapezoidal rule 52

69 ( ). / ( ) (( ( ) ( ) ( ) ( ) ) (( ) ) ( ( ) ( ) ( ) ( ) ) (( ) )) from basic function definition, ( ) ( ) ( ) then ( ). / ( ) (( ) ( ) (( ) ) (( ) )). / ( ) ( ) ( (( ) ) (( ) )) ( ). / ( ). / ( ) ( ) ( (( ) ) (( ) )) ( ). / using properties of integration for the first term, and Lemma (3.2.2) for the other one we get 53

70 . / ( ( ) ( ) ( ) ( ) ) ( (( ) ) (( ) )) ( (. / ) ) using trapezoidal rule we get. / (( ( ) ( ) ( ) ( ) ).( ) / ( ( ) ( ) ( ) ( ) ).( ) /) ( (( ) ) (( ) )) from basic function definition, we know that ( ) ( ) ( ). / ((( ) ) ( ).( ) /.( ) /) ( (( ) ) (( ) )) after simplifying we get ( )..( ) ( ) ( ) ( ) // In case of uniform fuzzy partition, the lemma can be written as follows. 54

71 Lemma [33] Let a uniform fuzzy partition of be given by * + * + and let ( ) be a twice continuously differentiable function on ( ) ( ). Then for each ( ) ( (( ) ( ) )) ( ) Lemma can be proved easily, since uniform partition is a partition with symmetry, then using Lemma and properties of uniform fuzzy partition ( ) ( ) ( ) ( ) ( ) ( ) so..( ) ( ) ( ) ( ) // ( ( ) ( ) ) and hence the Lemma holds. For functions which known only at some nodes, we use the discrete (direct) F-transform form which defined as follows. Definition 4.3.2[22] Let a fuzzy partition of be given by * + * + and let be known at nodes ( ) where. We say that a real matrix, -, - which is given by ( ) ( ) ( ) ( ) ( ) ( ) 55

72 is the discrete (direct) F-transform of with respect to the given fuzzy partition, and is called components of F-transform 4.4 F-transform inversion for functions with two variables The inverse F-transform of two variables function will be defined analogously to definition (3.4.1) as follows: Definition 4.4.1[33] Let, - be the F-transform of a function ( ) with respect to a given partition * + * +. Then the function ( ) ( ) ( ) ( ) is called the inverse F-transform of Stepnicka in [33] proved that the F-transform of multi-variable function converges uniformly to the original function. In this thesis, we consider functions with two variables, so we will show that the inverse F- transform of function with two variables uniformly converges to the original function. But first, we have to prove that the uniqueness property is hold for { }. Lemma 4.4.1[33] Let * + * + be a fuzzy partition of, then each { } ( ) ( ) ( ) is uniquely determined by -tuples, -of real numbers 56

73 Proof Let ( ) ( ) ( ) ( ) where s.t. now at fixed node ( are real numbers, where there exists and ) using basic function definition, we have that this implies that. ( ),and ( ) for ( ),and ( ) for but the node ( ) is arbitrary, then which contradict with the assumption, and hence the Lemma holds. Theorem [33] Let ( ), then for any there exist and a fuzzy partition * + * + of such that ( ) ( ) Proof First since is continuous, then for each there is such that for all implies ( ) ( ) Now let * + * + be a uniform fuzzy partition of, such that 57

74 and let ( ), - [ ] then, using basic function definition, we get ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) since is continuous, then ( ) ( ) ( ) using Lemma (4.2.2) and uniform fuzzy partition properties, we get ( ) ( ) ( ) similarly we get ( ) ( ) ( ) so from previous 58

75 ( ) now ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) from fuzzy partition definition we know that ( ) ( ) ( ) ( ) ( ) ( ) since ( ) is arbitrary, then the theorem holds ( ). Corollary Let ( ) and let {* + * +} ( be a ) sequence of uniform fuzzy of. Let { } ( ) be a sequence of the inverse F-transform, one for each fuzzy partition. Then, for any there exist such that for each ( ). ( ) 59

76 Corollary Let all the assumptions of corollary (4.4.1) be fulfilled. Then, the sequence { } ( ) of the inverse F-transform; one for each fuzzy partition, uniformly converges to. Analogously to chapter three we will illustrate the fact of uniform convergence using the next example. The example demonstrates visually the uniform convergence property as the number of partition is increased. Example consider, -, -, where ( ) ( ) F-transform is applied to ( ), it is simulated as a user-subroutine on the mathematical code MATLAB with double precision calculation, F- transform components and inverse F-transform of ( ) are computed from equations (4.3.1) and (4.4.1) respectively for different partitions and different basic functions. The obtained results in tables (4.4.1) and (4.4.2), and figs. (4.4.1) to (4.4.5) indicates that the inverse F-transform of ( ) is uniformly converges to ( ) as the number of partition increase. 61

77 Table 4.4.1: The exact and computed solutions using triangular shaped basic function x Y Exact solution Computed solution n=10 absolute absolute n=80 error error e e e e e e e e e e e e e e e e e e e e e e e e e-04 Table 4.4.2: The exact and computed solutions using sinusoidal shaped basic function x Y Computed solution Exact solution absolute absolute n=10 n=80 error error e e e e e e e e e e e e e e e e e e e e e e e e e-03 Fig ( ) ( ) ( ) 61

78 Fig The computed ( ) using 10 sinusoidal shaped basic functions on both axes. Fig The computed ( ) using 10 triangular shaped basic functions on both axes. 62

79 Fig The computed ( ) using 80 sinusoidal shaped basic functions on both axes. Fig The computed of ( ) using 80 triangular shaped basic functions on both axes. 63

80 Chapter Five Numerical Solution of Ordinary Differential Equations with F-transform 5.1 Introduction Differential equations are used to model problems that involve the change of some variable with respect to another in many real-life situations. Differential equations which model practical problems are usually too complicated to be solved analytically, so one of two approaches is taken to approximate the solution. The first approach is to simplify the differential equation to one that can be solved exactly, and then use the solution of the simplified equation to approximate the solution to the original equation. The other approach, involves finding methods for directly approximating the solution of the original problem. This approach is commonly taken since more accurate results and realistic error information can be obtained.[29] In this chapter, we will implement numerical F-transform algorithms to solve two types of Ordinary Differential Equations, Second Order Ordinary Differential Equations and Fractional Order Differential Equations. 5.2 Second Order Ordinary Differential Equations In this section, the F-transform numerical algorithm is implemented as a user subroutine for MATLAB [15]. The subroutine will be simulated and applied to second Order Ordinary Differential Equations. For simplicity, we will consider a general form of initial value problem "IVP" with constant coefficients, and we will also consider the case of uniform partition. Consider the second IVP of the form ( ) ( ) ( ) ( ) (5.2.1) ( ) ( ) (5.2.2) where is a continuous solution of equation (5.2.1),. 64

81 First we will define a uniform partition with equidistant step on, and define a basic function * + of, so we construct a fuzzy partition. Now by applying F-transform on equation (5.2.1) we get where, -, -, -, - ( ), -, - is the F-transform components of,, -, - is the F-transform components of,, -, - is the F-transform components of, and, -, - is the F-transform components of. in equation (5.2.3), and are replaced by their finite differences. ( ) ( ) ( ) and ( ) ( ) so ( ) ( ) ( ( ) ( ) ) ( ) ( ) ( ( ) ( ) ( ) the above equation can be written as: ( ) ( ) ( ) ) ( ) ( ) 65

82 similarly, ( ) ( ) by substituting (5.2.4) and (5.2.5) into (5.2.3), the following recursive equation will be obtained,( ) ( ) ( ) by applying the F-transform on the initial conditions, we get ( ) ( ) ( ) ( ) ( ) ( ) ( ( ( ) ( )) ) ( ) ( ) by simplifying we get ( ( ) ( ) ( ) ( ) ( ) ( ) ) ( ) ( ) equations (5.2.6) to (5.2.8) determine the F-transform components of, -. Finally; The F-transform inverse from definition (3.4.1) is used to get the continuous approximation of. 66

83 Example Consider the following IVP ( ) ( ), ( ) The exact solution of the above problem is The above example is simulated on Our MATLAB software program [38] with F transform user-subroutine with double precision calculations. Equations (5.2.6) to (5.2.8) are used to find the F-transform components of in our example with different partitions and different basic functions. Results in tables (5.2.1), (5.2.2) and in Figs. (5.2.1) to (5.2.4) show that the computed solution of the above example with the Sinusoidal shaped basic functions was not close to the exact solution, while the computed solution with triangular shaped basic functions are much better and the error is smaller. Increasing the number of partitions improve the computed solution and gets closer to the exact solution as increases. x Table 5.2.1: the exact and computed solutions for Computed solution Sinusoidal shaped basic triangular shaped basic Exact functions functions solution n=m=10 absolute error n=m=10 absolute error e e e e e e e e e e e e e e e e e e e e e e e e e+01 67

84 Table 5.2.2: the exact and computed solutions for Computed solution x Exact Sinusoidal shaped basic triangular shaped basic solution functions functions n=m=100 absolute error n=m=100 absolute error e e e e e e e e e e e e e e e e e e e e e e e e e+00 Fig The computed solution with triangular shaped basic functions for n=10 68

85 Fig The computed solution with sinusoidal shaped basic functions for Fig The computed solution with triangular shaped basic functions for n=100 69

86 Fig The computed solution with sinusoidal shaped basic functions Example Consider the following IVP ( ) ( ), ( ) The exact solution of the above example is The F-transform is applied to above problem, it s simulated as a usersubroutine on the mathematical code MATLAB with double precision calculations [38]. The F-transform components of are computed from equations (5.2.6) to (5.2.8) for different partitions and different basic functions. The obtained results in tables (5.2.3), (5.2.4), and in Figs. (5.2.5) to (5.4.9) indicates that computed solutions are closer to the exact solution as the partition number increases, the best results obtained are for. 71

87 Table 5.2.3: the exact and computed solutions for Computed solution x Exact solution Sinusoidal shaped basic triangular shaped basic functions functions n=m=10 Absolute error n=m=10 Absolute error e e e e e e e e e e+01 Table 5.2.4: the exact and computed solutions for Computed solution x Exact solution Sinusoidal shaped basic triangular shaped basic functions functions n=m=1000 Absolute error n=m=1000 Absolute error e e e e e e e e e e-01 Fig The computed solution with sinusoidal shaped basic functions for n=10 71

88 Fig The computed solution with triangular shaped basic functions for n=10 Fig The computed solution with triangular shaped basic functions for n=50 72

89 Fig The computed solution with triangular shaped basic functions for n=500 Fig The computed solution with triangular shaped basic functions for n=

90 5.3 Fractional order differential equations Our understanding of Nature depends basically on calculus, which in turn depends on the intuitive concept of the derivative. Its descriptive power comes from the fact that it analyses the behavior at scales small enough that its properties change linearly to avoid complexities which arise at large ones. Fractional calculus deals with the generalization of differentiation and integration of real orders [6], allowing calculations such as deriving a function to order. The term "fractional" is used to denote this kind of derivatives. Despite it seems not to have significant applications in fundamental physics, various mathematicians have built up a large body of mathematical knowledge on fractional integrals and derivatives [29]. Fractional calculus seems to be valuable, and has played a significant role in engineering, finance and applied sciences. There are many interesting applications of fractional calculus.[16,15] Several different first-order accurate numerical methods to solve fractional order differential equations have been presented before [36]. Many finite difference approximations for the fractional difference equations are based on some form of the Grünwald estimates, and these estimates are only first-order accurate. Meerschaert in [35] introduced a practical numerical algorithm for solving multidimensional fractional partial differential equations with variable coefficients, using modified Euler method. He used the shifted Grünwald finite difference approximation formula.[8] Kai Diethelm [7] has a pioneer work in presenting algorithms for solving linear and non-linear differential equations with fractional order used in modeling plasticity and visco-plasticity. In [36] Tadjeran and Meerschaert used finite difference to develop a stable numerical algorithm for solving fractional order differential equations with non-constant coefficients. In this section, we will develop F-transform algorithm for solving fractional order differential equations on the finite domain, -. For calculations simplicity, we will consider the following general form of fractional order differential equation with constant coefficients ( ) ( ) ( ) ( ) Where 74

91 ( ) First we will define a uniform partition with equidistant step on, and define a basic function * + of, so we construct a fuzzy partition. By applying F-transform on (4.3.1) we get the equation, -, -, - ( ) Where,, -, - is the F-transform components of,, -, - is the F-transform components of, and, -, - is the F-transform components of. So for we have ( ) Discrete approximation to the fractional derivative term in equation (5.3.2) will be derived from the following Grünwald formula [27] ( ) ( ) ( where ) is the Grünwald weights and computed with the recurrence relationships ( ) ( ) ( ) ( ) ( ) applying the F-transform to the Grünwald formula ( ) ( ( ) ) ( ) ( ) ( ) 75

92 ( ) ( ) ( ) ( ) ( ) ( ) where Using the initial condition ( ) ( ) ( ) ( ) By substituting (5.3.6) in (5.3.3), we get the recursive equation ( ( ) ) ( ) Where Using equations (5.3.7) and (5.3.8), we can easily get the F-transform components of, -. Finally, the F-transform inverse from definition (3.4.1) is used to get the continuous approximation of. Example 5.3.1[1] Consider the following fractional differential equation ( ) ( ) ( ) ( ) ( ) ( ) where the exact solution to this problem is given by ( ), which can be verified by direct differentiation of the given solution and substituting in the fractional differential equation and the initial conditions are clearly satisfied. 76

93 Tables (5.3.1) and (5.3.2) and Figs. (5.3.1) to (5.3.6) show the computed solution obtained by applying the F-transform technique on the above differential equation with fractional order and number of partitions using both sinusoidal and triangular shaped basic functions. The computed solution compared well with the exact solution of the fractional differential equation. Results show that computed solution gets closer to the exact solution as increasing. 77

94 Computed solution Sinusoidal shaped basic triangular shaped basic triangular shaped basic Exact x functions functions functions solution absolute absolute absolute n=10 n=10 n=100 error error error e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-003 Table 5.3.2: the exact and computed solutions at α Computed solution Sinusoidal shaped basic triangular shaped basic triangular shaped basic Exact x functions functions functions solution absolute absolute absolute n=10 n=10 n=100 error error error e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e

95 Fig The computed solution at functions for with Sinusoidal shaped basic Fig The computed solution at functions for with triangular shaped basic 79

96 Fig The computed solution at functions for with triangular shaped basic Fig The computed solution at functions for with Sinusoidal shaped basic 81

97 Fig The computed solution at functions for with triangular shaped basic Fig The computed solution at with triangular shaped basic functions for 0 81

98 Chapter Six Numerical Solution of Partial Differential Equations with F-transform 6.1 Introduction Partial Differential Equations "PDEs" arise in a number of physical problems, such as fluid flow, heat transfer, solid mechanics, and biological processes. These equations often fall into one of three types. Parabolic equations are most commonly associated with diffusion, while Hyperbolic equations are most commonly associated with advection, and Elliptic equations are commonly associated with steady-states of either parabolic or hyperbolic problems[37]. Not all problems fall easily into one of these three types, which forces scientists to deal with partial fractional differential equations [34]. In this chapter, numerical algorithms will be introduced for solving different types of PDEs Parabolic equations, Hyperbolic equations, Elliptic equations, and Partial factional order differential equations. The numerical algorithms based on using F-transform to approximate the solution of PDEs. The Numerical algorithms are simulated and implemented as a user subroutine to the mathematical code MATLAB with double precision calculations. 6.2 Parabolic equations Parabolic equation is normally used to describe a wide family of scientific problems, this type of PDE is used mainly to describe heat and diffusion problems. In [33] Stepnicka consider and the partition is uniform. In this thesis we consider the partition is non-uniform and. For calculations simplicity we will use the next form of the parabolic equation in our study. Let the domain, -, - be a continuous solution of the parabolic equation. 82

99 ( ) ( ) ( ) with the initial condition and boundary conditions ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Define the partition with steps on, -, and define a basic function * + of, -, and define the partition with steps on, -, and define a basic function * + of, -, so we construct a fuzzy partition. Now after applying F-transform on equation (6.2.1), the equation turned to the following algebraic equation where, -, -, - ( ), - [ ] is the matrix of the F-transform components of, and, - [ ] is the matrix of the F-transform components of, and 83

100 , - [ ] is the matrix of the F-transform components of ( ). So for we get ( ) The partial derivatives difference: ( ) will be approximated by their divided ( ( ) ( )) and ( ) * ( ) ( ) ( ) ( ) + Now we can approximate as follow ( ) ( ) ( ) ( ) * *. /. /. /. / ++ ( ) ( ) ( ) ( ) using definite integration properties, we get * ( ) ( ) + ( ) by the same way we can obtain 84

101 ( ( ) ) ( ) By substituting equations (6.2.6) and (6.2.7) in equation (6.2.5), we obtain the recursive equation ( ) ( ) ( ( ) ( ) ) ( ) In case of uniform fuzzy partition, then Where ( ) and equation (6.2.11) will be written as ( ) ( ) ( ) ( ) ( ) where ( ) ( ) Remark Note that the stability condition in (6.2.8) and in (6.2.9) must hold to be sure of the convergence of the numerical solution to the analytical one, and the stable decay of the errors in the arithmetical operations when we solve the numerical equation. [2, 3] To use the recursive equations (6.2.8), we have to get values of the components and 85

102 Here, we will use the initial and boundary conditions and lemma (4.3.1) to get these components. So ( ). (( ) ( ) ) ( ) ( ) / so ( ). (( ) ( ) ) ( ) / ( ) ( ) ( ) similarly, we get ( ) ( ) ( ) Now all the matrix entities, - ( ) [ ] can be easily found using the equations (6.2.10),(6.2.11) and the recursive equation (6.2.8). Finally, the F-transform inverse from definition (4.4.1) will be used to get the continuous approximation of. Example 6.2.1[13] Consider the following parabolic PDE with the initial condition 86

103 and the boundary conditions ( ).. //.. // ( ) ( ) The exact solution of the above problem is ( ). /. / Tables (6.2.1) and (6.2.2) and Figs.(6.2.2) to (6.2.7) show the computed solution which obtained by applying the F-transform method on the previous example using a non-uniform fuzzy partition with different basic functions and different partitions. On the other hand, tables (6.2.3) and (6.2.4) and Figs. (6.2.8) to (6.2.13) show the computed solution which obtained by applying the F-transform method on the previous example using a non-uniform fuzzy partition with different basic functions and different partitions. Results show that the computed solution from uniform partition gets closer to the exact solution faster than the non-uniform partition. Also using triangular shaped basic functions give better results than using sinusoidal shaped basic functions. 87

104 Table 6.2.1: the exact and computed solutions for non-uniform partitions Computed solutions x t Exact solution Sinusoidal shaped basic Sinusoidal and triangular triangular shaped basic functions basic functions functions n=m=6 absolute error n=m=6 absolute error n=m=6 absolute error e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-02 Table 6.2.2: the exact and computed solutions for non-uniform partitions Computed solutions x t Exact solution Sinusoidal shaped basic functions Sinusoidal and triangular basic functions triangular shaped basic functions n=10, m=40 absolute absolute absolute n=10, m=40 n=10, m=40 error error error e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-04 88

105 Table 6.2.3: the exact and computed solutions for uniform partitions Computed solutions X t Exact solution Sinusoidal shaped basic functions Sinusoidal and triangular basic functions triangular shaped basic functions n=m=6 absolute absolute absolute n=m=6 n=m=6 error error error e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-02 Table 6.2.4: the exact and computed solutions for uniform partitions Computed solutions X t Exact solution Sinusoidal shaped basic functions Sinusoidal and triangular basic functions triangular shaped basic functions n=20, m=10 absolute absolute n=20, m=10 absolute n=20, m=10 error error error e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-02 89

106 Fig The exact solution of ( ) Fig The computed solution of resulting from a non-uniform fuzzy partition with triangular shaped basic functions for 91

107 Fig The computed solution of resulting from a non-uniform fuzzy partition with sinusoidal shaped basic functions on -axis and triangular shaped basic functions on -axis for Fig The computed solution of resulting from non-uniform fuzzy partition with sinusoidal shaped basic functions for 91

108 Fig The computed solution of resulting from non-uniform fuzzy partition with triangular shaped basic functions for Fig The computed solution of resulting from non-uniform fuzzy partition with sinusoidal shaped basic functions on -axis and triangular shaped basic functions on -axis for 92

109 Fig The computed solution of resulting from non-uniform fuzzy partition with sinusoidal shaped basic functions for Fig The computed solution of resulting from uniform fuzzy partition with triangular shaped basic functions for 93

110 Fig The computed solution of resulting from uniform fuzzy partition with sinusoidal shaped basic functions on -axis and triangular shaped basic functions on -axis for Fig The computed solution of resulting from uniform fuzzy partition with sinusoidal shaped basic functions for 94

111 Fig The computed solution of resulting from uniform fuzzy partition with triangular shaped basic functions for Fig The computed solution of resulting from uniform fuzzy partition with sinusoidal shaped basic functions on -axis and triangular shaped basic functions on -axis for 95

112 Fig The computed solution of resulting from uniform fuzzy partition with sinusoidal shaped basic functions for 6.3 Hyperbolic Equations In this section, the F-transform will be applied to the Hyperbolic equation. This type of equations describe a number of interesting physical problems in different areas such as fluid dynamics, solid mechanics, and astrophysics [11]. In [33] stepnicka consider for simplicity, we would not consider this condition, also we will use different basic functions on x-axis and t-axis. To be more specific, we will take the wave equation as an example of the hyperbolic PDE. The wave equation form is given by ( ) ( ) ( ) ( ) 96

113 with the conditions ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) For calculation simplicity, we will define the uniform partition with equidistant steps on, -. Then, we define a basic function * + of, -, and define the uniform partition with steps on, -, and define a basic function * + of, -, to construct uniform fuzzy partition. Now after applying F-transform the equation (6.3.1) transfer to the algebraic equation, -, -, - where, -, - are matrices as in the previous section, and, - [ ] is the matrix of the F-transform components of. So for we get ( ) The partial derivatives difference: will be approximated by their finite ( ( ) ( ) ( ) ( ) 97

114 and for. ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ] ( ) ( ( ) ( )) so ( ) ( ( ) ( )) ( ) by the same way we can get ( ) ( ( ) ( ) ) ( ) 98

115 After substituting (6.3.6) and (6.3.5) in the equation (6.3.4), and then simplify it, we will get the following recursive equation: ( ) ( ) ( ) ( ) ( ) ( ) ( ) where To guarantee the stability and convergence of the numerical solution, the inequality must be achieved. [2, 3] By applying the F-transform and Lemma (4.3.2) on the boundary and initial conditions, we get ( ) ( ) ( ) and ( ) ( ) Remark When we use the recursive equation to find, we will need the value of which can be determined by the last condition ( ) ( ) ( ) We will replace the first partial derivative difference approximation by its three-point central ( ( ) ( )) by applying the F-transform on both sides of eqn. (6.3.10). 99

116 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ) ( ) so ( ) (6.3.11) Now using equations (6.3.8), (6.3.9), and (6.3.11), and the recursive equation (6.3.7) we can easily get all components in the matrix, - [ ] and finally transform it back to the space, -, - using the inverse F- transform in definition (4.4.1). Example Consider the following wave PDE with the boundary conditions ( ) ( ) and the initial conditions 111

117 ( ) ( ) ( ) ( ) The exact solution of the above problem is ( ) ( ) ( ( ) ( )) Tables (6.3.1), (6.3.2) and Figs. from (6.3.2) to (6.3.4) show the numerical solution which are obtained by applying the F-transform method on the previous example using uniform fuzzy partition with different basic functions, and Figs. from (6.3.5) to (6.3.7) show the numerical solution which are obtained by applying the F-transform method on the previous example using uniform fuzzy partition with different basic functions. The results show that the computed solution is converging to the exact solutions as the number of partition increases. 111

118 Table 6.3.1: The exact and computed solutions for x t Exact solution Computed solution using uniform partition Sinusoidal shaped basic functions Sinusoidal and triangular basic functions triangular shaped basic functions n=5,m=10 absolute error n=5, m=10 absolute error n=5,m=10 absolute error e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-02 Table 6.3.2: the exact and computed solutions for Computed solution using uniform partition x t Exact solution Sinusoidal shaped basic Sinusoidal and triangular triangular shaped basic functions basic functions functions n=40,m=70 absolute error n=40, m=70 absolute error n=40,m=70 absolute error e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e

119 Fig The exact solution of ( ) Fig The computed solution of resulting from uniform fuzzy partition with sinusoidal shaped basic functions for 113

120 Fig The computed solution of resulting from uniform fuzzy partition with triangular shaped basic functions for Fig The computed solution of resulting from uniform fuzzy partition with triangular shaped basic functions on -axis and triangular shaped basic functions on -axis for 114

121 Fig The computed solution of resulting from uniform fuzzy partition with sinusoidal shaped basic functions for Fig The computed solution of resulting from uniform fuzzy partition with triangular shaped basic functions for 115

122 Fig The computed solution of resulting from uniform fuzzy partition with triangular shaped basic functions on -axis and triangular shaped basic functions on -axis for 6.4 Elliptic equations In this section, we will discuss the Elliptic PDEs.. Elliptic PDEs. describe a number of interesting physical problems in diverse areas. Elliptic PDEs give rise to Boundary Value Problems (BVPs), that is only boundary conditions must be supplied to complete the specification of an elliptic PDEs. Elliptic PDEs. arise typically in the formation of static (time independent) problems in, for example, potential theory, Fluid mechanics, and Elasticity. Poisson s equation is the archetypical elliptic equation. Our emphasis in this section is on numerical solution of these important equations. For simplicity, we specifically will consider the Poisson equation as an example of the Elliptic PDEs, and use the resulted numerical algorithm in a MATLAB as a user-subroutine with double precision calculation, which will be simulated and applied on an example to ensure the efficiency of the algorithm. 116

123 Consider the Poisson s equation ( ) ( ) ( ) ( ) ( ) where, -, -, and the next Dirichlet boundary condition ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) For calculation simplicity we will define the uniform partition with equidistant steps on, -, and define a basic function * + of, -. We also define the uniform partition with steps on, -, and define a basic function * + of, -, to we construct uniform fuzzy partition of. By applying the F-transform on both sides of equation (6.4.1), we obtain, -, -, - ( ) where, -, - are matrices in section (3.2), and, - [ ] is the matrix of the F-transform components of, and finally The partial derivatives difference will be replaced by with their finite ( ) ( ( ) ( ) ( ) ( ) 117

124 ( ) ( ( ) ( ) ( ) ( ) which enable us to approximate and as follow: so ( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ] ( ) ( ( ) ( ) ) ( ) Similarly, we get ( ) ( ( ) ( )) ( ) by substituting equations (6.4.5) and (6.4.6) in 118

125 after simplifying, we get the recursive equation ( ( ) (( ) ( ) ) ) ( ( ) ( )) ( ( ( ) (( ) ( ) ) ) ( ( ) ( ) ) ( ) ( ) (( ) ( ) ) ) ( ) and using the boundary conditions and Lemma (4.3.2) we get for ( ( )) ( ) ( ( )) ( ) ( ( )) ( ) ( ( )) ( ) To use equations (6.4.7),,(6.4.11) in finding the components of, we will need an initial values for them. The average value of the boundary values will be used as an initial values of.[38] Finally, we will transform it back to the space using the inverse F- transform defined in (4.4.1). Example Consider the following Poisson equation ( ) ( ) ( ) 119

126 with the boundary conditions ( ) ( ) ( ) ( ) the exact solution of the above problem is ( ). The F -Transform algorithm is applied to the above example. The algorithm is simulated in MATLAB [38]. Equations from (6.4.7) to (6.4.11) are used to calculate the solution of the problem. Tables (6.4.1)and (6.2.2) and Figs. (6.4.2) to (6.4.5) show computed solutions for two partitions with different basic functions. It is clear that, the computed results compares well with the exact solution and the results obtained for (number of partitions) are more accurate and closer to the exact solution than. Table 6.4.1: The exact and computed solutions for Computed solution x y Exact solution Sinusoidal shaped basic functions triangular shaped basic functions n=m=10 Absolute Absolute n=m=10 error error e e e e e e e e e e e e e e e e e e e e e e e e e+01 Table 6.4.2: The exact and computed solutions for Computed solution x y Exact solution Sinusoidal shaped basic functions triangular shaped basic functions n=m=100 absolute absolute n=m=100 error error e e e e e e e e e e e e e e e e e e e e e e e e e

127 Fig The exact solution ( ) Fig The computed solution of ( ) for with triangular shaped basic functions. 111

128 Fig The computed solution ( ) for with sinusoidal shaped functions. Fig The computed solution ( ) for with triangular shaped functions. 112

129 Fig The computed solution ( ) for with sinusoidal shaped functions. 6.5 Partial Fractional Differential Equations The concept of differentiation and integration to non-integer order is by no means new, interest in this subject was evident almost as soon as the ideas of the classical calculus were known-leibniz(1859) mentions it in a letter to L'Hospital in 1695[29]. Partial fractional differential equations have an important role in mathematical modeling to describe complex processes in different branches of science "physics, biology. In this section, we will take the timedependent fractional diffusion equations as an example of the partial fractional differential equations [14]. ( ) ( ) ( ) ( ) 113

130 Where ( ) is the derivative of with respect to time of order in the sense of Caputo and. The time-dependent fractional diffusion equations have an important role in describing complex processes in physics, biology, chemistry and economics. We can see that when the equation is the heat "diffusion" equation, when the equation is called fractional sub-diffusion equation, when the equation is called fractional super-diffusion equation, and when the equation will be the Poisson equation.[8] Consider the fractional order differential equation of the form ( ) ( ) ( ) ( ) ( ) where, -, - and with the initial condition ( ) ( ) and the boundary conditions ( ) ( ) For simplicity, uniform partition on, - will be considered, where with steps on, -, and define a basic function * + of, -, we also define the partition with steps on, -, and define a basic function * + of, - to construct an uniform fuzzy partition of. Now, we will apply F-transform on ( to get ), and using the linearity property [ ], -, - ( ) where, - and, - are matrices as in section (6.2), and 114

131 [ ] [ ] is the matrix of the F-transform components of. Entries of equation (6.5.2) can be written in the form of linear combination ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) To use our algorithm for solving the above fractional problem numerically, the time fractional derivative will be replaced by its finite difference approximation ( ) ( ) ( ) ( where ) is the Grünwald weights and computed with the recurrence relationships. ( ) ( ) ( ) ( ) and the space second derivative will be also replaced by its central difference approximation ( ) ( ) ( ) ( ) ( ) 115

132 After substituting equations (6.5.4) and (6.5.5) into (6.5.3) and simplifying the equation, we will get the following form ( ) ( ) ( ) ( ) ( ) ( ) where and ( ) ( ) and ( ) are components in. We will transform the boundary and initial conditions as we did in previous sections, the boundary conditions and the initial condition ( ) ( ) ( ) ( ) ( ) ( ) equation (6.5.6) can be written in a matrix form using the matrix form as ( ) ( ) where ( ) is a square matrix with { ( ) ( ( ), and ( ( ) are vectors for The components of the F-transform of ( (6.5.7), (6.5.8) and(6.5.9) ) are computed from equations 116

133 , - ( ) [ ] The continuous approximation of ( ) can be obtained using equation (6.5.10) and finally transform it back to the space using the inverse F-transform from definition (4.4.1). Example Consider the time dependent fractional diffusion equation ( ) ( ) ( ) The differential equation is solved in, -, - with the initial condition ( ) ( ) and boundary conditions ( ) ( ) The exact solution of the above problem is ( ), ( )- ( ) where is one-parameter Mittag-Leffler function. Equations (6.5.7) :(6.5.9) are applied to find the F-transform components of for different partitions and values of. Tables (6.5.1) and (6.5.2) and Figs : show the exact solution and approximated computed solutions of the above fractional order differential equation for with different values of. Tables (6.5.3) and (6.5.4) and Figs : show the exact solution and the approximated computed 117

134 solutions of the above fractional order differential equation for with different values of. Tables (6.5.5) and (6.5.6) and Figs : show the exact solution and the approximated computed solutions of the above fractional order differential equation for with different values of. Table 6.5.1: The exact and computed solutions for α Computed solution x t Exact solution Sinusoidal shaped basic functions triangular shaped basic functions n=m=10 absolute absolute n=m=10 error error e e e e e e e e e e e e e e e e e e e e e e e e e-004 Table 6.5.2: The exact and computed for α Computed solution Sinusoidal shaped basic triangular shaped basic Exact x t functions functions solution absolute absolute n=m=20 n=m=20 error error e e e e e e e e e e e e e e e e e e e e e e e e e-004 Table 6.5.3: The exact and computed solutions for α Computed solution Sinusoidal shaped basic triangular shaped basic Exact x t functions functions solution absolute absolute n=m=10 n=m=10 error error e e e e e e e e e e e e e e e e e e e e e e e e e

135 Table 6.5.4: The exact and computed solutions for α Computed solution Sinusoidal shaped basic triangular shaped basic Exact x t functions functions solution n=30, absolute absolute n=30, m=20 m=20 error error e e e e e e e e e e e e e e e e e e e e e e e e e-009 Table 6.5.5: The exact and computed solutions for α Computed solution Sinusoidal shaped basic triangular shaped basic Exact x t functions functions solution absolute absolute n= m=10 n=m=10 error error e e e e e e e e e e e e e e e e e e e e e e e e e-003 Table 6.5.6: The exact and computed solutions for α Computed solution Sinusoidal shaped basic triangular shaped basic Exact x t functions functions solution absolute absolute n= m=100 n=m=100 error error e e e e e e e e e e e e e e e e e e e e e e e e e

136 Fig The exact solution of ( ) for Fig The computed solution ( ) for with triangular shaped functions 121

137 Fig The computed solution ( ) for with sinusoidal shaped functions Fig The computed solution ( ) for with triangular shaped functions 121

138 Fig The computed solution ( ) for with sinusoidal shaped functions Fig The exact solution of ( ) for α 122

139 Fig The computed solution ( ) for with triangular shaped functions Fig The computed solution ( ) for with sinusoidal shaped functions 123

140 Fig The computed solution ( ) for with triangular shaped functions Fig The computed solution ( ) for with sinusoidal shaped basic functions 124

141 Fig The exact solution ( ) for α Fig The computed solution ( ) for with triangular shaped basic functions 125

142 Fig The computed solution ( ) for with sinusoidal shaped functions Fig The computed solution ( ) for with triangular shaped functions 126

143 Fig The computed solution ( ) for with sinusoidal shaped basic functions 6.6 Summary F-transform have been applied to different types of PDEs. using uniform and non-uniform partitions with different values of, and using different basic functions. We used the obtained algorithms to solve numerical examples and compared the computed solutions with the exact solutions. In example we noticed that results from uniform partitions are better than results from non-uniform partitions. In each example we noticed that using triangular shaped basic functions on both axes gave better results than using sinusoidal shaped basic functions on at least one axis, which return to the sinusoidal function s behavior nature. Applied F-transform to fractional order differential equations for the first time was very efficiency numerical method, fractional order differential equations known by complicity and hard equations, using F- transform in fractional order equation gives a satisfied results with accepted errors. 127