Linear Fredholm Integro-Differential Equation. of the Second Kind. By Khulood Nedal Iseed Thaher. Supervised Prof. Naji Qatanani

Transcription

1 An-Najah National University Faculty of Graduate Studies Linear Fredholm Integro-Differential Equation of the Second Kind By Khulood Nedal Iseed Thaher Supervised Prof. Naji Qatanani This Thesis is Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science of Mathematics, Faculty of Graduate Studies, An-Najah National University, Nablus, Palestine. 206

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3 iii Dedication I dedicate this thesis to my beloved Palestine, my parents, my love my husband Ayman, my children Sandra and Kareem, my brother Khaled and my sisters Amany and Farah. Without their patience, understanding, support and most of all love, this work would not have been possible.

4 iv ACKNOWLEDGMENT First of all, I thank my God for all the blessing he bestowed on me and continues to bestow on me. I would sincerely like to thank and deeply greatful to my supervisor Prof. Dr. Naji Qatanani who without his support, kind supervision, helpful suggestions and valuable remarks, my work would have been more difficult. My thanks also to my external examiner Dr. Maher Qarawani and to my internal examiner Dr. Hadi Hamad for their useful and valuable comments. Also, my great thanks are due to my family for their support, encouragement and great efforts for me. Finally, I would also like to acknowledge to all my teachers in An-Najah National University department of mathematics.

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6 No vi Table of Contents Content Dedication Acknowledgement Declaration List of Figures List of Tables Abstract Introduction Chapter One: Mathematical Preliminaries Classification of integro-differential equations Types of integro-differential equations Linearity of integro-differential equations Homogenity of integro-differential equations Singularity of integro-differential equations Systems of Fredholm integro-differential equations Systems of Volterra integro-differential equations Kinds of kernels Chapter Two: Analytical Methods for Solving Fredholm Integro-Differential Equation Direct computation method Variational iteration method Adomian decomposition method Modified decomposition method Noise terms phenomenon Series solution method Chapter Three: Numerical Methods for Solving Fredholm Integro-Differential Equation B-spline scaling functions and wavelets Homotopy perturbation method (HPM) Legre polynomial method Taylor collocation method Chapter Four: Numerical Examples and Results The numerical realization of equation (4.) using the B-spline scaling functions and wavelets Page iii iv v viii ix x

7 vii The numerical realization of equation (4.) using the Homotopy perturbation method The numerical realization of equation (4.2) using the B-spline scaling functions and wavelets The numerical realization of equation (4.2) using the Homotopy perturbation method The numerical realization of equation (4.3) using the Legre polynomial method The numerical realization of equation (4.3) using the Taylor collocation method Conclusions References Appix Maple code for B-spline scaling functions and wavelets for example 4. Maple code for Homotopy perturbation method for example 4. Maple code for B-spline scaling functions and wavelets method for example 4.2 Maple code for Homotopy perturbation method for example 4.2 Matlab code for Legre polynomial method Matlab code for Taylor collocation method الملخص ب

8 No viii List of Figures Title The exact and numerical solutions of applying Algorithm 4. for equation (4.). The error resulting of applying algorithm 4. on equation (4.). The exact and numerical solutions of applying Algorithm 4.2 for equation (4.). The error resulting of applying algorithm 4.2 on equation (4.). The exact and numerical solutions of applying Algorithm 4.for equation (4.2). The error resulting of applying algorithm 4. on equation (4.2) The exact and numerical solutions of applying Algorithm 4.2 for equation (4.2). The error resulting of applying algorithm 4.2 on equation (4.2) The exact and numerical solutions of applying Algorithm 4.3 for equation (4.3). the error resulting of applying algorithm 4.3 on equation (4.2). The exact and numerical solutions of applying Algorithm 4.4 for equation (4.3). the error resulting of applying algorithm 4.4 on equation (4.3). Page

9 No ix List of Tables Title The exact and numerical solutions of applying Algorithm 4. for equation (4.). The exact and numerical solutions of applying Algorithm 4.2 for equation (4.). The exact and numerical solutions of applying Algorithm 4. for equation (4.2). The exact and numerical solutions of applying Algorithm 4.2 for equation (4.2). The exact and numerical solutions of applying Algorithm 4.3 for equation (4.3). The exact and numerical solutions of applying Algorithm 4.4 for equation (4.3). Page

10 x Linear Fredholm Integro-Differential Equation of the Second Kind By Khulood Nedal Iseed Thaher Supervisor Prof. Naji Qatanani Abstract In this thesis we focus on solving linear Fredholm integro-differential equation of the second kind due to it's wide range of physical applications. We will investigate some analytical and numerical methods to solve this equation. The discussed analytical methods include: Direct computation method, variational iteration method, Adomian decomposition method, modified decomposition method, noise terms phenomenon and series solution method. The numerical methods that will be presented here are: B-spline scaling function and wavelet method, Homotopy perturbation method, Legre polynomial method and Taylor collocation method. Particular numerical examples demonstrating these numerical methods have been implemented for solving linear Fredholm integro-differential of the second kind. A comparison has been drawn between these numerical methods. Our numerical results show that the Homotopy perturbation method and Legre polynomial method have proved to be the most efficient in comparison to the other numerical methods regarding their performance on the used examples.

11 Introduction The subject of integro-differential equations is one of the most important mathematical tools in both pure and applied mathematics. Integrodifferential equations play a very important role in modern science and technology such as heat transfer, diffusion processes, neutron diffusion and biological species. More details about the sources where these equations a rise can be found in physics, biology and engineering applications as well as in advanced integral equations books. (see [2, 4, 5, 8, 8, 9, 20, 22]). Some valid numerical methods, for solving Fredholm integro-differential equations have been developed by many researchers. In [7] Behiry and Hashish used wavelet methods for the numerical solution of Fredholm integro-differential equation. Lakestani, Razzaghi and Dehghan [25] applied linear semiorthogonal B-spline wavelets, specially constructed for the bounded interval to solve linear Fredholm integro-differential equation. In [6] Ji-Huan He solved linear Fredholm integro-differential equation by a Homotopy perturbation method. This method yields a very rapid convergence of the solution series in most cases, usually only few iterations leading to very accurate solutions. A Legre collocation matrix method is presented to solve high-order linear Fredholm integro-differential equations under the mixed conditions in terms of Legre polynomials were used in [42] by Yalcinbas, Sezer and Sorkan. In [23] Karamete and Sezer solved Fredholm integro-differential equation by a truncated Taylor series. Using the Taylor collocation points, this method transforms the

12 2 integro-differential equation to a matrix equation which corresponds to a system of linear algebraic equations with unknown Taylor coefficients. However many approaches for solving the linear and nonlinear kind of these equations may be found in [0], [], [3], [4], [5], [29], [30], [32], [36], [39] and [43]. In this work, some analytical methods have been used to solve the Fredholm integro-differential equation of the second kind. These methods are: direct computation method, variational iteration method, Adomian decomposition method, modified decomposition method, noise terms phenomenon and series solution method. For the numerical treatment of the Fredholm integro-differential equation of the second kind, we have implemented the following methods: B-spline scaling functions and wavelets, Homotopy perturbation method, Legre polynomial method and Taylor collocation method. This thesis is organized as follow: In chapter one, we introduce some basic concepts of integro-differential equations. In chapter two, we investigate some analytical methods to solve the Fredholm integro-differential equation. These have been mentioned before. In chapter three, we implement some numerical methods for solving Fredholm integrodifferential equation. These have mentioned before. Numerical examples and results are presented in chapter four and conclusions have been drawn.

13 Definition. [39] 3 Chapter One Mathematical Preliminaries An integro-differential equation is the equation in which the unknown function appears inside an integral sign and contains ordinary derivatives. A standard integro-differential equation is of the form: h( x ) ( n ) v ( x ) f ( x ) G ( x, y ) v( y ) dy, (.) g( x ) where ( n ) dv v ( x), n dx n n integer, is a constant, g( x ) and hx ( ) are limits of integration that may be both variables, constants, or mixed, G ( x, y ) is a known function of two variables x and y called the kernel or the nucleus of the integro-differential equation. The function v that will be determined appears under the integral sign, and sometimes outside the integral sign. The function f ( x ) and G ( x, y ) are given.. Classification of Integro-Differential Equations.. Types of integro-differential equations There are three types of integro-differential equations:

14 . Fredholm integro- differential equation 4 The Fredholm integro-differential equation of the second kind appears in the form: ( n v ) ( x ) f ( x ) G ( x, y ) b a v( y ). dy (.2) 2. Volterra integro-differential equation The Volterra integro-differential equation of the second kind appears in the form: ( n v ) ( x ) f ( x ) G ( x, y ) x a v( y ), dy (.3) where the upper limit of integration is variable. 3. Volterra-Fredholm integro-differential equation The Volterra-Fredholm integro-differential equation of the second kind appear in two forms, namely: and ( n ) x ( n ) v ( x ) f ( x ) G ( x, y ) v( y ) dy 2 G2 ( x, y ) v( y) dy (.4) a v ( x, y ) f ( x, y ) F ( x, y,,, v (, )) (.5) y d d, x, y [0, Y ] 0 b a

15 5 where f ( x, y ) and F ( x, y,,, v (, )) are analytic functions, D [0, Y ] and is a closed subset of n R, n,2,3. It is interesting to note that (.4) contains disjoint Volterra and Fredholm integrals, whereas (.5) contains mixed integrals. Other derivatives of less order may appear as well...2 Linearity of integro-differential equation Definition.2 [39] The integro- differential equation h( x ) ( n ) v ( x ) f ( x ) G ( x, y ) v( y ) dy, g( x ) is said to be linear if the exponent of the unknown function v( x ) under the integral sign is one and the equation does not contain nonlinear functions of v( x ), otherwise, the equation is called nonlinear...3 Homogeneity of integro-differential equation Definition.3 [39] The integro-differential equation h( x ) ( n ) v ( x ) f ( x ) G ( x, y ) v( y ) dy, g( x ) is said to be homogeneous if f ( x ) is identically zero, otherwise it is called nonhomogeneous.

16 ..4 Singularity of integro-differential equation 6 When one or both limits of integration become infinite or when the kernel becomes infinite at one or more points within the range of integration. For example, the integro-differential equation. dy v( y) ( n v ) ( x ) f ( x ) G ( x, y ) is a singular integro-differential equation of the second kind. (i) Singular integro-differential equation If the kernel is of the form H ( x, y ) G ( x, y ) x y (.6) where H ( x, y ) is differentiable function a x b, a y b with H ( x, y ) 0, then the integro-differential equation is said to be a singular H ( x, y ) equation with Cauchy kernel where the G ( x, y ) f ( y) x y understood in the sence of Cauchy Principal value (CPV) and the notation P.V b H ( x, y ) dy is usually used to denote this. Thus x y a b a dy is P.V. H ( x, y ) H ( x, y ) H ( x, y ) dy lim dy dy, x y x x y x y b x b a a x for example

17 x 2 v ( y ) v ( x ) x 7x x ln dy, 3 x 5 x y 7 x. (ii) Weakly singular integro-differential equation The kernel is of the form H ( x, y ) G ( x, y ) x y (.7) where H ( x, y ) is bounded ie.. several times continuously differentiable a x b and a y b with H ( x, y ) 0 and is a constant such that 0. For example, the equation v ( n ) x ( x) v( y) x 0 y dy, 0 is a singular integro-differential equation with a weakly singular kernel..2 Systems of Fredholm Integro-Differential Equations A system of Fredholm integro-differential equations of the second kind can be written as: b ( i ) u ( x ) f ( x ) ( G ( x, y ) u( y ) G ( x, y ) v( y)) a dy (.8) b ( i ) v ( x ) f ( x ) ( G ( x, y ) u( y ) G ( x, y ) v( y )) dy a The unknown functions ux ( ) and v( x ), that will be determined, occur inside the integral sign whereas their derivatives appear mostly outside the

18 integral sign. The kernels Gi ( x, y ) given as real-valued functions [39]., 8 Gi ( x, y ) and the functions f i ( x) are.3 Systems of Volterra Integro-Differential Equations A system of Volterra integro-differential equation of the second kind has the form: ( i ) u ( x ) f ( x ) ( G ( x, y ) x u y 0 ( ) G ( x, y ) v( y)) dy (.9) ( i ) x v ( x ) f ( x ) ( G ( x, y ) u( y ) G ( x, y ) dy v( y )). The unknown functions ux ( ) and v( x) that will be determined, occur inside the integral sign whereas their derivatives appear mostly outside the integral sign. The kernels Gi ( x, y ), G ( x, y ) and the functions f ( x) are given as real-valued functions [39]..4 Kinds of Kernels i i. Separable kernel A kernel G ( x, y ) is said to be separable or (degenerate) if it can be exp- ressed in the form ).0) n G ( x, y ) g k( x ) hk( y ), k

19 where the functions are linearly indepent. (see [39]). gk ( x) 9 and the functions hk ( y) 2. Symmetric (or Hermitian) kernel A complex-valued function G ( x, y ) is called symmetric (or Hermitian) if * (. ) G ( x, y ) G ( x, y ), where the asterisk denotes the complex conjugate. For a real kernel, we have G ( x, y ) G ( y, x ). (.2)

20 0 Chapter Two Analytical Methods for Solving Fredholm Integro-Differential Equation of the Second Kind There are many analytical methods for solving Fredholm integrodifferential equation of the second kind. In this chapter we will focus on the following methods: direct computation method, variational iteration method, Adomian decomposition method, modified decomposition method, noise terms phenomenon and series solution method. 2. Direct Computation Method This method can be used to solve the Fredholm integro-differential equation of the second kind directly instead of a series form. For the application of this method we consider the separable kernel of the form G( x, y ) g ( x ) h( y ), (2.) We consider the Fredholm integro-differential equation of the general form with the initial conditions ( n v ) ( x ) f ( x ) G ( x, y ) v( y) b a ( k v ) (0) b k, 0. k n dy, (2.2) Substituting (2.) into (2.2) gives b ( n v ) ( x ) f ( x ) g ( x ) h( y ) v( y ) dy. (2.3) a

21 Since the integral in equation (2.3) is a bounded integral and depent only on one variable That is y, then we can assign this integral by a constant, b h( y) ( ) a v y. dy (2.4) Thus equation (2.3) becomes ( n v ) ( x ) f ( x ) g ( x ). (2.5) Integrating both sides of (2.5) n conditions, we can find formula for means we can write times from 0 to v( x) x, also using the initial that deps on and x. This v ( x ) u( x, ). (2.6) Substituting (2.6) into the right side of (2.4), calculating the integral, also solving the resulting equation, we determine solution v( x) after substituting into (2.6)..We obtain the exact Example 2. Consider the Fredholm integro-differential equation v ( x ) 2 x v ( y ) dy with v (0) 0. (2.7) 0 This equation may be written as v ( x ) 2x, v (0) 0. (2.8)

22 2 obtained by setting v( y) dy. 0 (2.9) Integrating both sides of (2.8) from 0 to x, and by using the initial condition we obtain 2 (2.0) v ( x ) 6 x x. Substituting (2.0) into (2.9) and evaluating the integral yield ( ) v y dy 2 (2.) 2 0 hence we find 4. (2.2) The exact solution is given by 2 (2.3) v ( x ) 6x 4 x. 2.2 Variational Iteration Method (VIM) The variational iteration method (VIM) was established by Ji-Huan He [7]. The method provides rapidly convergent successive approximations of the exact solution only if such a closed form solution exists. We consider the Fredholm integro-differential equation of the general form

23 3 ( i v ) ( x ) f ( x ) G ( x, y ) b a v( y ). dy (2.4) The correction functional for this equation is given by x n n 0 ( i ) v ( x ) v ( x ) ( ) [ v n ( ) f ( ) G (, z ) b a dz. v n ( z ) ] d (2.5) The variational iteration method needs to determine the Lagrange multiplier ( ) via integration by parts and by using a restricted variation. The variational iteration formula, without restricted variation, should be used for the determination of the successive approximations of the solution v( x). The zeroth approximation v 0 v n ( x ), n 0 can be any selective function. However, using the given initial values v (0), v (0), are preferably used for the selective zeroth approximation v 0. Consequently, the solution is given by v ( x ) lim v ( x ). (2.6) n n Example 2.2 Consider the Fredholm integro-differential equation v ( x ) 2 x v ( y ) dy with v (0) 0. (2.7) 0 The correction functional for this equation is given by x ] v ( x ) v ( x ) [ v ( ) 2 v ( z ) dz d, (2.8) n n n n 0 0

24 4 as for first-order Fredholm integro-differential equation. This gives: v 0 ( x) 0 x v ( x ) v ( x ) [ v ( ) 2 v ( z ) dz ] d 6x 2 x v ( x ) v ( x ) [ v ( ) 2 v ( z ) dz ] 2 d 6x 2x x v ( x ) v ( x ) [ v ( ) 2 v ( z ) x v ( x ) v ( x ) [ v ( ) 2 v ( z ) x 0 0 v ( x ) v ( x ) [ v ( ) 2 v ( z ) x dz ] d dz ] d dz ] d v ( x ) v ( x ) [ v ( ) 2 v ( z ) n n n n x 3 x ( ) x x 2x x (2.9) 2 6x 2x x x 2 2 6x 2x x x x 2 4 dz ] d and so on. The VIM admits the use of v ( x ) lim v ( x ). (2.20) n n We obtain the exact solution 2 (2.2) v ( x ) 6x 4 x.

25 5 2.3 Adomian Decomposition Method The Adomian decomposition method (ADM) was introduced and developed by George Adomian [-3]. ADM gives the solution in an infinite series of components. The idea of Adomian decomposition method is transforming Fredholm integro-differential equation to an integral equation. We consider the second order Fredholm integro-differential equation of the second kind of the form. v ( x ) f ( x ) G ( x, y ) b a v( y ), dy (2.22) with the initial conditions v(0) b0, v(0) b. Integrating both sides of (2.22) from 0 to x twice, we get b 0 v( y) v ( x ) b b x L ( f ( x )) L ( G ( x, y ) a dy ) (2.23) where the initial conditions are used, and operator. v( x ) expressed as L is a two-fold integral v ( x ) v ( x ). (2.24) n 0 n Setting (2.24) into (2.23) gives: x v n( x ) b0 bx L ( f ( x )) L G ( x, y ) v n( y ) dy (2.25) n0 0 n0

26 6 or equivalently, L x ( G ( x, y ) 0 v ( x ) v ( x ) v ( x ) b b ( x ) L ( f ( x )) dy v 0 ( y ) ) L x ( G ( x, y ) 0 dy v ( y ) ) L x ( G ( x, y ) 0 v 2 ( y) dy ) (2.26) Note that, v 0 ( x) integral sign, that is is defined by all the terms that are not included under the 0 0 v ( x ) b b ( x ) L ( f ( x )) x k ( ) ( (, ) ( ) k v x L G x y 0 v y dy ), k 0. (2.27) Using (2.24), the obtained series converges to the exact solution if such a solution exists, see ([2], [39]). Example 2.3 Consider the Fredholm integro-differential equation v ( x ) 24 x v ( y ) dy with v (0) 0. (2.28) 0 Note that, the integral at the right side is equivalent to a constant. Integrating both sides of Eq. (2.28) from 0 to x, we get

27 7 2 v ( x ) v (0) x 2 x x v ( y ) dy (2.29) 0 using the initial condition 2 v ( x ) x 2 x x v ( y ) dy. (2.30) 0 Let v ( x ) v ( x ). (2.3) n 0 n Setting (2.3) into (2.30) gives 2 v n( x ) x 2 x x v n( y ) dy. (2.32) n0 0 The components v j, j 0 of v( x ) can be determined by using the recurrence relation 2, k v 0 ( x ) x 2x v ( x ) x v k( y ) dy, k 0. (2.33) 0 This gives v 2 0 ( x ) x 2x, 7 v ( x ) x v 0( y ) dy x, v 2( x ) x v ( y ) dy x, v 3( x ) x v 2( y ) dy x (2.34) Hence, the solution is

28 8 2 7 v ( x ) x 2 x x ( ), (2.35) which gives the exact solution 2 (2.36) v ( x ) 6x 2 x. 2.4 Modified Decomposition Method As shown before, The Adomian decomposition method provides the solu- tion in an infinite series of components. The method substitutes the decomposition series of v( x), given by v ( x ) v ( x ). (2.37) n 0 n The process of converting Fredholm integro-differential equation to an integral equation can be achieved by integrating both sides of the Fredholm integro-differential equation from 0 to convert it into the integral equation x as many times is needed to b v ( x ) f ( x ) G ( x, y ) v( y ) dy. (2.38) a The standard Adomian decomposition method introduces the recurrence relation v 0 ( x ) f ( x ) (2.39) b v k ( x ) G ( x, y ) v k ( y ) dy, k 0. a

29 The modified decomposition method presents a slight variation to the recurrence relation (2.38) to determine the components of 9 v( x) in an easier and faster manner. For many cases, the function f ( x) can be set as the sum of two partial functions, namely f ( x) and f 2 ( x). In other words f ( x ) f ( x ) f ( x ). (2.40) 2 The modified decomposition method admits the use of the modified recurrence relation v ( x ) f ( x ) 0 v ( x ) f 2( x ) G ( x, y ) b a v 0 ( y) dy (2.4) b v k ( x ) G ( x, y ) v k ( y) dy, k. a The difference between the recurrence relation (2.39) and the modified recurrence relation (2.4) is in the formation of v 0 ( x) and v ( x) only. The other components v j ( x ), j 2 remain the same in the two recurrence relations [39]. Example 2.4 Consider the Fredholm integro-differential equation 2 v ( x ) sin( x ) x x y v( y) dy with v (0) 0, v (0). (2.42) 0

30 Using the modified decomposition method. First, integrate both sides of 20 (2.42) from 0 to x twice, and using the given conditions, we obtain the Fredholm integral equation v ( x ) sin( x ) x x y 3! 3! 0 v( y ). dy (2.43) 3 f ( x ) sin( x ) x (2.44) 3! We split f ( x) into two parts, namely f ( x ) sin( x ) 3 f 2( x ) x, (2.45) 3! f ( x ) related to the zeroth component, v 0 ( x), and add f 2 ( x ) to the component v ( x ). Therefore, we obtain the modified recurrence relation v 0 ( x ) sin( x ) v ( x ) x x y 0 3! 3! v ( y) dy 0, (2.46) 0 all remaining components v 2( x ), v 3( x ), are zeros. This gives the exact solution as v ( x ) sin( x ) (2.47)

31 2 2.5 The Noise Terms Phenomenon The idea behind solving equations with noise terms if the noise terms appear between components of v( x), then the exact solution can be obtained by considering only the first two components v 0 ( x) and v ( x). The noise terms are defined as the identical terms, with opposite signs, that may appear between the components of the solution v( x). The conclusion made by [], [3], and [4] suggests that if we observe the appearance of identical terms in both components with opposite signs, then by canceling these terms, the remaining non-canceled terms of v 0 may in some cases provide the exact solution, that should be justified through substitution. It was formally proved that other terms in other components will vanish in the limit if the noise terms occurred in v 0 ( x) and v ( x ). Example 2.5 Consider the Fredholm integro-differential equation v ( x ) x xy v( y) dy with v (0) 0. (2.48) 3 0 Using the noise terms phenomenon. By integrating both sides of the equation (2.48) from 0 to x gives 2 2 ( ) v ( x ) v (0) x x x y v y dy, v (0) 0. (2.49) by using the initial condition

32 v ( x ) x x x y v( y) dy. (2.50) Let v ( x ) v ( x ). (2.5) n 0 n Substituting (2.5) into both sides of (2.50) yields n0 0 n0 2 2 v n( x ) x x x y ( v n( y )) dy. (2.52) 6 2 or equivalently v 0( x ) x x v ( x ) x (2.53) Notice that v ( x ) can be written as The noise terms 2 2 v ( x ) x x. (2.54) x 2 appear between the components 0 ( ) v x and v ( x ). By canceling the noise term form v 0 ( x ), we obtain the exact solution v ( x ) x. (2.55) 2.6 The Series Solution Method The series method is useful method that stems mainly form the Taylor series for analytic functions for Fredholm integro-differential equation.

33 23 Definition 2. [39] A real function v( x) has the Taylor series representation n ( k ) v ( b) k (2.56) k! k 0 v ( x ) ( x b), is called analytic if it has derivatives of all orders such that the Taylor series at any point b in it is domain converges to v( x) in a neighborhood of b. For simplicity, the generic form of the Taylor series at b 0 is given as v ( x ) a x. (2.57) n 0 n n We will assume that the Fredholm integro-differential equation of the second kind ( k v ) ( x ) f ( x ) G ( x, y ) b a v( y) dy, v j (0) a, 0 j ( k ) (2.58) j is analytic, and therefore possesses a Taylor series of the form given in (2.57), where the coefficients a n will be determined recurrently. Setting (2.57) into (2.58) yields ( k ) b n0 a 0 n n anx T ( f ( x )) G ( x, y ) an y dy, (2.59) n This is equivalent to

34 24 b 2 ( k ) 0 2 a 2 ( a a x a x ) T ( f ( x )) G ( x, y ) ( a a y a y ) dy 0 2 (2.60) where T ( f ( x )) is the Taylor series for f ( x). After performing integration in the right hand side of (2.60), we compare coefficients of power of x on both sides. Example 2.6 Consider the Fredholm integro-differential equation of the second kind v ( x ) 4 x ( x y ) v( y) dy with v (0) 2, (2.6) using the series method. Let n v ( x ) ( a x ). (2.62) n 0 n Inserting (2.62) into (2.6), yields n nanx 4 x (( x y ) n0 n 0 n an y ) dy. (2.63) a0 2 from the given initial conditions. Evaluating the integral at the right side gives

35 a 2a x 3a x a a3 a5 a7 (4 2 a0 a2 a4 a6 ) x (2.64) Comparing coefficients of like powers of x in both sides of (2.64) gives a 0, a2 6, an 0, n 2. (2.65) The exact solution is given by 2 x (2.66) v ( x ) 2 6. where we used a0 2 from the initial condition.

36 26 Chapter Three Numerical Methods for Solving Fredholm Integro-Differential Equation of the Second Kind There are many numerical methods available for solving Fredholm integrodifferential equation of the second kind. In this chapter we will discuss the following methods: B-spline scaling functions and wavelets, Homotopy perturbation method, Legre polynomial method and Taylor collocation method.. 3. B-Spline Scaling Functions and Wavelets on [0,] B-spline scaling functions and wavelets which are presented to approximate the solution of linear second order Fredholm integrodifferential equation [6], [7] and [9]. Their properties and the operational matrices of derivative for these functions are presented to reduce the solution of linear Fredholm integro-differential equations to a solution of algebraic equations. Consider the linear second-order Fredholm integro-differential equation of the form: 2 ( ) ( ) i i x y ( x ) f ( x ) K ( x, t ) () i 0 0 with y t dt, 0x ( 3.) y(0) y 0, y() y (3.2)

37 27 where i ( i 0,,2), f, and K, are given functions in y 2 [0,], L 0 and y are given real numbers and y is the unknown function to be found. The second-order B-splines scaling functions are defined as: i, j xi j, j x i j i ( x ) 2 ( x i j ), j x i j 2, j 0,,2 2 (3.4) 0, otherwise with the respective left- and right-hand side boundary scaling functions i, j 2 ( xi j), 0 xi, j ( x ) 0, otherwise i, j xi j, i j x, 2 ( x ) i j j 0, otherwise )3.5) )3.6) the actual coordinate position x is related to x i i according to x 2 x. i The second-order B-spline wavelets are given by i, j xi j, 4 7( xi j), 9 6( xi j), ( x ) 29 6( x i j ), 6 7 7( xi j), 3 ( x j), 0, i j x j 2 i j 2 x j j x j 3 2 i j 3 2 x j 2 j 2 x j 5 2 i j 5 2 x j 3 otherwise i i i )3.7) with the respective left- and right-hand side boundary wavelets:

38 i, j 6 23 x i, 4 7, x i ( x) 0 7 xi, 6 2 x i, 0, 0 x 2 i 2 x x 3 2 i i 28 j 3 2 x 2, j otherwise i (3.8) i, j 2 ( j 2 x ), j x i i j 2 0 7( j 2 xi ), j 2 x i j i ( x ) 4 7( j 2 x i ), j x i j 3 2, j ( j 2 xi ), j 3 2 x i j 2 0, otherwise. (3.9) From (3.4)-(3.9), we get more clear description for these two sets of equations i, j ( x ) i,2 j ( x ) i,2 j ( x ) i,2 j 2( x ), 2 2 i 2,3, j j 0,,2,,2 2 i, ( x ) i, ( x ) i,0 ( x ), i 2,3, (3.0) 2 2 i ( x ) i ( x ) i ( x ), i,2 i,2 i,2 i 2,3, 2 and 5 i, j ( x ) i,2 j ( x ) i,2 j ( x ) i,2 j 2( x ) j i,2 j 3( x) i,2 j 4( x), i 2,3,, j 0,,2,,

39 29 i, ( x ) i, ( x ) i,0 ( x ) i, ( x ) i,2 ( x ), i ( x ) i ( x ) i ( x ) i ( x ) i ( x ). i,2 i,2 2 i,2 3 i,2 4 i, )3.) We define (3.2) M ( )(,...,2 ),,, M M k x k M M M,,,0,2 T let Q T M M M 0 dx. )3.3) The entry ( QM ) of the matrix Q M in (3.3) is calculated from ij 0 ( x ) ( x ) dx M, i M, j )3.4) using (3.4)-(3.6), (3.2) and (3.4), we get a symmetric M (2 M ) (2 ) matrix for Q M which is given by: 2 QM M (3.5)

40 30 Finally, we want to define a vector M of order ( M ) (2 ) as: T M 2, 2,0 2,3 2, 2,2 3, 3,6 M, M,2 2 [,,...,,,...,,,...,,...,,..., ]. )3.6) The operational matrix of derivative The differentiation of the vector M and in (3.6) can be expressed as: D M M, D (3.7) where D and D are M M (2 2) (2 2) operational matrices of derivative of B-spline scaling functions and wavelets respectively. The matrix D T T D M ( t ) M ( t ) dt M ( t ) M ( t ) dt QM R Q M 0 0 (3.8) where R ( t ) ( t ) dt. 0 T M M )3.9) M M In (3.9), R is (2 ) (2 ) matrix given by

41 R 3, 2( ), 2( ), 2( ) M M t M t dt M t ( t ) dt M,2 0 0 M ( t ),2, 2( ) M ( ) M M t dt t ( t ) dt M M,2 M,2 0 0 )3.20) since the elements in the vector given in (3.2) are nonzero between k /2 M and M then for any entries of ( k 2) / 2, R jk,, we have M M M M M M ( k 2) 2 R ( t ) ( t ) dt ( t ) ( t ) dt j, k M, j M, k M, j M, k 0 k 2 ( k ) 2 ( k 2) 2 ( k ) 2 M M, j ( t ) M, k ( t ) dt M, j ( t ) M, k ( t ) dt k 2 ( k ) 2 M M M, k t dt M, k 2 ( ) 2 ( t ) dt k ( k 2) 2 M M M 2 ( k ) 2. )3.2) From (3.2), we get R (3.22)

42 The matrix D M 32 can be obtained by considering MG X M )3.23) where G is a M M (2 ) (2 ) matrix, which can be calculated as follows. Let and be defined as:,,,0,, j j j, j,2,,,0,, j j j. j,2 )3.24) (3.25) Using (3.0) and (3.24), we get (3.26) j j j with j (3.27) where ( j 2,3, ), is j j j matrix. From (3.) and (2 ) (2 ) (3.25), we have (3.28) j j j with j

43 )3.29) where ( j 2,3, ), is j j j 2 (2 ) matrix. From (3.26) and (3.28), we get..., j j j M M j j j M M. (3.30) Using (3.24) and (3.6), we have G 23 M 23 M M M M M 2 M M.. )3.3) From (3.7),(3.8), and (3.3), we get

44 34 )3.32) G GD GR ( Q ) GR ( Q ) G D, where M M M M M M M D GR ( Q ) G. (3.33) we want to explain the technique. We solve linear Fredholm integrodifferential equation by using B-spline wavelets. For this purpose we write (3.) as 2 i 0 ( i ) ( x ) y ( x ) f ( x ) z ( x ) i 0 x, )3.34) where z ( x ) K ( x, t ) y ( t ) dt. (3.35) 0 We define T y ( x ) C ( x ) (3.36) T z ( x ) K ( x, t ) C () t 0 dt, (3.37) where ( x ) is defined in (3.6 ), and C defined as c, c,..., c, d,..., d, d,..., d,..., d,..., d M 0 3 2, 2,2 3, 3,6 M, M,2 2 is M (2 ) unknown vector T We can approximate (3.37), using Newton-Cotes techniques as T z ( x ) K ( x, t ) C () t dt 0 n T w ik ( x, ti) C ( t i ) F ( x, C ), (3.38) i

45 35 where w i and t i are weight and nodes of Newton-Cotes integration techniques respectively. Using (3.32) and (3.36), we get T T y ( x ) C ( x ) C D ( x ), T y ( x ) C D ( x ),. (3.39) From (3.30), (3.34), and (3.35), we get T T 0 w 2 ( x ) C ( x ) ( x ) C D ( x ) ( x ) D ( x ) f ( x ) F ( x, C ). 2 w )3.40) Also, using (3.2) and (3.36), we have T C 0 y 0, T C y. )3.4) To find the solution y( x ), we first collocate (3.40) in xi M 2 (2i ) (2 2), i M,2,,2, the resulting equation gener- rates M 2 linear equations which can be solved using Newton's iterative method. The initial values required to start Newton's method have been chosen by taking y( x ) as linear function between y(0) y 0 and y() y. The total unknown for vector C is M 2. These can be obtained by using (3.40) and (3.4). 3.2 Homotopy Perturbation Method (HPM) The Homotopy perturbation method was introduced and developed by Jihvan He [6]. The Homotopy perturbation method couples a Homotopy technique of topology and a perturbation technique.

46 36 Consider the following Fredholm integro-differential equation of the second kind of the form v ( x ) f ( x ) G ( x, y ) 0 v( y ). dy (3.50) We define the operator L( u) u( x ) G ( x, y ) 0 u( y) dy f ( x ) 0 (3.5) where u( x ) v ( x ). Next we define the Homotopy H ( u, m ), m [0,] by H ( u,0) F( u), H ( u,) L( u), (3.52) where Fu ( ) is a functional operator. We construct a convex Homotopy of the form H ( u, m) ( m) F( u) ml( u). (3.53) This Homotopy satisfies (3.52) for m 0 and m respectively. The embedding parameter m monotonously increases from zero to unity as the trivial problem Fu ( ) 0 is continuously deformed to the original problem Lu ( ) 0. HPM uses the Homotopy parameter m as an expanding parameter to obtain u w mw m w m w (3.54)

47 when m, 37 (3.54) corresponds to (3.53) and becomes the approximate solution of (3.5), i.e., v lim u w w w w. m )3.55) The series (3.55) is convergent for most cases and the rate of convergence deps on Lu ( ). Assume F ( u) u( x ) f ( x ), and substituting (3.54) in (3.5) and equating the terms with identical power of m, we have 0 m : w ( x ) f ( x ) (3.56) 0 n (3.57) 0 n m : w ( x ) G ( x, y ) w ( y ) dy, n,2,... n Notice that the recurrence relation (3.57) is the same standard Adomian decomposition method as presented before in chapter two. 3.3 Legre Polynomial Method Orthogonal polynomials play a very important role in applications of mathematics, mathematical physics, engineering and computer science. One of the most common set of the Legre polynomials { P ( x ), P ( x ),, P ( x )} which are orthogonal on [,] and satisfy the 0 Legre differential equation N ( ) ( ) 2 ( ) ( ) ( ) 0, 2 x y x xy x n n y x x, n 0 and are given by the form

48 38 n 2 k n 2n 2k n 2k Pn ( x ) ( ) x n 2 k n n 0,, (3.58.a ) k 0 Moreover, recurrence formulas associated with derivates of Legre polynomials are given by the relation P ( x ) P ( x ) (2n ) P ( x ), n n n n. (3.58.b) Consider the following th m order linear Fredholm Integro-differential equation with variable coefficients m ( k ) k k 0 F ( x ) y ( x ) g ( x ) K ( x, t ) y ( t ) dt, x, t (3.59) with conditions m ( k ) ( k ) ( k ) a jk y ( ) b jk y () c jk y (0) j, j 0,,2,..., m k 0 (3.60) where the constants a,, jk b jk c jk, and j are stable constants. Our aim is to obtain a solution expressed in the form: N y ( x ) a P ( x ), x, (3. 6) n0 n n so that the Legre coefficients to be determined are the an ' s where n 0,,2,... N and the functions P ( x )( n 0,,2,, N ) are the Legre n polynomials defined by the formula (3.58.a). Here Fk ( x ), g( x ) and K ( x, t ) are functions defined in the interval x, t.

49 39 Fundamental matrix relations Equation (3.59) can be written as D ( x ) g ( x ) I ( x ) (3.62) f where m ( k ) D ( x ) Fk ( x ) y ( x ) k 0 and I ( x ) K ( x, t ) y ( t ) dt. f We convert the solution y( x ) and its derivative ( k y ) ( x ), the parts D( x ) and I ( x ), and the mixed conditions in (3.60) to matrix form. f Matrix relations for y( x ) and ( k y ) ( x ) The function y( x ) defined by (3.6) can be expanded to the truncated Legre series in the form N y ( x ) a P ( x ), x, (3.63) n0 n n formula (3.63) and its derivative can be written in the matrix forms, [ y ( x )] P( x )A and ( ) ( ) [ k k y ( x )] P ( x )A, (3.64) where P( x ) [ P ( x ) P ( x ). P ( x )] 0 N

50 40 ( k ) ( k ) P ( x ) [ P ( x ) ( k ) P ( x). ( k P ) ( x )] N and A [a 0 a. a N ], which A is the coefficient matrix. On the other hand, by using the Legre recursive formulas (3.58.a) and (3.58.b) and taking n 0,,..., N we can obtain the matrix equation () T P ( x) P( x), (3.65) where, for odd values of N for even values of N Also, it is clearly seen that the relation between the matrix Px ( ) and its derivative ( k P ) ( x ), from (3.65), is

51 () P ( x ) P( x ) 4 (2) () T T 2 P ( x ) P ( x ) P ( x )( ) T ( k ) (k) T k T k P ( x ) P ( x )( ) P ( x )( ), k 0,,2,... (3.66) Consequently, by substituting the matrix relations (3.66) into Eq. (3.64), we obtain the matrix relations for y( x) and ( k P ) ( x) as ( k ) T k [ y ( x )] P( x )( ) A. (3.67) Matrix representations based on collocation points The Legre collocation points defined by xi 2 i, i 0,,..., N, (3.68) N we substitute the collocation points (3.68) into Eq. (3.62) to obtain the system D ( x ) g ( x ) I ( x ) i i f i Represented in matrix equation as D G I f (3.69) where D ( x 0) g ( x 0) I f ( x 0) D ( x ) g ( x ) I f ( x ) D,G,I : : : D ( x N ) g ( x N ) I f ( x N )

52 42 Matrix relation for the differential part D( x) To reduce the D( x) part to the matrix form by means of the collocation points, we first write the matrix D defined in eq.(3.69) as m k ( k ) D F Y (3.70) k where Fk ( x 0) Fk ( x) 0 F k, 0 0 Fk( x N ) Y y ( k ) ( k ) ( k ) y x y ( k ) ( x 0) ( ). ( x N ) Using the collocation points x ( i 0,, N, in ) (3.67) we have the system of matrix equations i [ y ( x )] P( x )( ) A, k 0,,..., m ( k ) T k i i or Y ( k ) y ( x 0) Px ( 0) ( k ) ( k ) y ( x ( ) Px) T k T k [( ) A] P( ) A, : : ( k ) y ( x ) Px ( N ) N (3.7) where

53 43 P0 ( x 0) P ( x 0)... PN ( x 0) P0 ( x ) P ( x )... PN ( x ) P : : : P0( x N ) P( x N )... PN ( x N ) Thus, from the matrix forms (3.70) and (3.7), we get the fundamental matrix relation for the differential part m k 0 D( x) T k D Fk P( ) A. (3.72) Matrix relation for the Fredholm integral part I f ( x) The kernel function K ( x, t ) can be approximated by the truncated Legre series N N K ( x, t ) k P ( x ) P ( t ) (3.73) m0n0 t mn m n or N N K ( x, t ) k x t (3.74) m0n0 t m n mn where k mn t mn m n K (0,0) ; mn!! x t m, n 0,,..., N. We convert (3.73) and (3.74) to matrix forms and then equalize T T [ K ( x, t )] P( x )K P ( t ) X( x )K X ( t ) (3.75) l t

54 44 where P( x ) [ P ( x ) 0 P ( x) PN ( x)] N X( x ) [ x x ] l mn K [ k ], K [ k ]; l t t mn m, n 0,,..., N. On the other hand, by using the Legre recursive formula (3.58.a) for n 0,,..., N we can obtain the matrix equation T T T P ( x ) DX ( x ) or P( x ) X( x ) D (3.76) where for odd values of N and for even values of N see [42]. Therefore, substituting the matrix Px ( ) in (3.76) into the relation (3.75) and simplifying the result equation, we find the matrix relation T td K ( D ) K (3.77) l which is the relation between the Legre and Taylor coefficients of K ( x, t ). Substituting the matrix forms (3.75) and (3.64) corresponding to the functions K ( x, t ) and y() t into the Fredholm integral part I ( x ), we obtain f T f l l [ I ( x )] P( x )K P ( t )P( t )Adt P( x )K QA, (3.78) where T Q P ( t )P( t ) dt [ q mn ];

55 45, =0,,..., and the matrix K l is defined in (3.77). Using the collocation points x ( i 0,,, N ) defined in (3.68) in the relation (3.78) we obtain the system of the matrix equations I ( x ) P( x ) K QA; i 0,,..., N f i i l l i or briefly I PK QA; (3.79) f l which is the fundamental matrix relation for If ( x). Matrix relation for the mixed conditions We can obtain the corresponding matrix form for the conditions (3.60), by means of the relation (3.66), as m T k [ a jk P ( ) b jk P() c jk P(0)]( ) A j, j 0,,, m. (3.80) k 0 Method of solution Now, we are ready to construct the fundamental matrix equation corresponding to (3.59). For this purpose, substituting the matrix relations (3.72) and (3.79) into Eq. (3.69) then simplifying it, we obtain the matrix equation m T k {FkP( ) PKlQ}A G, ( 3.8) k 0

56 which corresponds to a system of ( N ) 46 unknown Legre coefficients Eq. (3.8) in the form ( N ) a 0,, algebraic equations for the a. a N Briefly, we can write WA= G or [W ; G] (3.82) where m T k pq k l k 0 W [ w ] F P( ) PK Q, p, q 0,,..., N G [ g( x ) 0 g( x) N T g( x )]. On the other hand, the matrix form (3.80) for the conditions (3.60) can be written as where U A [ ] or [U ; ], j 0,,..., m (3.83) j j m j jk jk jk k 0 j j T U [ a P ( ) b P () c P (0)]( ) k [ u j 0 u j. u jn ]. To obtain the solution of Eq. (3.82) under the conditions (3.83), by replacing the rows matrices (3.83) by the last m rows of the matrix (3.8) we have the new augmented matrix

57 [W;G] 47 w 00 w 0 w 0N ; g ( x 0) w 0 w w N ; g ( x ) ; w N m,0 w N m, w N m, N ; g ( x N m ) u00 u0 u0n ; 0 u0 u u N ; ; u m,0 u m, u m, N ; m (3.84) If rank W rank[w; G] N, then we can write A (W) G. Thus the coefficients a ( n 0,,..., N ) are uniquely determined by Eq. (3.84). 3.4 Taylor Collocation Method n Consider the mth -order linear Fredholm integro-differential equation m b ( k ) Pk ( x ) y ( x ) f ( x ) K ( x, t ) y ( t ) dt (3.85) k 0 a where the unknown functions Pk ( x ), f ( x ), K ( x, t ) are defined on a x, t b and λ is a real parameter where y( x) is the unknown function. With conditions m ( j ) ( j ) ( j ) [ aij y ( a) bij y ( b) cij y ( c)] i, i 0,,..., m j 0 )3.86)

58 48 where a c b, provided that the real coefficients a ij, b ij, c ij and i are appropriate constants. We assume that the solution of (3.85) be the truncated Taylor series N ( n ) y () c n y ( x ) ( x c) ; a x b (3.87) n! n 0 where N is chosen any positive integer such that N m. Then the solution (3.87) of Eq. (3.85) can be expressed in the matrix form [ y( x)] X M0 A where X [ x c ( x c) 2 N...( x c) ] (0) A [ y ( c) y () ( c) y (2) ( c) ( N )... y ( c)] and M= ! 0 0 0! ! N!. To obtain a solution, we can use the following matrix method, which is a Taylor Collocation method. This method is based on computing the Taylor

59 49 coefficients by means of the Taylor collocation points are thereby finding the matrix A containing the unknown Taylor coefficients. We substitute the Taylor collocation points b a x i a i ; i 0,,..., N; N x a 0, x N b (3.88) into (3.85) to obtain m ( k ) Pk ( x i ) y ( x i ) f ( x i ) I ( x i ) (3.89) k 0 such that b I ( x i ) K ( x, t ) y ( t ) dt a then we can write the system (3.8) in the matrix form m (0) () ( m) ( k ) 0 m k k 0 P Y P Y... P Y P Y F I )3.90) where P k = Pk ( x0) 0 0 Pk( x N )

60 f ( x 0) f ( x) F, f ( x N ) ( k ) y ( x 0) ( k ) ( k ) y ( x) Y, ( k ) y ( x N ) 50 I( x 0) I( x) I. I( x N ) Assume the k th derivative of the function (3.87) with respect to x has the truncated Taylor series expansion defined by (3.87): N ( n ) ( k ) y () c i nk y ( x ) ( x i c) ; ( n k)! nk a x b where ( k y ) ( x) ( k 0,, N ) are Taylor coefficients; y (0) ( x ) y ( x ). Then substituting the Taylor collocation points, we obtain ( k ) [ y ( x )] X M A, k 0,,, N (3.9) i x k i or the matrix equation ( k ) y CMk A (3.92) where X X C X x x x 0 N 0 ( x 0 c) ( x 0 c) ( x 0 c) 0 ( x c) ( x c) ( x 0 c) 0 ( x N c) ( x N c) ( x N c) N N N

61 ! ! M k ( N k)! Then Eq. (3.88) can be written as m ( P CM ) A F I k 0 k k )3.93) The kernel K ( x, t ) is expanded to construct Taylor series N N K ( x, t ) k ( x c) ( t c) n0m0 nm n m k nm K (0, 0) n! m! x t nm n m ( x c, t c ). The matrix representation of K ( x, t ) defined by [ K ( x, t )] X K T T )3.94) where X [ x c ( x c) 2 N...( x c) ] T [ t c ( t c) 2... N ( t c) ]

62 K 52 k k k k k k k k k N 0 N N 0 N NN In addition, the matrix representation of y( x) and y() t are [ y( x)] X M 0 A, [ yt ( )] T M 0 A. (3.95) Setting (3.94) and (3.95) into ( ) I x i defined in (3.89), we have b T [ I ( x )] {XKT TM A} dt XKHM A a 0 0 (3.96) T H [h nm ] T T dt, b a h nm ( b c) ( a c) n m n m n m n, m 0,,..., N from (3.96) we get the matrix I in the form I=CK H M 0 A. )3.97) Finally, substituting (3.97) in (3.93), we get the matrix equation m ( P CM - CKHM )A F k 0 k k 0 )3.98) which is the fundamental relation for solving of Fredholm integrodifferential equation defined in a x b. Equation (3.98) can be written as W A F )3.99)

63 which corresponds to a system of unknown Taylor coefficients 53 ( N ) algebraic equations with the m ij k k k 0 W [ w ] P CM - CKHM, 0 i, j 0,,, N. Also, for the conditions in (3.86). We assume y y ( j ) ( j ) ( a) PM j A ( b) QM j A ( j ) y ( c) RM j A (3.00) where P [ ( ac) ( ac) 2 N...( ac) ] Q [ ( b c) ( b c) 2 N...( b c) ] R =[ 0 0 0]. Setting (3.00) in (3.86), we get m j 0 { a P b Q c R}M A [ ] ij ij ij j i Let m i aij bij cij j ui 0 i j 0 U { P Q R}M [ u. u ], i 0,,..., m. (3.0) in Thus, the conditions (3.86) becomes

64 54 U A [ ] (3.02) i i also the augmented matrices representing them are [U ; ] [ u i i i 0 u i u in ; i ]. (3.03) Finally, the augmented matrix W;F is defined by the matrices: w w w w w w W w w w u u u u u u N 0 N N m,0 N m, N m, N N m,0 m, m, N f ( x 0) f ( x) F f ( x N m). 0 m If, W 0 we can write A W F Matrix Ais uniquely determined. Then (3.85) has a unique solution in the form (3.87).

65 55 Chapter Four Numerical Examples and Results To test the efficiency and accuracy of the numerical methods represented in chapter three we will consider the following numerical examples: Example 4. Consider the Fredholm integro-differential equation of the second kind 3x u ( x ) 3 e (2e ) x 3xt ut () dt with 3x Equation (4.) has the exact solution u ( x ) e. u (0). (4.) 4. The numerical realization of equation (4.) using the B-spline scaling functions and wavelets on [0,] The following algorithm implements the B-spline scaling functions and wavelets on [0,] using the Maple software Algorithm 4. [25]. Input the fixed positive integer M 2. Input the values of the matrix Q M 3. Input the values of the matrix R 4. Input the values of the matrix ( j ) 5. Input the values of the matrix ( j )

66 56 6. Calculate the matrix G, by using (3.24) and (3.38). 7. Calculate the matrix D 8. For i from 2 to M 2 2 2, X () 0, X M (2 ) M 2 X ( i ) (2i ) / (2 2) 9. Defined, ( x ),, ( ), i j i x i ( x ),2 i 0. Defined, ( x ),, ( x ),,2 i ( x ) i j i i. Input h ( b a), t ( i ) i * h M 2. For i from to 5, for j from to M 2, for k from 2 to M, for I K from to 2, define mut ( i, j ) ( mu(2, i 2, t ( j )), and mut ( i, j ) ( mu ( k, I 2,( j )) to calculate mutt mut, mut 3. Define the kernel 4. For i from to 5, for j from to M 2, for k from 2 to M, for I K from to 2, define mux ( i, j ) ( chi (2, i 2, X ( j )), and mux ( i, j ) ( mu ( K, I 2,X( j )) 5. Calculate muxx mux, mux 6. Calculate zz z integration 7. F (,), and calculate F(, i ) 8. Calculate C F * Matrix Inverse zz 9. The solution is U C * muxx

67 Input the exact solution u Transpose ( U ) u convert ( u, vector ) r : plot ( X, u ) r : plot ( u ( x )) Display ( rr, 2) 22. Absolute error = u u M Table 4. shows the exact and numerical solutions when applying algorithm 4. on equation (4.), and showing the resulting error of using the numerical solution. Table 4.: The exact and numerical solutions of applying Algorithm 4. on. equation (4.). x Exact solution u( x ) exp(3 x ) Numerical solution u M Absolute error u u = M

68 58 These results show the accuracy of the B-spline scaling functions and wavelets to solve equation (4.) with the max error = Figure 4. compares the exact solution u ( x ) 3x e with the approximate solution with M 4. Figure 4.: The exact and numerical solutions of applying algorithm 4. for equation (4.). Figure 4.2 shows the absolute error resulting of applying algorithm 4. on equation (4.). Figure 4.2: The error resulting of applying algorithm 4. on equation (4.).

69 The numerical realization of equation (4.) using the Homotopy perturbation method The following algorithm implements the Homotopy perturbation method using the Maple software Algorithm 4.2 () Input a,, n, G x y f ( x ), (, ) (2) Calculate wm[0] int (( f, x )) (3) Calculate wm[0]: y sub ( x y, wm[0]) (4) Calculate c a eval ( wm[0](0)) (5) Define ws[0] wm[0] c (6) Calculate w [0] subs ( x y, ws[0]) n, calculate (7) For i to wm[ i ] ( int ( G * w [ i ], y 0...)) ws[ i ] ( int ( wm[ i ],x )) w [ i ] subs ( x = y,ws[ i ]) (9) Define u n = combine( add ( ws[ k ],k 0..), (0) Define u : x exp( x ) r : plot ( u ( x )) r 2 n : plot ( u ( x )) () Absolute Error = u un For more details see [2, 43].

70 60 Table (4.2) shows the exact and numerical results when n 4, and showing the error resulting of using the numerical solution. Table (4.2): The exact and numerical solution of applying Algorithm 4.2 on equation (4.). x Exact solution u ( x ) e 3x Numerical solution u n ( x) Absolute Error = u u n These results show the accuracy of the Homotopy perturbation method to solve equation (4.) with the max error = Figure 4.3: shows both exact and numerical solutions with n 4. Figure 4.3: The exact and numerical solutions of applying algorithm 4.2