Homotopy Analysis Method for Solving Fuzzy Integro-Differential Equations

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1 Modern Applied Science; Vol. 7 No. 3; 23 ISSN E-ISSN Published by Canadian Center of Science Education Hootopy Analysis Method for Solving Fuzzy Integro-Differential Equations Ean A. Hussain & Ayad W. Ali Departent of Matheatics College of Science AL-Mustansiriyah University Baghdad Iraq Correspondence: Ean A. Hussain Departent of Matheatics College of Science AL-Mustansiriyah University Baghdad Iraq. E-ail: dr_eansultan@yahoo.co Received: January 2 23 Accepted: February 23 Online Published: February doi:.39/as.v7n3p URL: Abstract In this paper the nuerical solution of the linear nonlinear fuzzy Volterraintegro-differential equations have been investigated using the analytical ethod naely hootopy analysis ethod (HAM) then the proposed ethod is illustrated by solving two nuerical eaples. It was found that the HAM provides a siple way to adjust control the convergence region of solution series by introducing a nonzero auiliary paraeter. Keywords: fuzzy nubers fuzzy volterraintegro-differential equations hootopy analysis ethod. Introduction Integro-differential equations (IDEs) play an iportant role in any branches of linear nonlinear functional analysis their applications in the theory of engineering echanics physics cheistry astronoy biology econoics potential theory electrostatics. The fuzzy functions were introduced by Chang Zadeh (972). Later the concept of integration of fuzzy functions was introduced by Dobois Prade (982). In 992 Liao eployed the basic ideas of the hootopy a fundaental concept in topology differential geoetry to propose a general analytic ethod for linear nonlinear probles called Hootopy Analysis Method (HAM) (Liao 23; 24; 29 a & b). The hootopy analysis ethod (HAM) is a general analytic approach to get series solutions of various types of nonlinear equations including algebraic equations ordinary differential equations partial differential equations differential-integral equations differential-difference equation coupled equations of the. This ethod has been successfully applied to solve any types of linear nonlinear probles (Afroozi Vahidi & Saeidy 2; Abbasby Magyari & Shivanian 29; Abbasby Babolian & Ashtiani 29; Bataineh Noorani & Hashi 28; Dubois & Prade 982; Ghanbari 2; Liao & Tan 27). 2. Preliinaries In this section the ost basic notations used in fuzzy calculus are introduced. We start with defining a fuzzy nuber. Definition 2. (Goetschel & Voan 983) A fuzzy nuber is a ap u: R I [] which satisfies i. u is upper sei-continuous ii. u() outside soe interval [cd] iii. There eist real nubers a b such that ca b d where ) u() is onotonic increasing on [ca] 2) u() is onotonic decreasing on [bd] 3) u() a b. An equivalent paraetric definition of fuzzy nubers is given by Friedan Ma Kel (999) as follows: Definition 2.2 An arbitrary fuzzy nuber in paraetric for is represented by an ordered pair of functions (u( )u( )) which satisfying the following requireents: i. u( ) is a bounded left-continuous non-decreasing function over [] ii. u( ) is a bounded left-continuous non-increasing function over []

2 Modern Applied Science Vol. 7 No. 3; 23 iii. u( ) u( ). For arbitrary fuzzy nubers u (u( )u( )) v (v( ) v( )) real constant we define addition u v scalar ultiplication ku as (uv)( ) u( ) v( ) (uv)( ) u( ) v( ) () (ku)( ) ku( )(ku)( ) ku( )ifk (ku)( ) ku( )(ku)( ) ku( )ifk The collection of all such fuzzy nubers with addition ultiplication as defined by Equations () (2) is denoted by E is a conve cone. Net we will define the fuzzy function notation a etric D in E (Goetschel & Voan 986). Definition 2.3 For arbitrary fuzzy nubers u (u( )u( )) v (v( ) v( )) the quantity D(u v) a sup u( ) v( ) sup u( ) v( ) is the distance between u v. This etric is equivalent to the one used by Puri Ralescu (983) Kaleva (987). It is shown (Puri & Ralescu 986) that (E D) is a coplete etric space. Definition 2.4 A fuzzy function f:r E is said to be continuous if for arbitrary fied R there eists such that if then D(f ()f ( )) (3) Throughout this work we also consider fuzzy functions which are defined only over a finite interval [ab] (we siply replace R by [ab] in definition 2.4). Definition 2. (Mehrkanoon Suleian & Majid 29) The Seikkala derivative f() of a fuzzy function f is defined by [f ()] [f (; )f (; )] (] where prie sybol denote the derivative with respect to. Following the idea of Bede Gal in 24 2 Chalco-Cano Roan-Flores (28) define the fuzzy lateral H-derivative for a fuzzy function f:i E as follows: Definition 2.6 (Mehrkanoon Suleian &Majid 29) Let f:i E be a fuzzy function I R then f is differentiable at if (I) there eist an eleent f() E such that for all h sufficiently sall there are f( h) f( ) ; f( ) f( h) or (II) there eist an eleent f( ) f( h) f( h) f( ) f( ) f( h) (4) h h li li f ( ) h h E such that for all h f() (2) sufficiently sall there are f( h) f( ) ; f( h) f( ) f( ) f( h) li li f ( ) () h h h h where the relation (I) is the classical definition of the fuzzy H-derivative (or differentiability in the sense of Hukuhara). 3. HAM for Solving Fuzzy Volterra Integro-Differential Equations (FVIDEs) In this section we shall apply HAM for solving linear nonlinear fuzzy Volterra integro-differential equations (FVIDEs) F () f() K(t)F(t)dt (6) 6

3 Modern Applied Science Vol. 7 No. 3; 23 F () f() K(tF(t))dt (7) d under the initial condition F F() (F( ) F( )) () where F () F() f:[b] E is d continuous fuzzy function K is arbitrary continuous function over the regions {( t) t b} {( t F()) t b F() E } respectively F is to be deterined. In the following we shall follow the sae ethod that proposed by Bede (28) Chalco-Cano Roan-Flores (28) in the fuzzy differential equations to reduce FVIDEs to a crisp systes of integro-differential equations (IDEs) using relations (4) () in definition (2.6). Let us recall the proposed ethod. We denote the level set of F f K by [F()] [F(; )F(; )] [f()] [f(; )f(; )] [K(tF(t)] [K(tF(t ))K(tF(t ))] respectively. Then we have the following two cases: Case I If we consider F() in the first for (4) of definition (2.6) then we have to solve the following systes of crisp IDEs for the linear case for the nonlinear case. F(; ) f(; ) K(t)F(t; )dt F(; ) f(; ) K(t)F(t; )dt F(; ) f(; ) K(tF(t; ))dt F(; ) f(; ) K(tF(t; ))dt F(; ) F(; ) (8) F(; ) F(; ) (9) Case II If we consider F() in the second for () of definition (2.6) then we have to solve the following systes of crisp IDEs for the linear case F(; ) f(; ) K(t)F(t; )dt F(; ) f(; ) K(t)F(t; )dt F(; ) f(; ) K(tF(t; ))dt F(; ) f(; ) K(tF(t; ))dt F(; ) F(; ) () F(; ) F(; ) () for the nonlinear case. In our work we will consider case I hence we have to solve the systes (8) (9). For siplicity without lose of generality of the proble we assue that the kernels K(t) on K(t ()) K(t ()) on. 3. Linear Fuzzy VolterraIntegro-Differential Equations For solving syste (8) by HAM we first construct the zeroth-order deforation equations: 7

4 Modern Applied Science Vol. 7 No. 3; 23 ( p)l[ (p; ) w (; )] ph()[ (p; ) f (; ) K(t) (tp; )dt] ( p)l[ (p; ) w (; )] p H()[F (p; ) f (; ) K(t)F(tp; )dt] where p [] is the ebedding paraeter is nonzero auiliary paraeter L is an auiliary linear operator H() is an auiliary function w(; ) w(; ) are initial guesses of F(; ) F(; ) respectively (p; ) (p; ) are unknown functions. Differentiating the zeroth-order deforation Equations (2) ties with respect to the ebedding paraeter p dividing the by! finally setting p we obtain the so called th order deforation equations L[u (; ) u (; )] H()[u (; ) ( )f(; ) K(t)u (t; )dt] u (; ) L[u (; ) u (; )] H()[u (; ) ( )f(; ) K(t)u u (; ) (t; )dt] Choose the auiliary linear operator as L the initial guesses w u w u the auiliary function H(). By taking L on both sides of equations (3) (4) then we obtain the following iteration fors u(; ) f(s )ds u(; ) f(s )ds s s u (; ) ( )u (; ) K(st)u (t; )dtds u (; ) ( )u (; ) K(st)u (t; )dtds Hence the hootopy solution series is given by (2) (3) (4) () 2 (6) therefore the approiation solution series of order n is F(; ) u (; ) F(; ) u (; ) n F(; ) u (; ) n n F(; n ) u (; ) 3.2 Nonlinear Fuzzy Volterra Integro-Differential Equations For solving syste (9) by HAM we first construct the zeroth-order deforation equations where ( p)l[ (p; ) w (; )] ph()n[ (p; )] ( p)l[ (p; ) w (; )] p H()N[ (p; )] (7) (8) (9) 8

5 Modern Applied Science Vol. 7 No. 3; 23 N[ ( p; )] ( p; ) f (; ) K( t (t p; ))dt (2) N[ ( p; )] ( p; ) f (; ) K( t (t p; ))dt (2) The corresponding -th order deforation equations reads: L[u (; ) u (; )] H()R [u (; )] (22) u (; ) where where L[u (; ) u (; )] H()R [u (; )] (23) u (; ) R [u (; )] u (; ) ( )f (; ) K ( t; )dt R [u (; )] u (; ) ( )f (; ) K ( t; )dt K (t; ) K(t (tp; )) ( )! p K (t; ) K(t (tp; )) ( )! p p p (24) (2) Choose the auiliary linear operator as L the auiliary function H(). By taking L on both sides of equations (22) (23) we obtain the following iteration fors s u (; ) u (; ) f(s )ds K (st; )dtds s u (; ) u (; ) f(s )ds K (st; )dtds s u (; ) ( )u (; ) K (st; )dtds s u (; ) ( )u (; ) K (st; )dtds where K(st; ) K(stu(t; )) K(st; ) K(stu(t; )) Hence the hootopy solution series is given by (26) (27) 2 (28) therefore the approiation solution series of order n is F(; ) u (; ) u (; ) F(; ) u (; ) u (; ) F (; ) u (; ) u (; ) n n F n (; ) u (; ) u (; ) 4. Nuerical Eaples In this section we eaine our ethod by giving two nuerical eaples with known eact solutions. n (29) (3) 9

6 Modern Applied Science Vol. 7 No. 3; 23 Eaple 4. Consider the linear FVIDE with The eact solution in this case is given by Fro Equations () (6) we obtain K(t) t b 4 f(; ) ( ) f(; ) 2 (2 ) F(; ) F(; ) (2 ) u(; ) ( ) u(; ) ( ) ( ) ( ) u(; ) ( ){( ) ( ) ( )} ( ( ( ) ) ( ) ) 9 94 u(; ) ( 3 3 ) ( )( ) ( ) u(; ) ( ) ( ) ( ) ( ) ( ) u(; ) {( ) 2} u 2 (; ) ( ) {( ) 2 } { ( ) (2 ) } u(; 3 ) ( ) ( ) ( 4 ) (2 ) u(; 4 ) ( ) ( 8/ ) ( ) ( ) u(; ) ( ) ( ) ( ) ( 4/2727 ) ( ) ( ) Then we approiate F(; ) with n by 2

7 Modern Applied Science Vol. 7 No. 3; 23 F (; ) ( ) ( 2 ) ( ) ( ) ( ) with n by F(; ) ( ) ( 4 2 ) ( ) ( ) ( ) ( ) F(; ) We deterine the valid region of such that the hootopy solution series converges we plot the curve of approiate solution F(..) F(..) which is obtained by HAM as shown in Figure. Fro this figure we can find that the valid region of is the interval [..3]. The error between eact approiate solution of order obtained by HAM is shown in Table. Figure. Plot of the curve of F(..) F(..) dashed dotted line represents F(.;.) solid line represents F(.;.) Table. The error between eact approiate solution for different values of e e-.446e e- 2.e-3.6e e-3.279e-4 3.6e e-4 4.e-3 3.e e e-4.26e-2.948e-4 6.2e e e e e e-4 8.3e e-2. 9.e e e e-4.6e e-2.6.e e-4 3.9e- 4.4e-4.3e-2 9.6e-2.7.4e e e-4.982e-4.6e

8 Modern Applied Science Vol. 7 No. 3; e e-4.346e e-4.98e e e-.96e-.e e e e-4.446e-.6e-3 3.2e-2.98 Eaple 4.2 Consider the nonlinear FVIDE with 2 K(tF(t; )) F (t; ) b 2 K(tF(t; )) F (t; ) f(; ) ( 2 ) 2( ) The eact solution in this case is given by f (; ) ( ) 2(4 ) 2 2 F(; ) ( ) F(; ) (4 ) 3 2 Choose u(; ) u(; ) fro equation (28) we obtain s i i i s i i i u (; ) ( )u (; ) u (t; )u (t; )dtds u (; ) ( )u (; ) u (t; )u (t; )dtds 2 Fro this (27) we can obtain the first three ters of approiate hootopy solution series as u(; ) (2 ) ( ) u (; ) ( ) (2 ) ( ) ( ) u(; ) ( ) ( ) ( ) ( 2 2 ) u (; ) ( ) (32 3 ) u(; 2 ) ( )( ) ( )( 4 ) u(; 3 ) ( ) ( ) ( ) ( ) Then the approiate solutions F(; 3 ) F(; 3 ) are given by. 22

9 Modern Applied Science Vol. 7 No. 3; 23 F(; ) ( ) ( ) ( F(; 3 ) ( ) ( ) ) ( ) ( ) ( ) respectively. To deterine the valid region of the auiliary paraeter we plot the curves F 3(.;.) F 3(.;.) as shown in Figure 2. We can find that the region of is the closed interval [.4.6]. The error between eact approiate solution is given in Table 2. Figure 2. The curve of F 3 (.;.) F 3(.;.) dashed dotted line represents F 3(.;.) solid line represents F 3(.;.) 23

10 Modern Applied Science Vol. 7 No. 3; 23 Table 2. The error between eact approiate solutions e e e e e e-4.e e-3.3e-3.726e e e-4.3e-3 4.3e-3.3.e e e e e-4 3.e-3.e-2.4.9e e-3.4e e-6.2e-3 6.2e-3.87e e-26.39e-2 3.7e e- 3.3e-3.2e e e e-2 9.3e e-4 8.4e e-2.47e e-2 2.9e-2.3e-3.99e-2 4.6e e e-2 e-3 4.4e-2 9.2e e-2.62e-2 9.6e e It is worthy to point out that there are any applied probles that are odeled as integro-differential equations can be solved by the current ethod such as; steady state condition of biological species living together Mawell equation in integro-differential for chance to find tie gap in dense traffic etc.. Conclusions In this paper we solved successfully the fuzzy Volterra integro-differential equations using Hootopy Analysis Method (HAM). It is apparently seen that HAM is very powerful efficient technique in finding approiate solution for such equations. This ethod enjoys great freedo in choosing initial approiations auiliary linear operators auiliary functions. By eans of this kind of freedo a coplicated proble can be transferred into an infinite nuber of sipler linear subprobles that contain the auiliary paraeter which provides a convenient way to adjust control the convergence of the solution series this represents the ain advantage of this ethod. References Abbasby S. Babolian E. & Ashtiani M. (29). Nuerical solution of the generalized Zakharov equation by hootopy analysis ethod. Coune Nonlinear Sci. Nu. Siulat. 4(2) Abbasby S. Magyari E. & Shivanian E. (29). The hootopy analysis ethod for ultiple solutions of nonlinear boundary value probles. Coun Nonlinear Sci. Nu. Siulat. 4(9-) Afroozi G. Vahidi A. J. & Saeidy M. (2). Solving a class of two-diensional linear nonlinear integral equations by eans of the Hootopy Analysis Method. International J. of nonlinear sciences 9(2) Bataineh A. S. Noorani M. S. M. & Hashi I. (28). Approiate analytical solutions of systes of PDEs by hootopy analysis ethod. Cop. Math. With Appls. (2) Bede B. & Gal S. G. (24).Alost periodic fuzzy-nuber-valued functions. Fuzzy Set Syst. 47(3) Bede B. & Gal S. G. (2). Generalizations of the differentiability of fuzzy nuber valued functions with applications to fuzzy differential equations. Fuzzy Set Syst. (3) Bede B. (28). Note on Nuerical solutions of fuzzy differential equations by predictor-corrector ethod. Inforation Sciences 78(7) Chalco-Cano Y. & Roan-Flores H. (28). On new solutions of fuzzy differential equations. Chaos Solitons Fractals 38(2)

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