Iterative Algorithm to Solve a System of Nonlinear Volterra-Fredholm Integral Equations Using Orthogonal Polynomials

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1 Issue 5 volume 2, March.-April.215 Ieraive Algorihm o Solve a Sysem of Nonlinear Volerra-Fredholm Inegral Equaions Using Orhogonal Polynomials Dr. Sharefa Eisa Ali Alhazmi Umm Al-Qura Universiy College of Educaion for Girls a Al-Qunfudah Mahemaics deparmen, Macca, KSA sehazmi@uqu.edu.sa Absrac This paper focuses on numerical mehod used o solve a linear and nonlinear sysem of Volerra or Fredholm inegral equaions using Legendre and Chebyshev collocaion mehod. The mehod is based on replacemen of he unknown funcion by a reduced series o lengh size by discarding he high degree erms of he Legendre or Chebyshev polynomials. This lead o a sysem of linear or nonlinear algebraic equaions wih Legendre or Chebyshev coefficiens which will be solved using conjugae gradien or Newon ieraion mehod respecively. Thus, by solving he marix equaion, Legendre or Chebyshev coefficiens are obained. Some numerical examples are included o demonsrae he validiy and applicabiliy of he proposed echnique. Keywords: Sysem Volerra-Fredholm Inegral equaion, Legendre and Chebyshev polynomials, gradien mehod, Newon Mehod 1 Inroducion Volerra and Fredholm inegral equaions arise in many problems peraining o mahemaical physics like hea conducion problems. They plays an imporan role in many branches of linear and nonlinear analysis and heir applicaions in he heory of engineering, mechanics, physics, chemisry, asronomy, biology [3]-[7]. In [12]-[16], he auhors presened several mehod for solving linear and nonlinear Fredholm Inegral equaions. In [2, 1], chnii consider only he soluion of a linear Fredholm or Volerra-Fredholm inegral equaion wih singuler Kernel. There is no informaion wha happen for a sysem of nonlinear Volerra-Fredholm Inegral equaion wih singular kernel. To give a correc answer o his quesion, we will inroduce a echnique ha can be generalized o he case of singuler Kernel. Here, we will consider he case of a nonlinear Fredholm inegral equaions of he ype b Φ(x) + a K(x, s)f[s, Φ(s)]ds =, a x b, where K(x, s) and F(x, s) are known funcions, while Φ(x) is he unknown funcion. The funcion K(x, s) is known as he kernel of he Fredholm inegral equaion as play an imporan R S. Publicaion, rspublicaionhouse@gmail.com Page 563

2 Issue 5 volume 2, March.-April.215 role for solving he problem and i s can be smooh or singular funcion. A. Hammersein, who considered he case where K(x, s) is a symmeric and posiive Fredholm kernel, i.e. all is eigenvalues are posiive. If, in addiion, he funcion F(x, s) is coninuous and saisfies he condiion F(x, s) β 1 s + β 2, where β 1 and β 2 are posiive consans and β 1 is smaller han he firs eigenvalue of he kernel K(x, s), he Hammersein equaion has a leas one coninuous soluion. If, on he oher hand, F(x, s) happens o be a nondecreasing funcion of s for any fixed x from he inerval (a, b), Hammersein s equaion canno have more han one soluion. This propery holds also if F(x, s) saisfies he condiion F(x, s 1 ) F(x, s 2 ) β s 1 s 2, where he posiive consan β is smaller han he firs eigenvalue of he kernel K(x, s). Le us consider he following inegral equaion: Φ() = g() + η 1 K 1 (, s)f (Φ(s))ds + η 2 K 2 (, s)g(φ(s))ds, (1) where g, K 1 and K 2 are known funcions and η 1, η 2 are wo consans. and hen we will give more general case, and we will solve a sysem of nonlinear Volerra-Fredholm inegral equaion of he form, for i =,..., n: φ i (s) = f i (s) + j= s K ij (s, )F (φ i ())d + j= K ij (s, )G(φ i ())d (2) where F and G wo known nonlinear funcions, f i are known funcions and φ i are he unknown funcions mus be deermined. The paper is organized as follows. In secion 2, we presen some resuls abou Legendre polynomial. In secion 3, we ransform Fredholm inegral equaion o a sysem of algebraic equaions. In secion 4, some numerical examples are presened. We generalize our mehod o a nonlinear sysem 5, finally we conclude. 2 Legendre Polynomials Orhogonal polynomials are widely used in applicaions in mahemaics, mahemaical physics, engineering and compuer science. One of he mos common se of orhogonal polynomials is he Legendre polynomials. The Legendre polynomials P n saisfy he recurrence formula: (n + 1)P n+1 (x) = (2n + 1)xP n (x) np n (x), n N P (x) = 1 P 1 (x) = x An imporan propery of he Legendre polynomials is ha hey are orhogonal wih respec o he L 2 inner produc on he inerval [, 1]: P n (x)p m (x)dx = 2 2n + 1 δ nm R S. Publicaion, rspublicaionhouse@gmail.com Page 564

3 Issue 5 volume 2, March.-April.215 where δ nm denoes he Kronecker dela, equal o 1 if m = n and o oherwise. The polynomials form a complee se on he inerval [, 1], and any piecewise smooh funcion may be expanded in a series of he polynomials. The series will converge a each poin o he usual mean of he righ and lef-hand limis. There are many applicaion of he Legendre polynomial, for insance, when we solve Laplace s equaion on a sphere, we wan soluions ha will be valid a he norh and souh poles (whose polar coordinaes are cos θ = ±1), herefore, he physically meaningful soluions o Laplace s equaion on a sphere are he polynomials of he firs kind. The Legendre and Chebyshev polynomials are used also o solve several problems of differenial equaions or inegral equaions, for insance he Legendre pesudospecral mehod is used o solve he delay and he diffusion differenial equaions ([8], [9]). Oher orhogonal polynomial like Chebyshev polynomials are used o inroduce an efficien modificaion of homoopy perurbaion mehod [1]. Also, he polynomial approximaion is used o solve high order linear Fredholm inegro differenial equaions wih consan coefficien [13] and ohers ([11], [17]-[19]). The Legendre polynomial is used o approximae any coninuous funcion. If we noe by P he se all real-valued polynomials on [, 1], hen P is a dense subspace of he space of coninuous funcion on he same inerval equipped wih he uniform norm, his is a paricular case of he heorem of f he Sone-Weiersrass. basing on hese properies of he Legendre polynomials we will solve he Fredholm inegral equaion. 3 Sysem of algebraic equaions The Fredholm-Volerra inegral equaion (1) is considered in his paper. funcion Φ using infinie series of Legendre polynomials is given by: The expansion of he Φ() = α n P n (), (3) n= where α n = (Φ(), P n ()). If we runcae he infinie series in Equaion(3) o only N + 1 erms, we obain N Φ() α n P n () = α T P (), (4) where α and P are marices given by n= α = [α,..., α n ], P = [P,..., P N ] T. (5) Subsiuing he runcaed series (4) ino Equaion(1) we obain α T P () = g() + η 1 K 1 (, s)f (α T P (s))ds + η 2 K 2 (, s)g(α T P (s))ds. (6) Using he Legendre collocaion poins defined by and subsiuing he variable by i defined in (7) o ge i = i, i =, 1,..., N, (7) N j α T P ( j ) = g( j ) + η 1 K 1 ( j, s)f (α T P (s))ds + η 2 K 2 ( j, s)g(α T P (s))ds. (8) R S. Publicaion, rspublicaionhouse@gmail.com Page 565

4 Issue 5 volume 2, March.-April.215 The domain [, 1] discreized ino m equally spaced panels, or N + 1 grid poins, where he grid spacing is h = 1 N, rapezoidal rules gives K 1 ( j, s)f (α T P (s))ds = h N 2 (F(s ) + F(s N ) + 2 F(s k )), (9) where F(s) = K 1 ( j, s)f (α T P (s)), h = 1 N, for any ineger N, s i = ih, i =, 1,..., N and j K 2 ( j, s)g(α T P (s))ds = h N j 2 (G( s ) + G( s N ) + 2 G( s k )), (1) where G(s) = K 2 ( j, s)g(α T P (s)), h j = j N, for an arbirary ineger N, s i = ih. Now, we obain he following sysem: α T P ( j ) = g( j ) + η 1 h 2 ( F(s ) + F(s N ) + 2 N k=1 F(s k ) ) + η 2 h j 2 ( k=1 k=1 G( s ) + G( s N ) + 2 N k=1 G( s k ) (11) The sysem (11) is a (N + 1) linear or nonlinear algebraic equaions, which can be solved for α i, i =, 1,..., N. Therefore he unknown funcion Φ can be deermined. We recall ha he conjugae gradien mehod is an algorihm for he numerical soluion of paricular sysems of linear equaions, i s implemened as an ieraive algorihm used o solve a sparse sysems ha are oo large o be handled by a direc implemenaion or oher direc mehods such as he Cholesky decomposiion. Large sparse sysems can be obained when we solve a parial differenial equaions or opimizaion problems. In he oher hand, he nonlinear sysem will be solved using Newon s mehod. A special code wrien using MATLAB is implemened for solving nonlinear equaions using Newon s mehod. We noe ha Malab has is rouines for solving sysems of nonlinear equaions (one can look o fsolve in Malab) which is based on Newon s mehod. The linear and nonlinear sysem will be solved using he conjugae gradien mehod and Newon s mehod respecively. 4 Examples We confirm our heoreical discussion wih numerical examples in order o achieve he validiy, he accuracy. The compuaions, associaed wih he following examples, are performed by MATLAB 7. Example 1. Here, we will apply he echnique presened in previous secion o a linear inegral equaion, in order o show ha he mehod presened can be applied. We consider he equaion (1) wih g(x) = x 3 (6 2e)e x, η 1 = 1, η 2 = 1, The equaion (1) is as follows K 1 (x, y) = e (x+y), K 2 (x, y) =, F (Φ(y)) = Φ(y), G(Φ(y)) =. Φ(x) = x 3 (6 2e)e x + e (y+x) Φ(y)dy. (12) R S. Publicaion, rspublicaionhouse@gmail.com Page 566 )

5 Issue 5 volume 2, March.-April.215 The suggesed mehod will be considered wih N = 4, and he approximae soluion Φ(x) can be wrien in he following way 4 Φ 4 (x) = α i P i (x) = α T P (x). (13) i= Using he same echnique presened in previous secion and using Equaion(8) we obain 4 i= α i P i (x j ) (x 3 j (6 2e)e x j ) h m 2 (F(y ) + F(y m ) + 2 F(y k )) =, j =, 1, 2, 3, 4, (14) where F(y) = e (y+x j) 4 i= α ip i (y) and he nodes y l+1 = y l + h, l =, 1,..., N, y = and h = 1 N. Equaion(14) represens linear sysem of 5 algebraic equaions in he coefficiens α i, i =,..., 4, which will be solved by he conjugae gradien mehod and we ge he following coefficiens: α =.48, α 1 =.5955, α 2 =.15, α 3 =.3998, α 4 =.1. Hence, he approximae soluion of Equaion(13) is as follows: Φ(x) =.48P (x) P 1 (x).15p 2 (x) P 3 (x).1p 4 (x). corresponding o exac soluion Φ(x) = x 3. k= Exac soluion Approximae soluion 12 x Error example Figure 1: Error beween exac soluion and he presen mehod wih N = 4 In example 1, we have considered only N = 4 erms in he expansion of he soluion using Legendre polynomials, he Figure 1 gives he behavior of rue (exac) and he approximae soluion and he behavior of he error beween hem, we noice ha he echnique used is much more perinen and can be considered as a profiable mehod o solve he linear inegral equaions. R S. Publicaion, rspublicaionhouse@gmail.com Page 567

6 Issue 5 volume 2, March.-April.215 Example 2. Unlike, he firs example, here we will consider he case of a nonlinear inegral equaion wih: Consider he equaion (1) wih he following funcions and coefficiens Equaion(1) akes he following form g(x) = 2xe x e x + 1, η 1 = 1, η 2 =, K 1 (x, y) =, K 2 (x, y) = (x + y), F (Φ(y)) =, G(Φ(y)) = e Φ(y). Φ(x) = 2xe x e x + 1 (y + x)e Φ(y) dy. (15) The suggesed mehod will be considered wih N = 4, and he approximae soluion Φ(x) can be wrien in he following way Φ N (x) = The echnique presened can be applied and using Eq.(8) we obain 4 i= 4 α i P i (x) = α T P (x). (16) i= α i P i (x j ) f(x j ) + h m j 2 (F(y ) + F(y m ) + 2 F(y k )) =, j =, 1, 2, 3, 4, (17) where he nodes y l+1 = y l + h, l =, 1,..., m, y = and h j = j m, F(y) = (y + x j)e CT P (y). The Equaion (17) presens nonlinear sysem of N + 1 algebraic equaions in he coefficiens α i. By solving i by using he Newon ieraion mehod wih suiable iniial soluion we obain α =.2, α 1 =.9895, α 2 =.22, α 3 =.88, α 4 =.23. Hence, he approximae soluion obained from (16) wrien in he following way Φ(x) =.2P (x) P 1 (x) +.22P 2 (x).88p 3 (x) +.23P 4 (x). associaed o exac soluion Φ(x) = x. Figure 2 presens he behavior of he approximae soluion using he proposed mehod wih N = 4 and he exac soluion. From his Fig. 2 we noice ha he proposed mehod can be considered as an perinen mehod o solve he nonlinear inegral equaions. Example 3. Equaion (1) will be considered wih Equaion(1) can be wrien as follows: k=1 g(x) = xe + 1, η 1 =, η 2 = 1, K 1 (x, y) = y + x, K 2 (x, y) =, F (Φ(y)) = e Φ(y), G(Φ(y)) =. Φ(x) = xe + 1 The exac soluion associaed o his problem he linear funcion Φ(x) = x. (y + x)e Φ(y) dy. (18) R S. Publicaion, rspublicaionhouse@gmail.com Page 568

7 Inernaional Issue 5 volume 2, March.-April Exac soluion Approximae soluion.15 Error example Figure 2: Error beween exac soluion and he presen mehod wih N = 4 The suggesed mehod will be considered wih N = 3, and he approximae soluion Φ(x) can be wrien in he following way: 3 i= α i P i (x j ) g(x j ) + h m 2 (F(y ) + F(y m ) + 2 F(y k )) =, j =, 1, 2, 3, (19) where y l+1 = y l + h, l =, 1,..., N, y = and h = 1 N and F(y) = (y + x j).e ( 3 i= α ip i (y)). We have a sysem of nonlinear algebraic equaions (19). The soluion can be obained using Newon ieraion mehod wih suiable iniial soluion, and one can obain: k=1 α =.23, α 1 = 1.13, α 2 =., α 3 =.. The approximaed soluion derived in his example is: Φ(x) =.23P (x) P 1 (x) +.P 2 (x) +.P 3 (x). Figure 3 gives he behavior of he approximae soluion using he proposed mehod wih N = 3 and he exac soluion. Example 4. We consider here a nonlinear inegral equaion wih he following funcion and coefficiens: g(x) = x 2 x , η 1 = 1, η 2 =, K 1 (x, y) = y + x, Equaion(1) becomes K 2 (x, y) = y x, F (Φ(y)) = Φ(y), G(Φ(y)) = Φ 2 (y). Φ(x) = x 2 x (y + x)φ(y)dy + (y x)φ 2 (y)dy. (2) R S. Publicaion, rspublicaionhouse@gmail.com Page 569

8 Inernaional Issue 5 volume 2, March.-April Exac soluion Approximae soluion 2.6 x Error example Figure 3: Error beween exac soluion and he presen mehod wih N = 3 The echnique presened can be applied using only he firs five Legendre Polynomials funcion and he approximae soluion can be wrien as: Φ N (x) = 4 α i P i (x) = α T P (x). (21) i= The same echnique and using equaion (8) we ge 4 α i P i (x j ) g(x j ) h 2 (F(ȳ ) + F(ȳ N ) i= +2 N k=1 F(ȳ k)) h j 2 (G(y ) + G(y N ) +2 N k=1 G(y k)) =, (22) where ȳ l+1 = ȳ l + h, y l+1 = y l + h j, l =, 1,..., N, s = ȳ = h = 1 N, h j = x j N, and F(y) = (y + x j )( 4 i= α ip i (y)), G(y) = (x j y)(α T P (y)) 2. Equaion (22) is a nonlinear sysem of 5 algebraic equaions. The Newon ieraion mehod can be used and we obain he following coefficiens: α =.12, α 1 =.9987, α 2 =.39, α 3 =.7, α 4 =.17. Hence, he approximae soluion can be wrien using he firs five Legendre polynomial as: Φ(x) =.12P (x) P 1 (x).39p 2 (x) +.7P 3 (x).17p 4 (x). and i s will be compared o he exac soluion which is Φ(x) = x. Figure (4) presens he behavior of he approximae soluion and he exac soluion, we noice ha he proposed echnique can be well chosen as a perinen way o solve he nonlinear Volerra Fredholm inegral equaions. The figure 4 and Table (1) show he behavior of he exac soluion and he approximae soluion using N = 4. R S. Publicaion, rspublicaionhouse@gmail.com Page 57

9 Inernaional Issue 5 volume 2, March.-April Exac soluion Approximae soluion 6 x 1 3 Error example Figure 4: Error beween exac soluion and he presen mehod wih N = 4 5 Sysem of Nonlinear Volerra-Fredholm inegral equaions In his secion, we will develop a general mehod o solve a sysem of nonlinear Volerra-Fredholm inegral equaion sysem by using he Chebyshev polynomials insead of Legendre polynomials used in he firs par, bu he compuaion i s he same. We will solve he sysem of n + 1 nonlinear Volerra-Fredholm inegral equaions are defined as follows: for i =,..., n: φ i (s) = f i (s) + j= s K ij (s, )F (φ i ())d + j= K ij (s, )G(φ i ())d (23) where F and G wo known nonlinear funcions, f i are known funcions and φ i are he unknown funcions mus be deermined. The sysem (23) inroduce a n+1 Volerra-Fredholm inegral equaion wih n + 1 unknowns. We use Chebyshev polynomials for solving nonlinear Volerra-Fredholm inegral equaions, one can propose oher ype of polynomials o solve he same problem using he same echnique. The Chebyshev polynomials of he firs kind can be defined as he unique polynomials saisfying Then he inner produc is given by T n (cos(ϑ)) = cos(nϑ), n =, 1,..., < T i, T j >= T i (x)t j (x)w(x)dx (24) where w(x) = (1 x 2 ) /2 Wih respec o he inner produc which is defined in Chebyshev polynomials are orhogonal < T i, T j >= πδ ij (25) R S. Publicaion, rspublicaionhouse@gmail.com Page 571

10 Issue 5 volume 2, March.-April.215 Exac soluion Approximae soluion Error Table 1: Example 4. Comparison beween exac and approximae soluion where δ ij denoe he Kronecker s dela. We will use Chebyshev polynomials o approximae he unknown funcion, hey are used as a collocaion basis o solve sysem of nonlinear Volerra-Fredholm inegral equaion and reduce i o a linear or nonlinear sysem of algebraic equaions. Newon s ieraive mehod can be used for solving nonlinear algebraic sysem. If f defined on [, 1], hen we have f(s) = c i T i (s) and afer runcaion we have i= where f(s) = N c i T i (s) = C T (s) i= C = [c, c 1, c 2,..., c N ], T (s) = [T (s), T 1 (s), T 2 (s),..., T N (s)] where he coefficien c i are defined as { 1 1 c i = π (1 s2 ) /2 f(s)ds if i = (1 s2 ) /2 f(s)t i (s)ds if i > 2 π For simpliciy we will consider some special cases where F (φ i ) = φ p i and G(φ i) = φ q i wo ineger. In his case, we have where p, q F (φ i (s)) = φ p i (s) = C pt (s) where C can be obained from he vecor C. For example, in he case where p = 2 we have (Thanks o he sofware Maple for helping us o do he compuaion) R S. Publicaion, rspublicaionhouse@gmail.com Page 572

11 Issue 5 volume 2, March.-April.215 The kernel where C 2 = 1 2 K(x, ) = 2c 2 + c2 1 + c2 2 + c2 3 4c c 1 + 2c 1 c 2 + 2c 2 c 3 c c c 2 + c 1 c 3 2c 1 c 2 + 4c c 3 N i=1 j=1 N T i (s)k ij T j () K ij =< T i (s), < K(s, ), T j () >> So K becomes a (N + 1) (N + 1) marix wih elemens K ij Therefore K(s, ) = T (s)k(s, )T () For i, j =,..., N, we have he following relaion: and where φ i (s) = T (s)φ i = (φ i (s)) p = T (s) Φ ip K ij = T (s)k ij T () F ip = [f i, f i1, f i2,..., f in ] Some special formula will be used in our compuaion: x T n ()d = 1 2n T 1 n(x) 2(n 1) T n 2(x) + ()n 1 (n 1) 2 T (x), n 3 x x T ()d = T (x) + T 1 (x) T 1 ()d = 1 4 ( T (x) + T 2 (x)) hese inegral can be wrien using a marix form P : x T ()d = P T (x), where P is a (n + 1) (n + 1) marix defined as follows: P = T ()d = P T (1) () n (n) 2 2n () n 1 n 2 2(n) R S. Publicaion, rspublicaionhouse@gmail.com Page 573

12 Issue 5 volume 2, March.-April.215 using Chebyshev Polynomial we have he following relaion: T (s)t (s) C = C T (s) where C is a (n + 1) (n + 1) square marix given by 2c c 1... c i... c n c n C = 1 2c 1 2c + c 2... c k + c k+1... c n 2 + c n c n c n c n 2 + c n... c n k... 2c c 1 2c n c n... c n i... c 1 2c Now, he sysem of Volerra-Fredholm inegral equaion becomes: T (x) Φ ip = f i (x) + j= x T (x)k ij T () φ ipj d + j= T (x)k ij T ()T () φ iqj d his lead o he following sysem: for i =, 1, 2,..., n we have f i (x) = T (x) Φ i T (x)k ij φipj P (x) T (x)k ij φipj P (1) (26) j= he equaion 26 can be evaluaed a he collocaion poins {x k }, k =,..., N in he inerval [, 1] hen we ge a sysem of (n + 1) (N + 1) equaion wih (n + 1) (N + 1) unknowns. For k =,..., N and i =,..., n we are going o solve: f i (x k ) = T (x k ) Φ i T (x k )K ij φipj P (x k ) T (x k )K ij φipj P (1) (27) j= The relaion 27 leads o a linear or nonlinear sysem of equaions such ha he unknown coefficiens can be found. In he las of his paper and o confirm he sraegy proposed we presen some numerical example. We use Newon s ieraive mehod for solving he generaed nonlinear sysem, Maple and Malab are used in our case o do our numerical es. Consider he following nonlinear inegral equaions sysem: where φ 1 (x) = f 1 (x) + φ 2 (x) = f 2 (x) + x x x 2 3 φ 4 1()d + xφ 2 1()d + j= j= (3 x 2 )φ 7 2()d 3 2 xφ 5 2()d (28) f 1 (x) = 2 x x x x9 4 x 8 8/5 x 7 1/4 x 6 f 2 (x) = x x x5 4/3 x 4 1/2 x 3 and he exac soluion are φ 1 (x) = 2x + 1, φ 2 (x) = x 2 + x Table 2 gives a comparison beween he exac and approximaion soluion. R S. Publicaion, rspublicaionhouse@gmail.com Page 574

13 Issue 5 volume 2, March.-April.215 x i φ φ Error φ φ Error Conclusion Table 2: Comparison beween he approximae soluion and exac soluion We solved a sysem of Volerra-Fredholm inegral equaions by using Chebyshev collocaion mehod. The properies of Chebyshev or Legendre polynomials are used o reduce he sysem of Volerra Fredholm inegral equaions o a sysem of nonlinear algebraic equaions. The mehod presened in his paper based on he Legendre and Chebyshev polynomials is suggesed o find he numerical soluion which will be compared o he analyic soluion. The ieraive mehod conjugae gradien mehod and Newon s mehod are used o solve he linear and nonlinear sysem. Analyzing he numerical soluion and he exac soluion declare ha he echnique used is very effecive and convenien.the approach used is esed wih differen examples o show ha he accuracy improves wih increasing N. Moreover, using he obained numerical soluion, we can affirm ha he proposed mehod gives he soluion in an grea accordance wih he analyic soluion. In addiion, one can invesigae oher ype of a nonlinear Fredholm inegro differenial equaion wih singular kernel. R S. Publicaion, rspublicaionhouse@gmail.com Page 575

14 Issue 5 volume 2, March.-April.215 References [1] C. Chnii. On The Numerical Soluion of Volerra-Fredholm Inegral Equaions Wih Abel Kernel Using Legendre Polynomials. IJAST, Issue 3 volume 1,(213), [2] C.Chnii. Numerical Approximaions of Fredholm Inegral Equaions wih Abel Kernel using Legendre and Chebychev Polynomials. J. Mah. Compu. Sci. 3 (213), No. 2, [3] R.P. Agarwal, Boundary value problems for higher order inegro-differenial equaions, Nonlinear Anal. Theory Mehods Appl., 9 (1983) [4] L.C. Andrews, Special Funcions For Engineers and Applied Mahemaicians, Macmillan publishing company, New York, (1985). [5] E. Babolian, F. Faahzadeh, E. Golpar Raboky, A Chebyshev approximaion for solving nonlinear inegral equaions of Hammersein ype, Applied Mahemaics and Compuaion, 189 (27) [6] A.K. Borzabadi, A.V. Kamyad, H.H. Mehne, A differen approach for solving he nonlinear Fredholm inegral equaions of he second kind, Applied Mahemaics and Compuaion, 173 (26) [7] L.M. Delves, J.L. Mohamad, Compuaional Mehods for Inegral Equaions, Cambridge Universiy press, (1985). [8] M.M. Khader, A.S. Hendy, The approximae and exac soluions of he fracional-order delay differenial equaions using Legendre pseudo-specral mehod, Inernaional Journal of Pure and Applied Mahemaics, 74(3) (212) [9] M.M. Khader, N.H. Sweilam, A.M.S. Mahdy, An efficien numerical mehod for solving he fracional diffusion equaion, Journal of Applied Mahemaics and Bioinformaics, 1 (211) [1] M.M. Khader, Inroducing an efficien modificaion of he homoopy perurbaion mehod by using Chebyshev polynomials, Arab Journal of Mahemaical Sciences, 18 (212) [11] M.M. Khader, Numerical soluion of nonlinear muli-order fracional differenial equaions by implemenaion of he operaional marix of fracional derivaive, Sudies in Nonlinear Sciences, 2(1) (211) [12] S.T. Mohamed, M.M. Khader, Numerical soluions o he second order Fredholm inegrodifferenial equaions using he spline funcions expansion, Global Journal of Pure and Applied Mahemaics, 34 (211) ] [13] N. Kur, M. Sezer, Polynomial soluion of high-order linear Fredholm inegro-differenial equaions wih consan coefficiens, Journal of he Franklin Insiue, 345 (28) [14] M. Shahrezaee, Solving an inegro-differenial equaion by Legendre polynomials and Blockpulse funcions, Dynamical Sysems and Applicaions, (24) R S. Publicaion, rspublicaionhouse@gmail.com Page 576

15 Issue 5 volume 2, March.-April.215 [15] N.H. Sweilam, Fourh order inegro-differenial equaions using variaional ieraion mehod, Compu. Mahs. Appl., 54 (27) [16] N.H. Sweilam, M.M. Khader, R.F. Al-Bar, Homoopy perurbaion mehod for linear and nonlinear sysem of fracional inegro-differenial equaions, Inernaional Journal of Compuaional Mahemaics and Numerical Simulaion, 1 (28) [17] N.H. Sweilam, M.M. Khader, A Chebyshev pseudo-specral mehod for solving fracional order inegro-differenial equaions, ANZIAM, 51 (21) [18] N.H. Sweilam, M.M. Khader, Semi exac soluions for he bi-harmonic equaion using homoopy analysis mehod, World Applied Sciences Journal, 13 (211) 1 7. [19] S. Yousefi, M. Razzaghi, Legendre wavele mehod for he nonlinear Volerra-Fredholm inegral equaions, Mah. Comp. Simul., 7 (25) 1 8. R S. Publicaion, rspublicaionhouse@gmail.com Page 577