CUBO A Mathemati al Journal Vol.15, N o 03, (89�103). O tober 2013 Approximate solution of fra tional integro-di�erential equation by Taylor expansion and Legendre wavelets methods M.H.Saleh, S.M.Amer, M.A.Mohamed and N.S.Abdelrhman Mathemati s Department, Fa ulty of S ien e, Suez anal University, Ismailia. nsae191088�yahoo. om ABSTRACT This paper, deals with the approximate solution of fra tional integro-di�erential equa- tions of the type Dq ∗ y(t) = f(t) + p(t)y(t) + ∫t 0 k(t,s)y(s)ds, t ∈ I = [0,1] by Taylor expansion and Legendre wavelet methods.In addition, illustrative example are presented to demonstrate the e� ien y and a ura y of this methods. RESUMEN Este artí ulo onsidera la solu ión aproximada de e ua iones integro-diferen iales fra - ionales del tipo Dq ∗ y(t) = f(t) + p(t)y(t) + ∫t 0 k(t,s)y(s)ds, t ∈ I = [0,1] por expansiones de Taylor y métodos de Ondeletas de Legendre. Además, un ejemplo ilustrativo se presenta para mostrar la e� ien ia y pre isión de este método. Keywords and Phrases: Fra tional integro-di�erential equation, Caputo fra tional derivative, Taylor expansion method, Legendre wavelets method. 2010 AMS Mathemati s Subje t Classi� ation: 45B05 , 45Bxx , 65R10. 90 M.H.Saleh, S.M.Amer, M.A.Mohamed & N.S.Abdelrhman CUBO 15, 3 (2013) 1. Introdu tion We study the approximate solution of an integro-di�erential equation with fra tional derivative of the type Dq ∗ y(t) = f(t) + p(t)y(t) + ∫t 0 k(t,s)y(s)ds, t ∈ I = [0,1], y(0) = α , (1.1) where 0 < q < 1,α ∈ Re and the fun tions f,p,k are assumed to be su� iently smooth on their domains I and S (S = {(t,s) : 0 ≤ s ≤ t ≤ 1}.This kinds of equations arise in many modeling problems in mathemati al physi s su h as heat ondu tion in materials with memory .The existen e and uniqueness of solution of fra tional di�erential equation have been investigated in [5,7℄. Re ently some attentions have been paid to the numeri al solution of equation (1.1) . Rawashdeh [4℄ applied the ollo ation method to �nd a spline approximation, in [11℄ used the de omposition method to �nd an analyti solution. 2. Basi de�nitions De�nition2.1. The Riemann-Liouville fra tional integral of order q ≥ 0 of a fun tion f ∈ cα,α ≥ −1 is de�ned by : Jqf(x) = 1 Γ(q) ∫x 0 (x − s)q−1f(s)ds where the real fun tion f(x) ∈ Cα,α ∈ Re,x > 0 is said to be in spa e if there exist a real number P > α su h that f(x) = xpf1(x) where f1(x) ∈ C[0,∞). De�nition2.2. Let f ∈ Ck−1,k ∈ N. Then the Caputo fra tional derivative of f is de�ned by : Dq ∗ f(x) =    Jk−qf(k)(x) ifk − 1 < q < k, f(k)(x) ifk = q . To obtain a numeri al s heme for the approximation of Caputo derivative , we an use a repre- sentation that has been introdu ed by Elliots [2℄; Dq ∗ f(x) = 1 Γ(−q) ∫x 0 f(s) − f(0) (x − s)1+q ds, (1.2) , where the integral in equation (1.2) is a Hadamard �nite-part integral CUBO 15, 3 (2013) Approximate solution of fra tional integro-di�erential equation . . . 91 De�nition2.3. Dq, denotes the fra tional di�erential operator of order q, de�ned by [5℄ as: Dqy(x) =    1 Γ(n−q) d n dtn ∫t 0 y(s) (x−s)q−n+1 ds, 0 ≤ n − 1 < q < n, d n y(x) dxn , q = n De�nition2.4. The following fun tions ψk,n(t) = |a0| k 2 ψ(ak0t − nb0) , form a family of dis rete wavelets , where a0 > 1,b0 > 0 and n,k are positive integers and ψ is given fun tion alled mother wavelet . Moreover, the fun tions ψn,m(t) =    √ m + 1 2 2 k 2 pm(2 kt − n̂), n̂−1 2k ≤ t < n̂+1 2k , = 0 otherwise (1.3) are alled legendre wavelets polynomials where n̂ = 2n−1, n = 1, .......,2k −1, k ∈ N, t ∈ [0,1] and m is the order of the legendre polynomials pm . some basi properties of the aputo and fra tional operator an be found in [5℄. 3. Taylor expansion method We onsider the following fra tional integro-di�erential equation Dqy(t) = f(t) + p(t)y(t) + λ ∫t 0 k(t,s)y(s)ds, (3.1) subje ted to the initial onditions y(k)(0) = ck,k = 0,1, ....n−1, n−1 < q ≤ n, n ∈ N. To �nd the solution of Eq.(3.1) , we integrate both sides of Eq.(3.1) with respe t to s for n times by using de�nitions (2.2) , (2.3) . In−qy(t) = Inf(t) + In(p(t)y(t)) + λIn( ∫t 0 k(t,s)y(s)ds), (3.2) further ∫t 0 (t − s)n−q−1 Γ(n − q) y(s)ds = ∫t 0 (t − s)n−1 (n − 1)! f(s)ds + ∫t 0 (t − s)n−1 (n − 1)! p(s)y(s)ds + λ (n − 1)! ∫t 0 y(s) ∫t s k(x,s)(x − s)n−1dsdx + qn(t). (3.3). 92 M.H.Saleh, S.M.Amer, M.A.Mohamed & N.S.Abdelrhman CUBO 15, 3 (2013) Next , we assume that the desired solution y(s) is m + 1 times ontinuously di�erentiable on the interval I. Consequently , for y ∈ cm+1,y(s) an be represented in terms of the mth order Taylor expansion as y(s) = y(t) + y′(t)(s − t) + ... + y(m)(t) (s − t)m m! + y(m+1)(ξ) (s − t)m+1 (m + 1)! , (3.4) where ξ is between s and t.The Lagrange reminder y(m+1)(ξ) (s−t) m+1 (m+1)! is small for a large enough m provided that y(m+1)(s) is uniformly bounded fun tion for any m on the interval I.Consequently we will negle t the reminder and the trun ated Taylor expansions y(x) as y(s) ≈ m∑ j=0 y(j)(t) (x − t)j j! . (3.5) We noti e that the lagrange reminder vanishes for a polynomial of degree equal to or less than m, this is implying that the above mth order Taylor expansion is exa t . Substituting the approximate expression (3.5) for y(t) into Eq(3.2) , we get m∑ j=0 ∫t 0 (t − s)n−q−1 Γ(n − q) yj(t) (s − t)j j! ds = ∫t 0 (t − s)n−1 (n − 1)! f(s)ds + m∑ j=0 ∫t 0 (t − s)n−1 (n − 1)! p(s)yj(t) (s − t)j j! ds + m∑ j=0 λ (n − 1)! ∫t 0 yj(t) (s − t)j j! ∫t 0 k(x,s)(x − s)n−1dsdx + Qn(t), (3.6) or K00(t)y(t) + K01(t)y ′ (t) + .... + K0m(t)y (m) (t) = fn(t), (3.7) where K0j(t) = (−1)jtn+j−q (n + j − q)Γ(n − q)j! − λ (n − 1)!j! ∫t 0 (s − t)j ∫t 0 k(x,s)(x − s)n−1dsdx − (−1)j (n − 1)!j! ∫t 0 p(s)(t − s)n+j−1ds, j = 0,1, ...,m. (3.8) f(n)(t) = 1 (n − 1)! ∫t 0 (t − s)n−1f(s)ds + Qn(t). (3.9) Thus Eq.(3.3) be omes an mth order , linear , ordinary di�erential equation with variable oef- � ients for y(t) and its derivatives up to m . We will determine y(t), ...,ym(t) by solving linear equations instead of solving analyti ally ordinary di�erential equation . By integrating both sides of Eq.(3.3) with respe t to s and hanging the order of the integrations we shall obtain m independent linear equation for y(s), ...,ym(s). ∫t 0 (t − s)n−q Γ(n + 1 − q) y(s)ds = ∫t 0 (t − s)n n! y(s)ds + ∫t 0 (t − s)n n! f(s) + Qn(s)ds CUBO 15, 3 (2013) Approximate solution of fra tional integro-di�erential equation . . . 93 + λ n! ∫t 0 y(s) ∫t 0 k(x,s)(x − s)ndsdx (3.10) Where we have repla ed x with t . Applying the Taylor expansion again and substituting (3.5) for y(s) into Eq.(3.10) gives K10(t)y(t) + K11(t)y ′(t) + .... + K1m(t)y (m)(t) = fn+1(t), (3.11) K1j(t) = (−1)jtn+j+1−q (n + j + 1 − q)Γ(n + 1 − q)j! − λ n!j! ∫t 0 (s − t)j ∫t 0 k(x,s)(x − s)ndsdx − (−1)j n!j! ∫t 0 p(s)(t − s)n+jds, j = 0,1, ...,m. (3.12) f(n+1)(t) = ∫t 0 (t − s)n n! f(s) + Qn(t)ds. (3.13) Now we have another linear equation for y(j)(t), j = 0, ...,m with y(0)(t) = y(t). By repeating the above integration pro ess for i (i ∈ N+,1 < i ≤ m) times, we get Ki0(t)y(t) + Ki1(t)y ′(t) + .... + Kim(t)y (m)(t) = fn+i(t), , i ≤ m (3.14) where Kij(t) = (−1)jtn+j+i−q (n + j + i − q)Γ(n + i − q)j! − (−1)j (n + i − 1)!j! ∫t 0 p(s)(t − s)n+j+i−1ds − λ (n + i − 1)!j! ∫t 0 (s − t)j ∫t 0 k(x,s)(x − s)n+j−1dsdx, j = 0,1, ...,m. (3.15) f(r)(t) = ∫t 0 fr−1(s)ds, r > n + 1, r ∈ N +. (3.16) Consequently, Eqs.(3.7) , (3.11) and (3.14) form a system of m+1 unknown fun tions y(s), ....y(m)(s). This system an be written as Kmm(t)Ym(t) = Fm(t), (3.17) 94 M.H.Saleh, S.M.Amer, M.A.Mohamed & N.S.Abdelrhman CUBO 15, 3 (2013) where Kmm(t) is an (m + 1) × (m + 1) matrix fun tion in t , Ym(t) and Fm(t) are two ve tor de�ned as Kmm =                             K00(t) K01(t) . . . K0m(t) K10(t) K11(t) . . . K1m(t) . . . . . . . . . . . . Km0(t) Km1(t) . . . Kmm(t)                             , (3.18) Ym(t) =                             y(t) y′(t) . . . y(m)                             , Fm(t) =                             f(n)(t) f(n+1)(t) . . . f(n+m)(t)                             . (3.19) Using ramer's rule , we obtain the m th-order approximate solution as y(t) = detMmm(t) detKmm(t) . (3.20) CUBO 15, 3 (2013) Approximate solution of fra tional integro-di�erential equation . . . 95 where Mmm =                             fn(t) K01(t) . . . K0m(t) f(n+1)(t) K11(t) . . . K1m(t) . . . . . . . . . . . . fn+m(t) Km1(t) . . . Kmm(t)                             . (3.21) 4. Legendre wavelets method We onsider the following fra tional integro-di�erential equation Dq ∗ y(t) = f(t) + p(t)y(t) + ∫t 0 k(t,s)y(s)ds, t ∈ I = [0,1] y(0) = α (4.1) the exa t solution of Eq.(4.1) an be expanded as a legendre wavelets series as y(t) = ∞∑ n=1 ∞∑ m=0 cnmψn,m(t), where ψn,m(t) is given by Eq.(1.3).We approximate the solution y(t) by the trun ated series yK,M(t) = 2 k−1 ∑ n=1 M−1∑ m=0 cnmψn,m(t), (4.2) Then a total number of 2k−1M onditions exist for determination of 2k−1M oe� ients c10,c11, .....,c1M−1,c20, ....,c2M−1, ....,c2k−10, ....,c2k−1M−1. By the initial ondition we obtain , yK,M(0) = 2 k−1 ∑ n=1 M−1∑ m=0 cnmψn,m(0) = α. (4.3) 96 M.H.Saleh, S.M.Amer, M.A.Mohamed & N.S.Abdelrhman CUBO 15, 3 (2013) We must obtain 2k−1M − 1 extra onditions to re over the unknown oe� ients cnm. These onditions an be obtained by substituting Eq.(4.2) in Eq.(4.1). 2 k−1 ∑ n=1 M−1∑ m=0 cnmD q ∗ ψn,m(t) = f(t)+ 2 k−1 ∑ n=1 M−1∑ m=0 cnm p(t) ψn,m(t)+ 2 k−1 ∑ n=1 M−1∑ m=0 cnm ∫t 0 k(t,s) ψn,m(s)ds. (4.4) Now we assume Eq.(4.4) is exa t at 2k−1M − 1 points xi as : 2 k−1 ∑ n=1 M−1∑ m=0 cnmD q ∗ ψn,m(xi) = f(xi) + 2 k−1 ∑ n=1 M−1∑ m=0 cnm p(xi) ψn,m(xi) + 2 k−1 ∑ n=1 M−1∑ m=0 cnm ∫xi 0 k(xi,s) ψn,m(s)ds. (4.5) The best hoi e of the xi points are the zeros of the shifted hebyshev polynomials of degree 2k−1M − 1 in the interval [0,1] that is xi = si + 1 2 , si = ( (2i − 1)π 2k−1M − 1 ), i = 1, ...,2k−1M − 1. Approximating D q ∗ ψn,m using Diethhelm method [6℄ on the representation that has been given by Eq.(1.2) , we get Dq ∗ ψn,m(xi) = 1 Γ(−q) ∫xi 0 ψn,m(s) − ψn,m(0) (xi − s) 1+q ds = x −q i Γ(−q) ∫1 0 ψn,m(xi − xiw) − ψn,m(0) w1+q ds ≃ l∑ r=0 αr(ψn,m(xi − xir l ) − ψn,m(0)) where L ∈ N and the weights αr is given by q(1 − q)L−q Γ(−q) x −q i αr =    −1, if r = 0, 2r1−q − (r − 1)1−q − (r + 1)1+q, if r = 1,2, ..., l − 1, (q − 1)r−q − (r − 1)1−q + r1−q, if r = l CUBO 15, 3 (2013) Approximate solution of fra tional integro-di�erential equation . . . 97 Then Eq.(4.5) be omes 2 k−1 ∑ n=1 M−1∑ m=0 l∑ r=0 αr(ψn,m(xi − xir l ) − ψn,m(0))cnm = f(xi) + 2 k−1 ∑ n=1 M−1∑ m=0 cnm p(xi) ψn,m(xi) + 2 k−1 ∑ n=1 M−1∑ m=0 cnm ∫xi 0 k(xi,s) ψn,m(s)ds. (4.6) Combine Eq.(4.3) and Eq.(4.6) to obtain 2k−1M linear equations from whi h we an ompute the unknowns oe� ients , cnm 5. Numeri al examples To show the e� ien y and the a ura y of the proposed methods , we onsider here some fra tional integro-di�erential equations . Now we shall solve some examples by Taylor expansion and Legendre wavelet methods and ompare the results in tables . All results are obtained by using Maple 15. Example 1. Consider the following Fra tional integro-di�erential equation D0.75 ∗ y(t) = 2t1.25 Γ(2.25) − t4 − t5 4 + t2y(t) + ∫t 0 tsy(s) ds , (5.1) with the initial ondition y(0) = 0 and the exa t solution y(t) = t2, with K=1 and M=6 . Table 1. The results of Example 1. 98 M.H.Saleh, S.M.Amer, M.A.Mohamed & N.S.Abdelrhman CUBO 15, 3 (2013) t Exa t Taylor method Abs.E Legendre method Abs.E 0.1 0.01 0.009999996 4.086580027 × 10 −9 0.041204470 0.312044705 × 10 −2 0.2 0.04 0.039999686 3.137931423 × 10 −7 0.157896393 0.1178963927 0.3 0.09 0.089996776 3.224235647 × 10−6 0.270260094 0.1802600943 0.4 0.16 0.159983397 1.660282668 × 10 −5 0.3523832431 0.1923832431 0.5 0.25 0.249941528 5.847208088 × 10−5 0.429628144 0.179628144 0.6 0.36 0.359839516 1.604836601 × 10 −4 0.576003047 0.216003047 0.7 0.49 0.489634434 3.655662632 × 10 −4 0.911533447 0.421533447 0.8 0.64 0.639288648 7.113517648 × 10 −4 1.599633386 0.959633386 0.9 0.81 0.808822770 1.177230376 × 10 −3 2.844476755 2.034476755 Figure 1: Results of Example 1 CUBO 15, 3 (2013) Approximate solution of fra tional integro-di�erential equation . . . 99 Example 2. Consider the following Fra tional integro-di�erential equation D0.25 ∗ y(t) = 6t2.75 Γ(3.75) − t4 − t2et 5 y(t) + ∫t 0 etsy(s) ds , (5.2) with the initial ondition y(0) = 0 and the exa t solution y(t) = t3, with K=2 and M=2 . Table 2. The results of Example 2. t Exa t Taylor method Abs.E Legendre method Abs.E 0.1 0.001 0.001000055 5.5082 × 10−8 0.7846772718 × 10−2 6.846772718 × 10−3 0.2 0.008 0.008001757 1.756874 × 10 −6 0.015693545 7.69354543 × 10 −3 0.3 0.027 0.027011518 1.151813 × 10 −5 0.02354031813 3.45968187 × 10 −3 0.4 0.064 0.064035232 3.523224 × 10 −5 0.03138709084 3.261290916 × 10 −2 0.5 0.125 0.125051836 5.18360 × 10 −5 -1.668332670 1.793332670 0.6 0.216 0.215957399 4.26011 × 10−5 -0.882599797 1.098599797 0.7 0.343 0.342472414 5.275860 × 10 −4 -0.096866924 0.439866924 0.8 0.512 0.510019153 1.980847 × 10−3 0.688865948 0.176865948 0.9 0.729 0.723579567 5.4204329 × 10 −3 1.474598821 0.745598821 100 M.H.Saleh, S.M.Amer, M.A.Mohamed & N.S.Abdelrhman CUBO 15, 3 (2013) Figure 2: Results of Example 2. Example 3. Consider the following Fra tional integro-di�erential equation D0.5 ∗ y(t) = 2t1.5 Γ(2.5) + t0.5 Γ(1.5) +t(2−3 ost−tsint+t2 ost)−( ost−sint)y(t)+ ∫t 0 etsy(s) ds , (5.3) with the initial ondition y(0) = 0 and the exa t solution y(t) = t2 + t with K=2 and M=2 . Similarly as in examples 1,2 applying the Taylor expansion method and Legendre wavelet method of the A omparison between the exa t solution and the approximate solution and the absolute error (Abs.E) are given in table 3 and Figures 3. CUBO 15, 3 (2013) Approximate solution of fra tional integro-di�erential equation . . . 101 Table 3. The results of Example 3. t Exa t Taylor method Abs.E Legendre method Abs.E 0.1 0.11 0.109999426 5.740 × 10 −7 0.1373123038 2.73123038 × 10 −2 0.2 0.24 0.239986839 1.31607 × 10−5 0.2746246075 3.46246075 × 10−2 0.3 0.39 0.389914938 8.50616 × 10 −5 0.4119369112 2.19369112 × 10 −2 0.4 0.56 0.559673632 3.263682 × 10 −4 0.5492492149 1.07507851 × 10 −2 0.5 0.75 0.749057525 9.424746 × 10 −4 0.649795587 0.100204413 0.6 0.96 0.957730079 2.2699214 × 10 −3 0.995295259 3.5295259 × 10 −2 0.7 1.19 1.185190201 4.809799 × 10−3 1.340794930 0.150794930 0.8 1.44 1.430747819 9.252181 × 10 −3 1.686294602 0.246294602 0.9 1.71 1.693507524 1.6492476 × 10−2 2.031794273 0.321794273 Figure 3: Results of Example 3. with Taylor expansion and Legendre wavelet method. 102 M.H.Saleh, S.M.Amer, M.A.Mohamed & N.S.Abdelrhman CUBO 15, 3 (2013) 6. Con lusion In this paper , we have applied the Legendre wavelet and Taylor expansion methods for solving the fra tional integro-di�erential equation . 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