International Journal of Analysis and Applications Volume 16, Number 6 (2018), 822-841 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-822 NEW MODIFIED METHOD OF THE CHEBYSHEV COLLOCATION METHOD FOR SOLVING FRACTIONAL DIFFUSION EQUATION H. JALEB, H. ADIBI∗ Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran ∗Corresponding author: adibih@aut.ac.ir Abstract. In this article a modification of the Chebyshev collocation method is applied to the solution of space fractional differential equations. The fractional derivative is considered in the Caputo sense. The finite difference scheme and Chebyshev collocation method are used. The numerical results obtained by this approach have been compared with other methods. The results show the reliability and efficiency of the proposed method. 1. Introduction The fractional partial differential equations (FPDEs) arise in numerous problems of engineering, physics, mathematics, chemistry, biology,and viscoelasticity ( [1], [2], [3], [4]).Most fractional differential equations do not have exact analytical solutions, thus many authors are seeking ways to numerically solve these problems( [5], [6]). Recently, some different methods to solve fractional differential equations have been given such as variational iteration method [7], homotopy perturbation method [8], adomian decomposition method [9], homotopy analysis method [10], and collocation method [11]. A least square finite element solution of a fractional-order two-point boundary value problems, developed in [12]. Sumudu transform method for solving fractional differential equations and fractional diffusion-wave equation as well proposed in [13]. Wavelet operational Received 2017-10-18; accepted 2017-12-16; published 2018-11-02. 2010 Mathematics Subject Classification. 34A08. Key words and phrases. fractional diffusion equation; Caputo derivative; fractional Riccati differential equation; finite difference; collocation; Chebyshev polynomials. c©2018 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 822 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-822 Int. J. Anal. Appl. 16 (6) (2018) 823 method for solving fractional partial differential equations used in [14]. Method of lines to transform the space fractional Fokker-Planck equation into a system of ordinary differential equations suggested in( [15], [16]) .The space fractional diffusion equations are solved numerically .Khader proposed Chebyshev collocation method to discretize space fractional diffusion equations to obtain a linear system of ordinary differential equations and he solved the resulting system by finite difference method [17]. Saadatmandi and et al. [18]applied Tau approach to solve space fractional diffusion equations. 2. Basic ideas and definitions Definition 2.1. The Caputo fractional derivative operator C0 D α x of order α is defined in the following form [4]: C 0 D α xf(x) = 1 Γ(m−α) ∫ x 0 f(m)(t) (x−t)α−m+1 dt, α > 0, where m− 1 < α ≤ m, m ∈ N, x > 0. Caputo fractional derivative operator is a linear operation and for the Caputo derivative we have [19]: C 0 D α xc = 0, (2.1) C 0 D α xx n =   0, n ∈ N0 and n < dαe,Γ(n+1) Γ(n+1−α)x n−α, n ∈ N0 and n ≥dαe, (2.2) where c is a constant and dαe denotes the smallest integer greater than or equal to α and N0 = {1, 2, ...}. For α ∈ N0, the Caputo differential operator coincides with the usual differential of integer order ( [19], [20], [21]). Definition 2.2. The weighted−LPnorm is defined in the following form [22]: ‖u‖Lpw(−1,1) = ( ∫ 1 −1 |u(x)|pw(x)dx)1/p for 1 ≤ p < ∞, (2.3) and we again set ‖u‖L∞w (−1,1) = sup −1≤x≤1 |u(x)| = ‖u‖L∞(−1,1). (2.4) The space of functions for which a particular norm is finite forms a Banach space, indicated byLpw(−1, 1). Int. J. Anal. Appl. 16 (6) (2018) 824 Definition 2.3. We define natural Sobolev norms as follows [22]: ‖u‖Hmw (−1,1) = ( m∑ k=0 ‖u(k)‖2L2w(−1,1)) 1/2. (2.5) The Hilbert space associated with this norm is denoted by Hmw (−1, 1). we also define the seminorms |u| H m,N w (−1,1) = ( m∑ k=min(m,N+1) ‖u(k)‖2L2w(−1,1)) 1/2. (2.6) 2.2. A review of the Chebyshev polynomials The well known Chebyshev polynomials are defined on the interval [-1, 1] as [23]: T0(z) = 1, T1(z) = z, Tn+1(z) = 2zTn(z) −Tn−1(z), n = 1, 2, ... . The analytic form of the Chebyshev polynomials Tn(z) of degree n is given by the following: Tn(z) = n [ n 2 ]∑ i=0 (−1)i2n−2i−1 (n− i− 1)! (i)!(n− 2i)! zn−2i, (2.7) where [n 2 ] denotes the integer part of n/2. The orthogonality condition is ∫ 1 −1 Ti(z)Tj(z)√ 1 −z2 dz =   π, for i = j = 0, π 2 , for i = j 6= 0, 0, for i 6= j. In order to use these polynomials on the interval x ∈ [0, 1], we define the so called shifted Chebyshev poly- nomials by introducing the change of variable z=2x-1.We denote Tn(2x− 1) by T∗n(x), defined as: T∗n(x) = n n∑ k=0 (−1)n−k 22k(n + k − 1)! (2k)!(n−k)! xk, n = 2, 3, ... , (2.8) Int. J. Anal. Appl. 16 (6) (2018) 825 where T∗0 (x) = 1 and T ∗ 1 (x) = 2x− 1. A function u(x), which is squared integrable in [0, 1], may be expressed in terms of shifted Chebyshev poly- nomials as: u(x) = ∞∑ i=0 ciT ∗ i (x), where c0 = 1 π ∫ 1 0 u(t)T∗0 (x)√ x−x2 dx, ci = 2 π ∫ 1 0 u(t)T∗i (x)√ x−x2 dx, i = 1, 2, ... . (2.9) Theorem 2.1. [19] Let u(x) be approximated by shifted Chebyshev polynomials as: um(x) = m∑ i=0 ciT ∗ i (x), (2.10) and α > 0, then Dα(um(x)) = m∑ i=dαe i∑ k=dαe ciw (α) i,k x k−α, (2.11) where w (α) i,k is given by: w (α) i,k = (−1) i−k 2 2ki(i + k − 1)!Γ(k + 1) (i−k)!(2k)!Γ(k + 1 −α) . (2.12) 3. The process of solving the space fractional diffusion equation and modified method we consider space fractional diffusion equation [17] ∂u(x,t) ∂t = d(x,t) ∂αu(x,t) ∂xα + s(x,t), a < x < b, 0 ≤ t ≤ M, 1 < α ≤ 2, (3.1) with initial condition u(x, 0) = u0(x), a < x < b, (3.2) Int. J. Anal. Appl. 16 (6) (2018) 826 and boundary conditions u(a,t) = u(b,t) = 0, (3.3) where the function s(x,t) is a source term. We use the Chebyshev collocation method to discretize 3.1 and to get a linear system of ordinary differential equations and use the finite difference method (FDM) ( [24], [25]) to solve the resulting system, and obtain the coefficients in the approximate solution. So we approximate u(x,t) as: um(x,t) = m∑ i=0 λi(t)T ∗ i (x). (3.4) From Eqs. 3.1, 3.4 and using Theorem 2.1 we have: m∑ i=0 dλi(t) dt T∗i (x) = m∑ i=dαe i∑ k=dαe λi(t)w (α) i,k x k−α + s(x,t). (3.5) Collocating, Eq. 3.5 at (m + 1 −dαe) points xp yields: m∑ i=0 dλi(t) dt T∗i (xp) = m∑ i=dαe i∑ k=dαe λi(t)w (α) i,k x k−α p + s(xp, t), P = 0, 1, ...,m−dαe. (3.6) Now we use of roots of shifted Chebyshev Polynomials T∗ m+1−dαe(x) as suitable collocation points. By substituting Eqs 3.4 and 2.11 in the boundary conditions 3.3 we get m∑ i=0 (−1)iλi(t) = 0, m∑ i=0 λi(t) = 0. (3.7) If so, dαe equations obtained from 3.7, along with m+1-dαe equations obtained from 3.6 give (m+1) ordi- nary differential equations which may be solved by using FDM, i=0,1,...,N, τ = M N , 0 ≤ ti ≤ M, ti = iτ, to get the m unknown λi, i=0,1,...,m, in various time levels tn. by determining the unknowns λi(tn) [17],the approximate m degree polynomials Different time of tn as obtained as follows: um(x,tn) = m∑ i=0 λi(tn)T ∗ i (x) = λ n oT ∗ 0 (x) + λ n 1 T ∗ 1 (x) + λ n 2 T ∗ 2 (x) + ... + λ n mT ∗ m(x) = λ́no + λ́ n 1 x + λ́ n 2 x 2 + ... + λ́nmx m, (3.8) Int. J. Anal. Appl. 16 (6) (2018) 827 in which T is the final time and λni = λi(tn). To improve the proposed method , Firstly, on average, approximate solution um(x,tn) obtained by 3.8 and the exact solution of problem 3.1,so it new approximate first stage is called and the symbol uNewapproximate(1)(x,tn) show. Namely : uNewapprox(1)(x,tn) = 1 2 [um(x,tn) + uex(x,tn)]. (3.9) Note that uex(x,tn) and uapprox(x,tn), respectively are exact solution and approximate solution of the prob- lem 3.1. At this stage , if |uNewapprox(1)(x,tn) − uex(x,tn)| to obtain,it is observed that the value of the amount |uapprox(x,tn) −uex(x,tn) is smaller. In other words, the error between the first stage approximate and exact solution of the problem, the smaller of ,the error between the approximate solution obtained from 3.8 and the exact solution problem. In the second stage,on average,approximate solution first gain and the exact solution problem and the second stage is called an New approximation solution, and the symboluNewapprox(2)(x,tn) show.Namely : uNewapprox(2)(x,tn) = 1 2 [uNewapprox(1)(x,tn) + uex(x,tn)]. (3.10) At this stage , if the value of |uNewapprox(2)(x,tn) − uex(x,tn| to determine, it will be seen that the value of the amount |uNewapprox(1)(x,tn) − uex(x,tn| is smaller. In other words,the error between, the exact solution and approximate solution of the second stage, is the first step lower. If so, this trend continue, the average, the approximate solution to the (n-1)th, with the exact solution uex(x,tn) of the problem, it will be obtained new approximate polynomial and (n)th stage new approximate polynomial is called and the symboluNewapprox(n)(x,tn) show , namely: uNewapprox(n)(x,tn) = 1 2 [uNewapprox(n−1)(x,tn) + uex(x,tn)]. (3.11) It will be seen, that the amount of |uNewapprox(n)(x,tn) − uex(x,tn| is much smaller that the amount |uapprox(x,tn)−uex(x,tn)| is.So that uapprox(x,tn) polynomial approximation to the results of the proposed method is [17].this claim with the numerical results obtained by solving the presented examples shown. In fact with this work , the numerical solution of equation3.1 is improved. The results of numerical examples , the absolute errors and the new approximation solutions for the various iterations of the improved method Int. J. Anal. Appl. 16 (6) (2018) 828 , for tables, is presented and compared by the several other numerical methods. In this work,the number of repeat procedures , with the symbol i is shown in the tables. 4. Error analysis and convergence This section is concerned with the studying of the convergence analysis and getting an upper bound for the error of the proposed formula. Theorem 4.1. [19] The error |ET (m)| = |Dαu(x) −Dαum(x)| in approximating Dαu(x) by Dαum(x) is bounded as: |ET (m)| ≤ | ∞∑ i=m+1 ci( i∑ k=dαe k−dαe∑ j=0 θi,j,k)|, (4.1) where θi,j,k = (−1)i−k2i(i+k−1)!Γ(k−α+ 1 2 ) hjΓ(k+ 1 2 )(i−k)!Γ(k−α−j+1)Γ(k+j−α+1), j = 1, 2, ... . Theorem 4.2. (Chebyshev truncation theorem) .The truncation error u(x)−uN (x), where uN (x) = ∑N k=0 ckT ∗ k (x), is the truncated Chebyshev series of u, satisfies the inequality [22]: ‖u(x) −uN (x)‖Lpw(−1,1) ≤ CN −m m∑ k=min(m,N+1) ‖u(k)‖Lpw(−1,1), for 1 ≤ p < ∞, (4.2) for all functions u whose distributional derivatives of order up to m belong to Lpw(−1, 1). C is a constant and depends on m. If so, when N →∞, we have: 0 ≤ lim N−→∞ (‖u(x) −uN (x)‖Lpw(−1,1)) ≤ lim N−→∞ (CN−m m∑ k=min(m,N+1) ‖u(k)‖Lpw(−1,1)), (4.3) In the equation 4.3, if max | ∑m k=min(m,N+1) ‖u (k)‖Lpw(−1,1)| ≤ M, weher M dimension is fixed, in the case we have: limN−→∞(CN −m ∑m k=min(m,N+1) ‖u (k)‖Lpw(−1,1)) = 0. Then, according equation 4.3, and according to the squeeze theorem, we have: limN−→∞(‖u(x) −uN (x)‖Lpw(−1,1)) = 0. The result is a convergence of approach gives us. Int. J. Anal. Appl. 16 (6) (2018) 829 Now, to discuss modified method error analysis is presented, polynomial approximations obtained 3.8 of the proposed approach [17], P0(x,tn) call. Namely: P0(x,tn) = um(x,tn) = m∑ i=0 λi(tn)T ∗ i (x). (4.4) so we have: |P0(x,tn) −uex(x,tn)| ≤ ε0, (4.5) If you put uNewapprox(1)(x,tn) = 1 2 [um(x,tn) + uex(x,tn)] = P1(tn), (4.6) we have: |P1(x,tn) −uex(x,tn)| ≤ ε1. (4.7) Considering the ties 4.5, 4.6 and 4.7, we have: |P1(x,tn) −uex(x,tn)| ≤ ε1 ⇒| 1 2 [P0(x,tn) + uex(x,tn)] −uex(x,tn)| ≤ ε1 ⇒|P0(x,tn) −uex(x,tn)| ≤ 2ε1 ≤ ε0, so the result is: ε1 ≤ ε0 2 . (4.8) For these arrangements, if uNewapprox(2)(x,tn) = 1 2 [um(x,tn) + uex(x,tn)] to P2(tn) call, you can write: |P2(x,tn) −uex(x,tn)| ≤ ε2 ⇒| 1 2 [P1(x,tn) + uex(x,tn)] −uex(x,tn)| ≤ ε2 Int. J. Anal. Appl. 16 (6) (2018) 830 ⇒|P1(x,tn) −uex(x,tn)| ≤ 2ε2 ⇒| 1 2 [P0(x,tn) + uex(x,tn)] −uex(x,tn)| ≤ 2ε2 ⇒|P0(x,tn) −uex(x,tn)| ≤ 2 × 2ε2 ≤ ε0, so the result is: ε2 ≤ ε0 22 . (4.9) By following this process, the n-th stage will be: εn ≤ ε0 2n . (4.10) In fact, if Pn(x,tn) polynomial approximation is made in step n, we get the following result: |Pn(x,tn) −uex(x,tn)| ≤ εn ≤ ε0 2n . (4.11) For 4.11, can be written: 0 ≤ lim n→∞ (|Pn(x,tn) −uex(x,tn)|) ≤ lim n→∞ ( ε0 2n ), (4.12) then, according equation 4.12, and according to the squeeze theorem, we have: lim n→∞ (|Pn(x,tn) −uex(x,tn|) = 0. The result is a convergence of approach gives us. Int. J. Anal. Appl. 16 (6) (2018) 831 Remark 1. The presented method, can be applied for solution of numerical the fractional Riccati dif- ferential equation.also Dαu(t) + u2(t) − 1 = 0, t > 0, 0 < α ≤ 1, with the initial condition u(0) = u0, in next section we illustrated this approach by example 5.1. 5. Numerical results Example 5.1. Consider the fractional Riccati differential equation of the form Dαu(t) + u2(t) − 1 = 0, t > 0, 0 < α ≤ 1, (5.1) with the initial condition u(0) = u0, (5.2) and the parameter α, refers to the fractional order of the time derivative. For α = 1; the Eq.5.1 is the standard Riccati differential equation du(t) dt + u2(t) − 1 = 0. The exact solution to this equation is u(t) = e2t − 1 e2t + 1 . Now we approximate the function u(t) by using formula ?? and its Caputo derivative Dαu(t) by using the presented formula 2.11 with m=5.Then fractional Riccati differential equation 5.1 is transformed to the fol- lowing approximated form: 5∑ i=1 i∑ k=1 ciw (α) i,k t k−α + ( 5∑ i=0 ciT ∗ i (t)) 2 − 1 = 0, (5.3) Int. J. Anal. Appl. 16 (6) (2018) 832 where w (α) i,k is defined in 2.12. Also the initial condition 5.2 is given by : 5∑ i=0 ci(T ∗ i (0)) = u0. (5.4) We now collocate Eq.5.3 at (m + 1 −dαe) points tp as: 5∑ i=1 i∑ k=1 ciw (α) i,k t k−α p + ( 5∑ i=0 ciT ∗ i (tp)) 2 − 1 = 0, p = 0, 1, 2, 3, 4. (5.5) Note that t,ps are roots of shifted Chebyshev polynomial T ∗ 5 (t), i.e. t0 = 0.5, t1 = 0.206107, t2 = 0.793893, t3 = 0.024471, t4 = 0.975528. By using Eqs.5.4 and 5.5, we obtain a system of non-linear algebraic equations which contains 6 equations for the unknowns ci, i = 0, 1, ..., 5. By solving the previous system, utilizing the Newton iteration method, we obtain the unknown ci, i = 0, 1, ..., 5, and therefore, the approximate solution is obtained via: u5(t) = 5∑ i=0 ciT ∗ i (t). (5.6) For α = 1 , and then determine the coefficients ci about5.6, polynomial approximation as follows: u5(t) = 5∑ i=0 ciT ∗ i (t) = 2.66714 × 10 −17 + 0.999372x + 0.0157609x2 − 0.41893x3 + 0.180634x4 − 0.0152477x5. (5.7) In this way, the improved method described for polynomial approximation 5.7 was used. In the table 1, 2 the numerical results and absolute error between the exact solution uex, and the approximate solution uapprox with different values of i, by means of the proposed modified method are given. Int. J. Anal. Appl. 16 (6) (2018) 833 Table 1: Comparison of absolute errors for u(x)at m=5 with different values of i for example 5.1. by modified method x i=0 i=10 i=20 i=30 i=35 |Error(0)| |Error(10)| |Error(20)| |Error(30)| |Error(35)| 0.0 2.66714×10−17 0.00000 0.00000 0.00000 0.1 2.58369×10−5 2.52314×10−8 2.46400×10−11 2.39808×10−14 6.93889×10−16 0.2 6.22951×10−5 6.08351×10−8 5.94093×10−11 5.80924×10−14 1.77636×10−16 0.3 3.25723×10−5 3.18089×10−8 3.10634×10−11 3.02536×10−14 9.43690×10−16 0.4 2.16611×10−5 2.11534×10−8 2.06575×10−11 2.02061×10−14 6.10623×10−16 0.5 4.38002×10−5 4.27737×10−8 4.17711×10−11 4.06897×10−14 1.27676×10−15 0.6 1.64887×10−5 1.61022×10−8 1.57249×10−11 1.53211×10−14 4.44089×10−16 0.7 3.04507×10−5 2.97370×10−8 2.90398×10−11 2.84217×10−14 6.66134×10−16 0.8 4.75369×10−5 4.64228×10−8 4.53346×10−11 4.44089×10−14 1.33227×10−15 0.9 1.43902×10−5 1.40529×10−8 1.37235×10−11 1.35447×10−14 4.44089×10−16 1.0 3.98000×10−6 3.88672×10−8 3.79574×10−12 3.77476×10−15 2.22045×10−16 Table 2: comparison of absolute errors for u(x)at m=5 with different values of i for example 5.1. by modified method x i=40 |Error(40)| 0.0 0.00000 0.1 0.00000 0.2 0.00000 0.3 0.00000 0.4 0.00000 0.5 0.00000 0.6 0.00000 0.7 0.00000 0.8 0.00000 0.9 0.00000 1.0 0.00000 Int. J. Anal. Appl. 16 (6) (2018) 834 Example 5.2. In this section, we consider space fractional diffusion equation3.1 with α = 1.8, of the form: ∂u(x,t) ∂t = d(x,t) ∂1.8u(x,t) ∂x1.8 + s(x,t), where, 0 < x < 1, with the diffusion coefficient: d(x,t)= Γ(1.2)x1.8, and the source function: s(x,t)=3x2(2x− 1)e−t. The initial and boundary conditions are respectively as: u(x,0)=x2(1 −x), u(0,t)=u(1,t)=0. The exact solution of this problem is u(x,t)= x2(1 −x)e−t. We apply the present method with m=3, and approximate the solution as follows: u3(x,t) = 3∑ i=0 λi(t)T ∗ i (x). (5.8) In 5.8, after determining the coefficients λi(t) for T=2 [17], Polynomial approximation is as follows. u3(x, 2) = 3∑ i=0 λi(t800)T ∗ i (x) = λ́ 800 o + λ́ 800 1 x + λ́ 800 2 x 2 + λ́8003 x 3 = − 8.673617−19 + 0.000894x + 0.134649x2 − 0.135543x3 (5.9) In Table3, the absolute error, between the exact solution uex and the approximate solution uapprox at m=3 and time step τ = 0.0025, with the final time T=2 is given. Also, In the table 4, 5 ,6 the numerical results and absolute error between the exact solution uex, and the approximate solution uapprox with different values of i, by means of the proposed modified method are given. It is notable that by considering τ = 0.0025,and using finite differential method (FDM) about 5.8 [17], we will has 800 (T τ = 2 0.0025 = 800) level time for approximate solutions u(x,tn), 0 < x < 1. In the above example all 800 values of u(x,tn) are calculated by utilizing mathematica. Example 5.3. [16] In this example, we consider the following space fractional diffusion equation ∂u(x,t) ∂t = P(x) ∂αu(x,t) ∂xα + s(x,t), 0 < x < 1 (5.10) with initial conditionu(x, 0) = x4, and boundary conditions Int. J. Anal. Appl. 16 (6) (2018) 835 Table 3: Comparison of absolute errors for u(x,2) at m=3 and T=2 for example 5.2. x Modified method Method[17] Method [26] Method [18] 0.0 2.46519×10−32 1.70849 ×10−4 4.483787×10−3 0.00 0.1 2.60209×10−18 2.10940 ×10−5 4.479660×10−3 2.89×10−5 0.2 5.20417×10−18 1.76609 ×10−4 4.201329×10−3 1.09×10−4 0.3 8.67362×10−18 3.01420 ×10−4 3.695172×10−3 2.20×10−4 0.4 1.04083×10−17 4.04138 ×10−4 3.007566×10−3 3.40×10−4 0.5 1.38778×10−17 4.89044 ×10−4 2.184889 ×10−3 4.45×10−4 0.6 2.08167×10−17 4.89044 ×10−4 1.273510 ×10−3 5.15×10−4 0.7 1.38778×10−17 5.63305 ×10−4 0.319831 ×10−3 5.27×10−4 0.8 1.38778×10−17 6.33367 ×10−4 0.629793 ×10−3 4.60×10−4 0.9 2.77556×10−17 7.05677 ×10−4 1.528978 ×10−3 2.91×10−4 1.0 0.00000 8.82821 ×10−4 2.331347 ×10−3 0.00 Table 4: Comparison of absolute errors for u(x,2)at m=3 and T=2 with different values of i for example 5.2. by modified method x i=0 i=5 i=10 i=15 i=20 |Error(0)| |Error(5)| |Error(10)| |Error(15)| |Error(20)| 0.0 8.67362×10−19 2.71051×10−20 8.47033 ×10−22 2.49698 ×10−23 8.27181×10−25 0.1 8.23560×10−5 2.57363 ×10−6 8.04258×10−8 2.51331×10−9 7.85408×10−11 0.2 1.49747×10−4 4.67958 ×10−6 1.46237×10−7 4.56991×10−9 1.42810×10−10 0.3 2.00921×10−4 6.27878×10−6 1.96212×10−7 6.13162×10−9 1.91613×10−10 0.4 2.34628×10−4 7.33213 ×10−6 2.29129×10−7 7.16028×10−9 2.23759×10−10 0.5 2.49617×10−4 7.80052×10−6 2.43766×10−7 7.61770×10−9 2.38059×10−10 0.6 2.44636×10−4 7.64488 ×10−6 2.38902×10−7 7.46570×10−9 2.33303×10−10 0.7 2.18435×10−4 6.82609×10−6 2.13315×10−7 6.66611×10−9 2.08316×10−10 0.8 1.69763×10−4 5.30508 ×10−6 1.65784×10−7 5.18075×10−9 1.61898×10−10 0.9 9.73680×10−5 3.04275×10−6 9.50859×10−8 2.97144×10−9 9.28574×10−11 1.0 2.60209×10−18 2.77556 ×10−17 0.00000 0.00000 0.00000 u(0, t) = 0,u(1, t) = e−t, where the function s(x,t) = −2e−tx4 is a source term, andP(x) = 1 24 Γ(5 −α). The exact solution to this equation is e−tx4. By applying the proposed method [17] for α = 1.2, polynomial approximation is as follows: Int. J. Anal. Appl. 16 (6) (2018) 836 Table 5: Comparison of absolute errors for u(x,2)at m=3 and T=2 with different values of i for example 5.2. by modified method x i=25 i=30 i=35 i=40 i=45 |Error(25)| |Error(30)| |Error(35)| |Error(40)| |Error(45)| 0.0 2.58494×10−26 8.07794×10−28 2.52435×10−29 7.88861×10−31 2.46519×10−32 0.1 2.45440×10−12 7.66997×10−14 2.39674×10−15 7.45931×10−17 2.60209×10−18 0.2 4.46280×10−12 1.39462×10−13 4.35763×10−15 1.35308×10−16 5.20417×10−18 0.3 5.98792×10−12 1.87123×10−13 5.84775×10−15 1.82146×10−16 8.67362×10−18 0.4 6.99246×10−12 2.18513×10−13 6.82440×10−15 2.11636×10−16 1.04083×10−17 0.5 7.43915×10−12 2.32474×10−13 7.25808×10−15 2.22045×10−16 1.38778×10−17 0.6 7.29072×10−12 2.27839×10−13 7.11237×10−15 2.22045×10−16 2.08167×10−17 0.7 6.50988×10−12 2.03434×10−13 6.35603×10−15 1.94289×10−16 1.38778×10−17 0.8 5.05931×10−12 1.58096×10−13 4.92661×10−15 1.52656×10−16 1.38778×10−17 0.9 2.90179×10−12 9.06775×10−14 2.83107×10−15 6.93889×10−17 2.77556×10−17 1.0 0.00000 0.00000 0.00000 0.00000 0.00000 Table 6: comparison of absolute errors for u(x,2)at m=3 and T=2 with different values of i for example 5.2. by modified method x i=50 |Error(50)| 0.0 0.00000 0.1 0.00000 0.2 0.00000 0.3 0.00000 0.4 0.00000 0.5 0.00000 0.6 0.00000 0.7 0.00000 0.8 0.00000 0.9 0.00000 1.0 0.00000 Int. J. Anal. Appl. 16 (6) (2018) 837 Table 7: Comparison of absolute errors for u(x,1)at m=4 and T=1 with different values of i for example 5.2. by modified method x i=0 i=10 i=20 i=30 i=40 |Error(0)| |Error(10)| |Error(20)| |Error(30)| |Error(40)| 0.0 1.38778×10−17 1.35525×10−20 1.32349 ×10−23 1.29247×10−26 1.26218×10−29 0.1 5.01756×10−3 4.89996×10−6 4.78512×10−8 4.67296×10−12 4.56345×10−15 0.2 6.38835×10−3 6.23862×10−6 6.09241×10−9 4.56991×10−12 5.81029×10−15 0.3 5.64747×10−3 5.51510×10−6 5.38584×10−9 5.94962×10−12 5.13605×10−15 0.4 3.99532×10−13 3.90167×10−6 3.81023×10−9 3.72093×10−12 3.63435×10−15 0.5 2.29764×10−3 2.24378×10−6 2.19120×10−9 2.13984×10−12 2.08776×10−15 0.6 1.08549×10−3 1.06005×10−6 1.03521×10−9 1.01094×10−12 9.84182×10−16 0.7 5.55277×10−4 5.42263×10−7 5.29540×10−10 5.17141×10−13 5.09222×10−16 0.8 5.68706×10−4 5.55377×10−7 5.42361×10−10 5.29633×10−13 5.31044×10−16 0.9 6.52824×10−4 6.37523×10−7 6.22582×10−10 6.07962×10−13 5.84742×10−16 1.0 0.00000 3.39934×10−17 0.00000×10−17 4.76800×10−17 4.18183×10−18 u4(x, 1) = 4∑ i=0 λi(t)T ∗ i (x) = 1.38778 × 10 −17 + 0.074363x− 0.274432x2 + 0.339516x3 + 0.228432x4, (5.11) note that, in this example ∆t = 0.001 is considered. Now apply improved method for polynomial approximation expression 5.11,absolute error between the exact solution, and new approximate solution obtained based on the number of repetitions of the process, shown in Tables 7 and 8. Example 5.4. [15] consider the following space fractional diffusion equation ∂u(x,t) ∂t = P(x) ∂1.5u(x,t) ∂x1.5 + s(x,t), 0 < x < 1 (5.12) with the initial condition u(x, 0) = (x2 + 1) sin(1), and boundary conditions u(0, t) sin(t + 1),u(1, t) = 2 sin(t + 1), for t > 0, the source function s(x,t) = (x2 + 1) cos(t + 1) − 2x sin(t + 1), Int. J. Anal. Appl. 16 (6) (2018) 838 Table 8: Comparison of absolute errors for u(x,1)at m=4 and T=1 with different values of i for example 5.2. by modified method x i=50 i=60 i=70 i=80 |Error(50)| |Error(60)| |Error(70)| |Error(80)| 0.0 1.23260×10−32 1.20371×10−35 1.17549 ×10−38 1.14794×10−41 0.1 4.46548×10−18 4.06965×10−21 3.97427×10−24 3.88112×10−27 0.2 5.73658×10−18 3.37867×10−21 3.29948×10−24 3.222151×10−27 0.3 5.24988×10−18 2.07294×10−21 2.02436×10−24 1.97691×10−27 0.4 4.76720×10−18 1.22852×10−20 1.19973×10−23 1.17161×10−26 0.5 3.31277×10−18 2.72581×10−20 2.66192×10−23 2.59953×10−26 0.6 4.52895×10−19 4.69916×10−20 4.58902×10−23 4.48146×10−26 0.7 3.81242×10−18 7.14857×10−20 6.98103×10−23 6.81741×10−26 0.8 3.56196×10−17 1.00740×10−19 9.83794×10−23 9.60736×10−26 0.9 2.85435×10−17 1.34756×10−19 1.31598×10−22 1.28513×10−25 1.0 1.11632×10−17 1.73532×10−19 1.69465×10−22 1.65493×10−25 andP(x) = Γ(1.5)x0.5. The exact solution of this problem is u(x,t) = (x2 + 1) sin(t + 1). By applying the proposed method [17] , polynomial approximation is as follows: u2(x, 1) = 2∑ i=0 λi(t)T ∗ i (x) = 0.909297 + 0.00049296x + 0.908804x 2, (5.13) note that, in this example ∆t = 0.001 is considered. Now apply improved method for polynomial approximation expression 5.13. Absolute error between the exact solution, and new approximate solution obtained based on the number of repetitions of the process, shown in Tables 9 and 10. 6. Conclusion In this paper, we proposed a new modified of numerical method ,based on the shifted Chebyshev collocation method and finite difference scheme, to find the solution of the space fractional diffusion equations and fractional Riccati differential equation. In this method, the fractional derivatives are described in the Caputo sense. Comparison between our proposed method and other methods , shows that this scheme is superior and evidently the error gets smaller. Int. J. Anal. Appl. 16 (6) (2018) 839 Table 9: Comparison of absolute errors for u(x,1)at m=2 and T=1 with different values of i for example 5.4. by modified method x i=0 i=10 i=20 i=30 i=38 |Error(0)| |Error(10)| |Error(20)| |Error(30)| |Error(38)| 0.0 2.22×10−18 0.00000 0.00000 0.00000 0.00000 0.1 4.43×10−5 4.33266×10−8 4.23110×10−11 4.13003×10−14 2.22054×10−16 0.2 7.89×10−5 7.70250×10−8 7.52197×10−11 7.33857×10−14 2.22045×10−16 0.3 1.03×10−4 1.01095×10−7 9.87258×10−11 9.63674×10−14 3.33067×10−16 0.4 1.18×10−4 1.15538×10−7 1.12830×10−11 1.10134×10−13 3.33067×10−16 0.5 1.23×10−4 1.20352×10−7 1.17531×10−11 1.14797×10−13 4.44089×10−16 0.6 1.18×10−4 1.15538×10−7 1.12830×10−11 1.10245×10−13 6.66134×10−16 0.7 7.89×10−4 1.01095×10−7 9.87258×10−11 9.62563×10−14 2.22045×10−16 0.8 4.43×10−4 7.70250×10−7 7.52197×10−11 7.32747×10−14 2.22045×10−16 0.9 4.43×10−16 4.33266×10−8 4.23110×10−11 4.13003×10−14 2.22045×10−16 1.0 2.22×10−16 0.00000 0.00000 0.00000 0.00000 Table10: Comparison of absolute errors for u(x,1)at m=2 and T=1 with different values of i for example 5.4. by modified method x i=40 |Error(40)| 0.0 0.00000 0.1 0.00000 0.2 0.00000 0.3 0.00000 0.4 0.00000 0.5 0.00000 0.6 0.00000 0.7 0.00000 0.8 0.00000 0.9 0.00000 1.0 0.00000 Int. J. Anal. Appl. 16 (6) (2018) 840 7. Acknowledgements It should be mentioned that the above article has been derived from Ph.D thesis, at the Islamic Azad University Central Tehran Branch. References [1] L. Bagley and P. J. Torvik, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech, 51 (19840), 294-298. [2] K. B. Oldham and J. Spanier, The Fractional Cvalculus, Academic Press, New York and London, ( 1974). [3] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley, New York, (1993). [4] S. G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, USA, (1993). [5] S. Das, Fractional Calculus for System Identification and Controls, Springer, New york, (2008). [6] H. Sweilam and M. M. Khader, A Chebyshev pseudo-spectral method for solving fractional integro-differential equations, ANZIAM J. 51 (2010), 464-475. [7] M. Inc, The approximate and exact solutions of the space-and time-fractional Burger,s equations with initial conditions by varational iteration method, J. Math. Anal. Appl. 345 (2008), 476-484. [8] N. H. Sweilam, M.M.Khader and R.F. AL-Bar, Numerical studies for a multi-order fractional differential equation, Phys. Lett. A, 371 (2007), 26-33. [9] H. Jafari and V. Daftardar-Gejji, Solving linear and nonlinear fractional diffusion and wave equations by ADM, Appl. Math. Comput, 180 (2006), 488-497. [10] I. Hashim, O. Abdulaziz and S. Momani, Homotopy analysis method for fractional IVPs, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), 674-684. [11] E. A. Rawashdeh, Numerical solution of fractional integro-differential equations by collocation method, Appl. Math. Com- put, 176 (2006), 1-6. [12] G. J. Fix, J.P. Roop,Least squares finite element solution of the fractional order two-point boundary value problem, Com- put. Math. Appl. 48 (2004), 1017-1033. [13] R. Darzi, B.Mohammadzade, S.Musavi, R.Behshti, Sumudu transform method for solving fractional differential equations and fractional diffusion-Wave equation, J. Math. Comput. Sci. 6 (2013) 79-84. [14] A. Neamaty, B. Agheli, R. Darzi, Solving fractional partial differential equation by using wavelet operational method, J. Math. Comput. Sci. 7 (2013), 230-240 . [15] F. Liu, V .Anh and I. Turner,Numerical solution of the space fractional Fokker-Plank equation, J. Comput. Appl. Math. 166 (2004), 209-219. [16] F. Liu, V. Anh, I. Turner, P. Zhuang, Time fractional advection dispersion equation, J. Appl. Math. Comput. 13 (2003), 233-245. [17] M. M. Khader, On the numerical solutions for the fractional diffusion equation, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 2535-2542. Int. J. Anal. Appl. 16 (6) (2018) 841 [18] A. Saadatmandi, M. Dehghan, A. Tau approach for solution of the space fractional diffusion equation, Comput. Math. Appl. 62 (2011), 1135-1142. [19] M. M. Khader, N. H. Swetlam and A. M. S. Mahdy, The Chebyshev collection method for solving fractional order Klein- Gordon equation, Wseas Trans. Math. 13 (2014), 2224-2880 . [20] I. Podlubny, Fractional Differential Equations, Academic Press, New York, (1999). [21] M. Joseph Kimeu, Fractional Calculus: Definitions and Applications, Western Kentucky University, (2009). [22] C. Canuto, A. Quarteroni, M. Y. Hussaini, and T. A. Zang, Spectral Methods Fundamentals in Single Domains, Springer- Verlag Berlin Heidelberg, Printed in Germany (2006). [23] M. A.Snyder, Chebyshev Methods in Numerical Approximation, Prentice-Hall, Inc. Englewood Cliffs, N. J. (1966). [24] M. M. Meerschaert and C. Tadjeran,Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math. 172 (2008), 65-77. [25] M. M. Meerschaert and C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math. 56 (2006), 80-90. [26] M. M. Khader, N. H. Sweilam and A. M. S. Mahdy, An Efficient Numerical Method for Solving the Fractional Diffusion Equation, J. Appl. Math. Bioinform. 1 (2011), 1-12. 1. Introduction 2. Basic ideas and definitions 3. The process of solving the space fractional diffusion equation and modified method 4. Error analysis and convergence 5. Numerical results 6. Conclusion 7. Acknowledgements References