In present time significant attention has been given to study non-integer order partial differential equations. The current article is devoted to find numerical solutions to the following class of time–space fractional partial differential.
Journal of Advanced Research 25 (2020) 39–48 Contents lists available at ScienceDirect Journal of Advanced Research journal homepage: www.elsevier.com/locate/jare Stable numerical results to a class of time-space fractional partial differential equations via spectral method Kamal Shah a, Fahd Jarad b, Thabet Abdeljawad c,d,e,⇑ a Department of Mathematics, University of Malakand, Chakdara Dir (lower), Khyber Pakhtunkhawa, Pakistan Çankaya University, Department of Mathematics, 06790 Etimesgut, Ankara, Turkey c Department of Mathematics and General Sciences, Prince Sultan University, P.O Box 66833, Riyadh 11586, Saudi Arabia d Department of Medical Research, China Medical University, Taichung 40402, Taiwan e Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan b g r a p h i c a l a b s t r a c t a r t i c l e i n f o Article history: Received April 2020 Revised May 2020 Accepted 20 May 2020 Available online 19 June 2020 Keywords: Fractional partial differential equations Caputo fractional derivative Shifted Jacobin polynomials Operational matrices Numerical solution Stability a b s t r a c t In this paper, we are concerned with finding numerical solutions to the class of timespace fractional partial differential equations: Dpt ut; xị ỵ jDpx ut; xị ỵ sut; xị ẳ gt; xị; < p < 2; t; xị ẵ0; ẵ0; 1; under the initial conditions u0; xị ẳ hxị; ut 0; xị ẳ /xị; and the mixed boundary conditions ut; 0ị ẳ ux t; 0ị ẳ 0; where Dpt is the arbitrary derivative in Caputo sense of order p corresponding to the variable time t Further, Dpx is the arbitrary derivative in Caputo sense with order p corresponding to the variable space x Using shifted Jacobin polynomial basis and via some operational matrices of fractional order integration and differentiation, the considered problem is reduced to solve a system of linear equations The q Peer review under responsibility of Cairo University ⇑ Corresponding author at: Department of Mathematics and General Sciences, Prince Sultan University, P O Box 66833, Riyadh 11586, Saudi Arabia E-mail addresses: kamalshah408@gmail.com (K Shah), fahd@cankaya.edu.tr (F Jarad), tabdeljawad@psu.edu.sa (T Abdeljawad) https://doi.org/10.1016/j.jare.2020.05.022 2090-1232/Ó 2020 The Authors Published by Elsevier B.V on behalf of Cairo University This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) 40 K Shah et al / Journal of Advanced Research 25 (2020) 39–48 used method doesn’t need discretization A test problem is presented in order to validate the method Moreover, it is shown by some numerical tests that the suggested method is stable with respect to a small perturbation of the source data gðt; xÞ Further the exact and numerical solutions are compared via 3D graphs which shows that both the solutions coincides very well Ó 2020 The Authors Published by Elsevier B.V on behalf of Cairo University This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Introduction In present time significant attention has been given to study non-integer order partial differential equations In fact, it was shown that in many situations, derivatives of non-integer order are very effective for the description of many physical phenomena (see, for example [3,17,20,23]) The current article is devoted to find numerical solutions to the following class of time–space fractional partial differential equations: Dpt ut; xị ỵ jDpx ut; xị ỵ sut; xị ẳ gt; xị; < p < 2; t; xị ẵ0; ẵ0; 1; 1ị under the initial conditions u0; xị ẳ hxị; ut 0; xị ẳ /xị; 2ị and the mixed boundary conditions ut; 0ị ẳ ux t; 0ị ẳ 0; R; Dpt 3ị where ðs; jÞ R  denotes the Caputo fractional derivative of order p with respect to the variable time t; Dpx denotes the Caputo fractional derivative of order p with respect to the variable space x; ut is the derivative of u with respect to the variable time t; ux is the derivative of u with respect to the variable space x, and h; / : ½0; 1 ! R; g : ½0; 1  ½0; 1 ! R are given functions The modeling of some real world problems by using differential equations is a warm area of research in last many years Here we, remark that partial differential equations have important applications in many branches of science and engineering For instance heat transfer is a very important branch of mechanical and aerospace engineering analyses because many machines and devices in both these engineering disciplines are vulnerable to heat An engineer can predict about with possible shape changes of the plate in vibrations from the simulation results of the aforesaid equations Many engineering problems fall into such category by nature, and the use of numerical methods will to find their solutions are important for engineers In particularly time–space one dimensional equation has many applications For concerned applications detail, we recommend few article as [9–11] Conventionally, numerous techniques were developed to find approximate solutions to different classes of fractional partial differential equations such as homotopy analysis method [8], He’s variation iteration method [5], Adomian decomposition method [7], homotopy perturbation method [1], Fourier transform method [29], Laplace and natural transform methods [26,27] But all these method have their own advantages and disadvantages in application point of view For example, homotopy methods depend on a small parameter which restricted these methods Similarly the methods that are involving integral transform also are limited in applications In last few decades, some interesting numerical schemes based on radial basis functions (RBFs) and meshless techniques were introduced These methods require collocation and (RBFs) to solve fractional partial differential equations [22,28] Recently, numerical schemes based on operational matrices have attracted the attention of many researches The mentioned techniques provide highly accurate numerical solutions to both linear and nonlinear ordinary as well as partial differential equations of classical and fractional order In the mentioned schemes, some operational matrices of fractional order integration and differentiation are constructed, which play central roles to find approximate solutions for the considered problems In the most existing works, the mentioned matrices are obtained using a certain polynomial basis and a Tau-collocation method (see, for example [4,12,13,21,24,25,30]) However, in these methods, discretization is required, which needs extra memory Further for discretization and collocation extra amount of memory should be utilized To overcome this difficulty, in [14,15], the authors constructed the operational matrices without discretizing the data and omitting collocation method to compute numerical solutions for both ordinary as well as partial fractional differential equations Motivated by the above cited works, in this paper, a numerical solution to (1)–(3) is computed using shifted Jacobin polynomial basis and some operational matrices of fractional order integration and differentiation without actually discretizing the problem The Jacobi polynomials are more general polynomials and including ‘‘Legendre polynomials, Gegenbauer polynomials, Zernike and Chebyshev polynomials” as special cases The concerned polynomials have numerous applications in Quantum physics, fluid mechanics and solitary theory of waves, see detail [16] The used method reduces (1)–(3) to a system of linear algebraic equations of the form given by HTk2 A ¼ B; where the matrix H is the unknown which may be determined while the other matrices A; B are known coefficient matrices of 2 dimension k  k and  k respectively Here it is remarkable that the obtained system of algebraic equations is then solved by Gauss elimination method through Matlab for the unknown matrix H Further we demonstrate that by computational software, the solution is easily obtained up to better accuracy The computations in our work are performed using Matlab-16.‘‘The paper is organized as follows In Section 2, we recall briefly some necessary definitions and mathematical preliminaries about fractional calculus In Section 3, we recall some basic properties on Jacobi polynomials, which are required for establishing main results In Section 4, The shifted Jacobi operational matrices of fractional derivatives and fractional integrals are obtained Section is devoted to the numerical scheme, which is based on operational matrices In Section 6, numerical experiments are presented Also in the same section, we study the stability of the method with respect to a perturbation of the source data Conclusion is made in Section 7.” Basic materials Some fundamentals notions, definitions and results are recalled here from [6,17,18] Definition A real function f ðxÞ; x > is said to be in space C l ; l R, if and only if, there exists a real number m > l such that f xị ẳ xm gxị; x > 0; where g Cð0; 1Þ 41 K Shah et al / Journal of Advanced Research 25 (2020) 39–48 Definition A real function f ðxÞ; x > is said to be in space n C np ; l R; n N ¼ N [ 0, if and only if, f C l and ð-;xÞ QL;i Definition Corresponding to arbitrary order p > 0, for a function f C l ; l P À1, arbitrary order integral is recalled as Ip f xị ẳ Rx Cpị I f xị ẳ f xị: Cp ỵ c ỵ 1ị x pỵc : 4ị hi ẳ Definition Let function f C nÀ1 , then corresponding to order p ðn À 1; n; n ẳ ẵp ỵ 1, the arbitrary derivative in Caputo sense is provided by nị Ckỵ1ị pỵc C1ỵkpị x 5ị l P À1 and f C nl Then nÀ1 X xi iị f 0ỵ ị ; x P 0: i! iẳ0 -;xị nomials The famous Jacobi polynomials Pi the interval ẵ1; as ẳ 1; ẳ yị are defined over 7ị i ẳ 2; 3; ; -ỵxỵ2 y ỵ -x : 2 By means of the substitution yỵ1 ẳ Lt , we get a revised version of the concerned polynomials called the shifted Jacobi polynomials -;xị over the interval ẵ0; L A general term QL;i gested polynomials on ẵ0; L, with -;xị tị ẳ - ẳ x ẳ in (8) Ciỵ2ịC12ị 12;12ị QL;i tị, Ciỵ32ị is known as Ci ỵ 1ịCa ỵ 12ị -12;x12ị tị: QL;i Ci ỵ a ỵ 12ị tị of degree i of the sug- - > À1; x > is as: i X 1ịin Ci ỵ x þ 1ÞCði þ n þ - þ x þ 1Þ Cn ỵ x ỵ 1ịCi ỵ - ỵ x ỵ 1ịi nị!n!Ln nẳ0 tn ; C2i ỵ 1ịị! 12;12ị ðtÞ: Q ðCð2i À 1ÞÞ! L;i ðv iÞ sitting - ¼ À1 ; x ¼ 12 in (8), we have fourth kinds shifted Chebyshev polynomials as -ỵxỵ2i1ịẵ-2 x2 ỵy-ỵxỵ2i2ị-ỵxỵ2i2ị -;xị Pi1 yị 2i-ỵxỵiị-ỵxỵ2i2ị xỵi1ị-ỵxỵ2iị -;xị -iỵi1ị -ỵxỵiị-ỵxỵ2i2ị P iÀ2 ðyÞ; is called shifted Chebyshev poly- ðiiiÞ In same line one has UL;i tị ẳ VL;i tị ẳ Here we provide fundamental characteristic of the Jacobi poly- ð-;xÞ P1 yị ;1ị Ciỵ1ịC12ị QL;i2 tị, Ciỵ12ị iiị TL;i tị ẳ 6ị Derivation of Shifted Jacobi polynomials from fundamental Jacobi Polynomials -;xị P0 yị - ẳ x ẳ in (8) v ị Further if one sit - ẳ 12 ; x ¼ À1 in (8), we get third kind shifted Chebyshev polynomials as and QL;i Here for the readers we provide few special cases from shifted Jacobi basis as: GL;i tị ẳ D I f xị ẳ f ðxÞ; where ð10Þ Chebyshev polynomials of second kind when - ẳ x ẳ 12 in (8) iv ị Also if we sit - ¼ x in (8) we get shifted Gegenbauer (Ultraspherical) polynomials as p p ¼ if i – j; nomials by sitting if k P ½p þ 1; Lemma Let n À < p n; n N; ð-;xÞ Pi ðyÞ hi if i ẳ j; L-ỵxỵ1 Ci ỵ - ỵ 1ịCi ỵ x ỵ 1ị : 2i ỵ - ỵ x ỵ 1ịi!Ci ỵ - ỵ x ỵ 1ị ting where k N0 We have the following properties Ip Dp f xị ẳ f xị ẳ 0;0ị if k ẵp; D x ẳ 9ị iị LL;i tị ẳ QL;i tị is the shifted Legendre polynomials by sit- ðxÞ: For a power function for order p n 1; n; n ẳ ẵp ỵ 1, the arbitrary derivative in Caputo sense, one has ( ð-;xÞ tịdt ẳ RL;j such that Cc ỵ 1ị p k -;xị tịW L tị ẳ L tị- t x is the weight function, and ð-;xÞ RL;j and ðDp f ịxị ẳ Inp f -;xị tịQL;j -;xị Ip Iq f xị ẳ Ipỵq f xị; I x ẳ -;xị QL;i and xL For f C l ; l P À1; p; q P 0and c > À1, we have p c L 0 Ci ỵ - ỵ 1ị : C- ỵ 1ịi! Result regarding orthogonality of the said polynomials is Z ðx À lÞpÀ1 f ðlÞdl; ðLÞ ẳ 8ị W L;i tị ẳ C2i ỵ 1ịị! 12;12ị ðtÞ: Q ðCð2i À 1ÞÞ! L;i Here we claim that performing numerical computation with shifted Jacobi polynomials means that the above special cases are also considered Some time the shifted jacobi polynomials are also called hypergeometric polynomials which constitute a big class of orthogonal polynomials These polynomials are orthogonal with respect to some weight function, for more detail (see [19]) Assume that UðtÞ is a square integrable function with respect to -;xị on ẵ0; L Then it can be expressed in the weight function xL terms of shifted Jacobi polynomials as Utị ẳ X -;xị Dj QL;j tị; jẳ0 where -;xị QL;i 0ị ẳ 1ịi Ci ỵ x ỵ 1ị ; Cx ỵ 1ịi! from which the coefficients Dj can be computed easily using the orthogonality condition (9) Onward we are switching over to shifted Jacobi polynomials of two variable instead of one (see [2]) 42 K Shah et al / Journal of Advanced Research 25 (2020) 39–48 ð-;xÞ Definition Let fQL;i tịg iẳ0 be the sequence of one variable shifted Jacobi polynomials on ½0; L The notions -;xị fQL;i;j t; xịg i;jẳ0 for two variable shifted Jacobi polynomials which are defined on ½0; L  ẵ0; L by -;xị -;xị QL;i;j t; xị ẳ QL;i ð-;xÞ ðtÞQL;j ð-;xÞ weighted function ð-;xÞ ð-;xÞ ðt; xÞ ¼ W L ð-;xÞ ðtÞW L ¼ ¼ R L ð-;xÞ ð-;xÞ ð-;xÞ QL;i;j ðt; xÞQL;k;l ðt; xÞW L ðt; xÞdtdx ð-;xÞ ð-;xÞ ð-;xÞ ð-;xÞ ð-;xÞ ð-;xÞ QL;i ðtÞQL;j ðxÞQL;k ðtÞQL;l ðxÞW L ðtÞW L ðxÞdtdx 0 R L ð-;xÞ ð-;xÞ ð-;xÞ ð-;xÞ QL;i ðtÞQL;k ðtÞW ðL-;xÞ ðtÞdt  QL;j ðxÞQL;l ðxÞW Lð-;xÞ ðxÞdx -;xị RL;l ẳ 14ị where Ipt is the RiemannLiouville fractional integral of order p > with respect to the variable time t, and Mpk2 Âk2 is the square matrix of size k , given by ð-;xÞ Uðt; xị ẳ X X -;xị Di;j QL;i;j t; xị; t; xị ẵ0; L ẵ0; L; 11ị i¼0 j¼0 hi hj L r ¼ ki þ j þ 1; i; j; a; b N: Z L ð-;xÞ ð-;xÞ QL;i;j ðt; xÞUðt; xÞW L ðt; xÞdtdx: Proof Let ða; bÞ be a fixed pair of positive integers such that a; b N Then -;xị -;xị -;xị Ipt QL;a;b t; xị ẳ Ipt QL;a ðtÞ QL;b ðxÞ: On the other hand, we have where the notions Di;j are Jacobi coefficients provided by Z v ẳ ka ỵ b ỵ 1; otherwise: assume that a square integrable function Uðt; xÞ with respect to the 12ị -;xị Ipt QL;a Cn ỵ x ỵ 1ịCa ỵ - ỵ x ỵ 1ịa nị!n!Ln : From Property (4), we obtain Ipt tn ¼ kÀ1 X kÀ1 X ð-;xÞ Uðt; xÞ ’ U k ðx; yÞ ¼ Di;j QL;i;j ðt; xÞ ¼ HTk2 Uk2 ðt; xÞ; n! Cp ỵ n ỵ 1ị tnỵp ; which yields iẳ0 jẳ0 where -;xị Ipt QL;a ẳ D0;0 ; D0;1 ; ; D0;kÀ1 ; ; DkÀ1;0 ; DkÀ1;1 ; ; DkÀ1;kÀ1 Þ and -;xÞ ð-;xÞ ð-;xÞ Uk2 ðt; xÞ ¼ QðL;0;0 ðt; xÞ; QL;0;1 ðt; xÞ; ; QL;0;kÀ1 ðt; xÞ; ; T ð-;xÞ ð-;xÞ ð-;xÞ QL;kÀ1;0 ðt; xÞ; QL;kÀ1;1 ðt; xÞ; ; QL;k1;k1 t; xị : a X 1ịan Ca ỵ x ỵ 1ịCa ỵ n ỵ - ỵ x ỵ 1ị tị ẳ nẳ0 Truncated the series (11) up to their K-terms which can be expressed as: HTk2 Keeping in mind the above definitions, notions, one has the results presented here as: Mpv ;r ẳ W a;b i; jị; hi hj if i; jị ẳ k; lị; on ẵ0; L  ½0; L is expressed in terms of the weight function W L considered polynomials as Di;j ẳ 2i ỵ - ỵ x ỵ 1ịi!Lp : Ci ỵ - ỵ 1ị with where -;xị lẳ0 Cn ỵ p ỵ l ỵ x ỵ 1ị Cn ỵ p ỵ l ỵ x ỵ - ỵ 2ị Mpk2 k2 ẳ M pv ;r ị16v ;r6k2 ; -;xị -;xị RL;l ; ẳ RL;k RL;k Cl ỵ x ỵ 1ịi lị Ipt ðUk2 ðt; xÞÞ ’ M pk2 Âk2 Uk2 ðt; xÞ; t; xị ẵ0; L ẵ0; L; RL RL i X 1ịil Ci ỵ l ỵ - ỵ x ỵ 1ị Lemma From vector function given in (13) as Uk2 ðt; xÞ, we have ðxÞ; ðt; xÞ ½0; L  ½0; L: Indeed, from (9), we have RL RL Gi;j;b ẳ di;b xị; t; xị ẵ0; L ẵ0; L: The family fQL;i;j t; xịgi;jẳ0 is orthogonal with respect to the WL and a X 1ịan Ca ỵ x ỵ 1ịCa ỵ n ỵ - þ x þ 1Þ tnþp : C ðn þ x þ 1ÞCða þ - þ x þ 1Þða À nÞ!Ln Cp ỵ n ỵ 1ị nẳ0 tị ẳ Therefore, we have 13ị -;xị QL;a;b t; xị ẳ a X Da;n;p Ln nẳ0 -;xị tnỵp QL;b -;xị Approximating t nỵp QL;b Construction of required matrices corresponding to arbitrary order derivatives and integrals -;xị tnỵp QL;b xị xị: 15ị xị in terms of said polynomials, one has kÀ1 X kÀ1 X -;xị -;xị Si;j;b QL;i tịQL;j xị; 16ị iẳ0 jẳ0 Here in this part, let N ¼ f0; 1; ; k À 1g, some results are: For p > and i; j; a; b N, let dj; b ¼ if b ¼ j; if b – j where Si;j;b ¼ W a;b ði; jị ẳ where Da;n;p 1ịan Ca ỵ x ỵ 1ịCa ỵ n ỵ - ỵ x ỵ 1ị ẳ Cn þ x þ 1ÞCða þ - þ x þ 1Þða nị!Cp ỵ n ỵ 1ị Z Z L 0 L -;xị -;xị QL;i;j t; xịtnỵp QL;b -;xị xịW L ðt; xÞdtdx: On the other hand, we have and uman¼0 Da;n;p Gi;j;b ; hi hj Si;j;b ¼ hi hj Z L -;xị t nỵp QL;i -;xị ðtÞW L Z ðtÞdt L ð-;xÞ QL;j ð-;xÞ ðxÞQL;b ð-;xÞ ðxÞW L Therefore, using the orthogonality condition (9), we obtain Si;j;b ẳ dj;b hi Z L -;xị tnỵp QL;i ð-;xÞ ðtÞW L ðtÞdt : ðxÞdx 43 K Shah et al / Journal of Advanced Research 25 (2020) 3948 Further, we have Z L -;xị t nỵp QL;i -;xị tịW L tịdt ẳ i X 1ịil Ci ỵ x ỵ 1ịCi ỵ l ỵ - ỵ x ỵ 1ị lẳ0 Z Cl ỵ x ỵ 1ịCi þ - þ x þ 1Þði À lÞ!l!Ll L  t nỵpỵlỵx L tị- dt: Using the change of variable s ¼ Lt , we obtain R L nỵpỵlỵx RL -ỵ1ị1 t L tị dt ẳ Lnỵpỵlỵxỵ1 snỵpỵlỵxỵ-ỵ1ị1 sị ds ẳ Lnỵpỵlỵxỵ-ỵ1 Bn þ p þ l þ x þ 1; - þ 1Þ; where B is the beta function Next, using the property Bx; yị ẳ CxịCyị ; x > 0; y > 0; Cx ỵ yị L where Ipx is the Riemann–Liouville fractional integral of order p > with respect to the variable time x, and N pk2 Âk2 is the square matrix of size k , given by Npk2 k2 ẳ Npv ;r ị16v ;r6k2 ; v ẳ ka ỵ b ỵ 1; r ẳ ki ỵ j þ 1; i; j; a; b For p > and i; j; a; b N, let Cn ỵ p ỵ l ỵ x ỵ 1ịC- ỵ 1ị : Cn ỵ p ỵ l ỵ x ỵ - ỵ 2ị Hence, -;xị t nỵp QL;i -;xị tịW L tịdt ẳ i X 1ịil Ciỵxỵ1ịCiỵlỵ-ỵxỵ1ị lẳ0 Clỵxỵ1ịCiỵ-ỵxỵ1ịilị!l! Cnỵpỵlỵxỵ1ịC-ỵ1ị nỵpỵlỵxỵ-ỵ1 ; Cnỵpỵlỵxỵ-ỵ2ị L which yields di;b hi Si;j;b ẳ 17ị k 1: tnỵpỵlỵx L tị- dt ẳ Lnỵpỵlỵxỵ-ỵ1 Ipx Uk2 t; xịị N pk2 k2 Uk2 t; xị; t; xị ẵ0; L ẵ0; L; Npv ;r ẳ Xa;b i; jị; RL Lemma Let Uk2 ðt; xÞ be the vectorial function defined by (13) Then with we obtain Z Following the same arguments used in the proof of Lemma 2, we obtain the following result > : a X if a ẳ 0; 1; ; ẵp; if a ẳ ẵp ỵ 1; ẵp ỵ 2; ; k 1; nẳẵpỵ1 where Da;n;p ẳ 1ịan Ca ỵ x ỵ 1ịCa ỵ n ỵ - þ x þ 1Þ Cðn þ x þ 1ÞCða þ - ỵ x ỵ 1ịa nị!C1 ỵ n pị and i X 1ịil Ciỵxỵ1ịCiỵlỵ-ỵxỵ1ị lẳ0 W a; b i; jị ẳ > < Ii;j;b ẳ dj;b Clỵxỵ1ịCiỵ-ỵxỵ1ịilị!l! lẳ0 Cnỵpỵlỵxỵ1ịC-ỵ1ị nỵpỵlỵxỵ-ỵ1 : Cnỵpỵlỵxỵ-ỵ2ị L Using (10), we obtain i X 1ịil Ciỵlỵ-ỵxỵ1ị Clỵxỵ1ịilị!l! Cnpỵlỵxỵ1ịC-ỵ1ị Cnpỵlỵxỵ-ỵ2ị 2iỵ-ỵxỵ1ịi! : Ciỵ-ỵ1ịLp The following result holds Si;j;b ¼ Ln Gi;j;b : On the other hand, from (15) and (16), we obtain ð-;xÞ Ipt QL;a;b ðt; xÞ ’ a kÀ1 X kÀ1 X X -;xị -;xị Da;n;p Gi;j;b QL;i tịQL;j xị; nẳ0 iẳ0 jẳ0 that is, Dpt Uk2 t; xịị Rpk2 k2 Uk2 t; xị; t; xị ẵ0; L ẵ0; L; ð18Þ ð-;xÞ Ipt QL;a;b ðt; xÞ ’ where Rpk2 Âk2 is the square matrix of size k , given by kÀ1 X kÀ1 X -;xÞ Xa;b ði; jÞQðL;i;j t; xị; iẳ0 jẳ0 Rpk2 k2 ẳ Rpv ;r Þ16v ;r6k2 ; with which yields (16) For p > and i; j; a; b N, let Rpv ;r ẳ W a;b i; jị; v ẳ ka ỵ b ỵ 1; r ẳ ki ỵ j ỵ 1; i; j; a; b k À 1: if a ¼ i; di; a ¼ if a i and Xa;b i; jị ẳ Proof Let ða; bÞ be a fixed pair of positive integers such that a; b f0; 1; ; k À 1g Then b X Db;n;p GÃi;j;a ; -;xị -;xị -;xị Dpt QL;a;b t; xị ẳ Dpt QL;a tị QL;b xị: nẳ0 where Db;n;p ẳ Lemma Let Uk2 ðt; xÞ be the vectorial function defined by (13) Then 1ịbn Cb ỵ x ỵ 1ịCb ỵ n þ - þ x þ 1Þ Cðn þ x þ 1ịCb ỵ - ỵ x ỵ 1ịb nị!Cp ỵ n ỵ 1ị and On the other hand, we have -;xị Dpt QL;a tị ẳ a X 1ịan Ca ỵ x ỵ 1ịCa ỵ n ỵ - ỵ x ỵ 1ị nẳ0 Gi;j;a ẳ di;a j X 1ịjl Cjỵlỵ-ỵxỵ1ị lẳ0 We consider two cases Case.1 a ¼ 0; 1; ; ½p In this case, from (1), we have Clỵxỵ1ịjlị!l! Cnỵpỵlỵxỵ1ịC-ỵ1ị 2jỵ-ỵxỵ1ịj!Lp Cnỵpỵlỵxỵ-ỵ2ị Cjỵ-ỵ1ị Cn ỵ x ỵ 1ịCa ỵ - ỵ x ỵ 1ịa nị!n!Ln : Dpt tn ¼ 0; n ¼ 0; 1; 2; 3; ; a: Dpt t n : 44 K Shah et al / Journal of Advanced Research 25 (2020) 3948 Si;j;b ẳ Ln Ii;j;b : Therefore, -;xị Dpt QL;a t; xị ẳ 0: 19ị On the other hand, from (20) and (21), we obtain Case.2 a ẳ ẵp þ 1; ½p þ 2; ; k À In this case, from (1), we have Dpt QL;a;b t; xị Dpt t n ẳ 0; n ¼ 0; 1; 2; 3; ; ½p that is, -;xị -;xị Cn ỵ 1ị ẳ C1 ỵ n À pÞ t nÀp Dpt QL;a;b ðt; xÞ ’ ; n ẳ ẵp ỵ 1; ẵp ỵ 2; ; a: -;xị a X 1ịan Ca ỵ x ỵ 1ịCa ỵ n ỵ - ỵ x ỵ 1ị -;xị tnp QL;b xị: C n ỵ x ỵ 1ịCa ỵ - ỵ x ỵ 1ịa nị!C1 ỵ n pịLn nẳẵpỵ1 la;b i; jị ẳ Then, we obtain a X Da;n;Àp nÀp ð-;xÞ t QL;b ðxÞ: Ln nẳẵpỵ1 -;xị Dpt QL;a;b t; xị ẳ -;xị Approximating t nÀp QL;b ð20Þ ðxÞ ’ ðxÞ in terms of the considered polynomials, kÀ1 X kÀ1 X ð-;xÞ ð-;xÞ Si;j;b QL;i ðtÞQL;j ðxÞ; ð21Þ Z Z L L Si;j;b ¼ hi hj L ð-;xÞ ð-;xÞ ð-;xÞ ð-;xÞ ðxÞW L ðt; xÞdtdx: ð-;xÞ ðtÞW L Z L ðtÞdt ð-;xÞ ð-;xÞ ð-;xÞ QL;j ðxÞQL;b ðxÞW L ðxÞdx Due to orthogonality condition (9), one has Si;j;b ¼ dj;b hi Z L ð-;xÞ ð-;xÞ t nÀp QL;i ðtÞW L ðtÞdt : L ð-;xÞ t nÀp QL;i ð-;xÞ tịW L tịdt ẳ i X 1ịil Ci ỵ x þ 1ÞCði þ l þ - þ x þ 1Þ lẳ0 Z L t npỵlỵx L tị- dt: Upon substitution s ¼ Lt , one has L t npỵlỵx L tị- dt ẳ Lnpỵlỵxỵ-ỵ1 Cn p ỵ l ỵ x ỵ 1ịC- ỵ 1ị : Cn p ỵ l ỵ x ỵ - þ 2Þ t nÀp ð-;xÞ ð-;xÞ QL;i ðtÞW L tịdt ẳ i X 1ịil Ciỵxỵ1ịCiỵlỵ-ỵxỵ1ị lẳ0 Clỵxỵ1ịCiỵ-ỵxỵ1ịilị!l! Cnpỵlỵxỵ1ịC-ỵ1ị npỵlỵxỵ-ỵ1 ; Cnpỵlỵxỵ-ỵ2ị L 23ị where Spk2 k2 is the square matrix of size k , given by Spk2 Âk2 ¼ ðSpv ;r Þ16v ;r6k2 ; Spv ;r ¼ la;b i; jị; v ẳ ka ỵ b ỵ 1; r ẳ ki ỵ j ỵ 1; i; j; a; b k À 1: General algorithm for numerical results In this section, using the previous obtained results, the problem of finding an approximate solution to (1)–(3) is reduced to solving a certain algebraic equation Let < p < We write Dpt uðt; xÞ in the form: Dpt ut; xị ẳ HTk2 Uk2 t; xị; 24ị di;b hi HTk2 with size  k Thus one has Ipt Dpt ut; xịị ẳ HTk2 Ipt Uk2 ðt; xÞÞ: Using (6) and Lemma 4, we obtain uðt; xị ẳ u0; xị ỵ tut 0; xị ỵ HTk2 M pk2 k2 Uk2 t; xị; which yields Si;j;b ẳ Dpx ðUk2 ðt; xÞÞ ’ Spk2 Âk2 Uk2 ðt; xÞ; t; xị ẵ0; L ẵ0; L; where function vector Uk2 ðt; xÞ is given in (13) and unknown matrix Hence, RL Lemma Let Uk2 ðt; xÞ be the vectorial function defined by (13) Then Cl ỵ x þ 1ÞCði þ - þ x þ 1Þði À lÞ!l!Ll  Z Following the same arguments used in the proof of Lemma, we obtain the following result with Further, we have Z Clỵxỵ1ịjlị!l! Cnpỵlỵxỵ1ịC-ỵ1ị 2jỵ-ỵxỵ1ịj! : Cnpỵlỵxỵ-ỵ2ị Cjỵ-ỵ1ịLp QL;i;j t; xịtnp QL;b tnp QL;i j X 1ịjl Cjỵlỵ-ỵxỵ1ị lẳ0 On the other hand, we have Z Db;n;Àp Ij;i;a if b ẳ ẵp ỵ 1; ẵp ỵ 2; ; k 1; nẳẵp 1ịbn Cb ỵ x þ 1ÞCðb þ n þ - þ x þ 1Þ Cn ỵ x ỵ 1ịCb ỵ - ỵ x ỵ 1ịb nị!C1 ỵ n pị Ii;j;a ẳ di;a where hi hj > : if b ¼ 0; 1; ; ½p; b X and i¼0 j¼0 Si;j;b ¼ > < where Db;n;Àp ¼ one has ð-;xÞ ð22Þ Finally, (19) and (22) yield (18) For p > and i; j; a; b N, let Dpt QL;a;b ðt; xÞ t nÀp QL;b kÀ1 X kÀ1 X ð-;xÞ W a;b ði; jÞQL;i;j ðt; xÞ: iẳ0 jẳ0 Therefore, ẳ -;xị Da;n;p Ii;j;b QL;i;j t; xị; iẳ0 jẳ0 nẳẵpỵ1 and Dpt t n k1 X k1 X a X i X 1ịil Ciỵxỵ1ịCiỵlỵ-ỵxỵ1ị which yields from the considered initial conditions (2) lẳ0 ut; xị ẳ hxị þ t/ðxÞ þ HTk2 M pk2 Âk2 Uk2 ðt; xÞ: Clỵxỵ1ịCiỵ-ỵxỵ1ịilị!l! Cnpỵlỵxỵ1ịC-ỵ1ị npỵlỵxỵ-ỵ1 : Cnpỵlỵxỵ-ỵ2ị L Using (10), we obtain On the other hand, using (11), we may write hðxÞ þ t/ðxÞ in the form: 45 K Shah et al / Journal of Advanced Research 25 (2020) 3948 hxị ỵ t/xị ẳ Z Tk2 Mk2 ; gt; xị ẳ Z Tk2 where is a matrix of size  k The coefficients of the matrix can be computed using (29) Therefore, we obtain ut; xị ẳ HTk2 M pk2 k2 ỵ Z Tk2 ịUk2 t; xị: Z Tk2 25ị gt; xị ẳ Q Tk2 Uk2 t; xị; 26ị where Q Tk2 is a matrix of size  k that can be computed using (29) Now, using (1), (24), (25), and (26), we obtain i 1h Dpx ut; xị ẳ Q Tk2 Uk2 t; xị j HTk2 M pk2 k2 ỵ Z Tk2 Uk2 ðt; xÞ À HTk2 Uk2 ðt; xÞ ; s that is, 1h s i Q Tk2 À j HTk2 Mpk2 k2 ỵ Z Tk2 HTk2 Uk2 t; xị; Next, we obtain Ipx Dpx ut; xịị ẳ 1h s ð32Þ The exact solution (29)–(31) is given by uà t; xị ẳ t2 x2 ; t; xị ẵ0; ẵ0; 1: For t; xị ẵ0; ẵ0; 1, we denote by Et; xị the absolute error at the point ðt; xÞ, that is, Similarly, we may write gðt; xÞ in the form: Dpx uðt; xÞ ẳ p p x2 t ỵ t xị; t; xị ẵ0; ẵ0; 1: C1:5ị i Q Tk2 j HTk2 Mpk2 k2 ỵ Z Tk2 À HTk2 Ipx ðUk2 ðt; xÞÞ Using (6) and Lemma 3, we obtain ut; xị ẳ ut; 0ị þ ux ðt; 0Þx i 1h þ Q Tk2 j HTk2 M pk2 k2 ỵ Z Tk2 HTk2 Npk2 k2 Uk2 t; xị; Et; xị ẳ ju t; xị ut; xịj; t; xị ẵ0; 1  ½0; 1: The absolute errors at different points t; xị in the case k ẳ and -; xị ẳ 0; 0ị are shown in Table The absolute errors at different points ðt; xÞ in the case k ẳ and -; xị ẳ 0:5; 1ịare shown in Table Observe that in both cases, at every pair of point ðt; xÞ, the computed approximate solution is equal to the exact solution with a negligible amount of absolute error Next, we fix -; x; kị ẳ 0; 0; 4Þ, we compare our result with the exact solution at some fixed values of t, i.e t ¼ 0:1; t ¼ 0:25; t ¼ 0:5; t ¼ 0:75, and display the result in Fig We repeat the same experience with -; x; kị ẳ 0; 0:1; 4ị As it is shown by Fig 2, the obtained result is satisfactory Now, in order to check the stability of the approximated solution, a perturbation term is introduced in the source function gðt; xÞ More precisely, we consider problem (29)–(31) with the perturbed source g ðt; xÞ given by g t; xị ẳ gt; xị ỵ tx; t; xị ẵ0; ẵ0; 1; s 33ị which yields from the boundary conditions (3) ut; xị ẳ 1h s i Q Tk2 À j HTk2 M pk2 k2 ỵ Z Tk2 HTk2 Npk2 k2 Uk2 t; xÞ: ð27Þ Using (25) and (27), by identification, we obtain HTk2 M pk2 k2 ỵ Z Tk2 ẳ 1h s i Q Tk2 À j HTk2 M pk2 k2 ỵ Z Tk2 HTk2 Npk2 k2 ; which yields the algebraic equation HTk2 A ẳ B; 28ị where A is the square matrix of order k given by A ẳ M pk2 k2 ỵ jM pk2 k2 ỵ Ik2 k2 ịNpk2 k2 s and B is the matrix of size  k given by B¼ s ðQ Tk2 À jZ Tk2 ÞNpk2 Âk2 À Z Tk2 : Table Absolute errors in the case -; x; kị ẳ 0; 0; 4ị (0.25, (0.25, (0.25, (0.25, (0.50, (0.50, (0.50, (0.50, (0.75, (0.75, (0.75, (0.75, (1.00, (1.00, (1.00, (1.00, ðt; xÞ uà ðt; xÞ 0.25) 0.50) 0.75) 1.00) 0.25) 0.50) 0.75) 1.00) 0.25) 0.50) 0.75) 1.00) 0.25) 0.50) 0.75) 1.00) 0:00390625 0:01562500 0:03515625 0:06250000 0:01562500 0:06250000 0:14062500 0:25000000 0:03515625 0:14062500 0:31640625 0:56250000 0:06250000 0:25000000 0:56250000 1:00000000 uðt; xÞ Eðt; xÞ 0:00385100 0:01548100 0:03481800 0:06174500 0:01548000 0:06256000 0:14080000 0:24943000 0:03482000 0:14080000 0:31733000 0:56302000 0:06170000 0:24940000 0:56300000 1:00110000 0:00005525 0:00014400 0:00033825 0:00075500 0:00014500 0:00006000 0:00017500 0:00057000 0:00033625 0:00017500 0:00092375 0:00052000 0:00080000 0:00060000 0:00050000 0:00110000 Here, Ik2 Âk2 denotes the identity matrix of size k The algebraic Eq 2 (28) is equivalent to a system of k linear equations with k variables, which can be solved using Matlab Finally, after solving (28), the numerical solution to (1)–(3) can be computed using (25) Numerical experiments This portion is devoted to present a test problem Therefore, consider the given problem as 1:5 D1:5 t uðt; xị ỵ Dx ut; xị ẳ gt; xị; t; xị ẵ0; ẵ0; 1; 29ị under the initial conditions u0; xị ẳ ut 0; xị ẳ 30ị and the mixed boundary conditions ut; 0ị ẳ ux t; 0ị ẳ 0; where the source term gt; xị is given by ð31Þ Table Absolute errors in the case -; x; kị ẳ 0:5; 1; 4ị t; xị u ðt; xÞ (0.25, 0.25) (0.25, 0.50) (0.25, 0.75) (0.25, 1.00) (0.50, 0.25) (0.50, 0.50) (0.50, 0.75) (0.50, 1.00) (0.75, 0.25) (0.75, 0.5) (0.75, 0.75) (0.75, 1.00) (1.00, 0.25) (1.00, 0.50) (1.00, 0.75) (1.00, 1.00) 0:00390625 0:01562500 0:03515625 0:06250000 0:01562500 0:06250000 0:14062500 0:25000000 0:03515625 0:14062500 0:31640625 0:56250000 0:06250000 0:25000000 0:56250000 1:00000000 uðt; xÞ Eðt; xÞ 0:00422600 0:01579700 0:03562900 0:06510300 0:01580000 0:06279000 0:14089000 0:24954000 0:03563000 0:14089000 0:31685000 0:56231000 0:06510000 0:24950000 0:56230000 1:00930000 0:00031975 0:00017200 0:00047275 0:00260300 0:00017500 0:00029000 0:00026500 0:00046000 0:00047375 0:00026500 0:00044375 0:00019000 0:00260000 0:00050000 0:00020000 0:00930000 46 K Shah et al / Journal of Advanced Research 25 (2020) 39–48 Fig Exact and approximate solutions at different values of t that is ðt ¼ 0:1; t ¼ 0:25; t ¼ 0:5; t ẳ 0:75ị in the case -; x; kị ¼ ð0; 0; 4Þ Fig Exact and approximate solutions at different values of t that is ðt ¼ 0:1; t ẳ 0:25; t ẳ 0:5; t ẳ 0:75ị in the case -; x; kị ẳ 0; 0:1; 4ị where > We denote by u the numerical solution of the perturbed problem For t; xị ẵ0; ẵ0; 1, we denote by E t; xị the absolute error at the point ðt; xÞ, that is E t; xị ẳ ju t; xị u t; xịj; t; xị ẵ0; ẵ0; 1; where u is the approximate solution without noise (the approximate solution for ẳ 0ị The absolute errors E t; xị for ẳ 0; 0:1 at different points t; xị in the case k ẳ and -; xị ẳ 0; 0Þ are shown in Table The Table Absolute errors in the case -; x; k; ị ẳ 0; 0; 4; 0:01Þ (0.25, (0.25, (0.25, (0.25, (0.50, (0.50, (0.50, (0.50, (0.75, (0.75, (0.75, (0.75, (1.00, (1.00, (1.00, (1.00, ðt; xÞ uà ðt; xÞ 0.25) 0.50) 0.75) 1.00) 0.25) 0.50) 0.75) 1.00) 0.25) 0.50) 0.75) 1.00) 0.25) 0.50) 0.75) 1.00) 0:00385100 0:01548100 0:03481800 0:06174500 0:01548000 0:06256000 0:14080000 0:24943000 0:03482000 0:14080000 0:31733000 0:56302000 0:06170000 0:24940000 0:56300000 1:00110000 absolute errors E ðt; xị for ẳ 0; 0:05 at different points t; xị in the case k ẳ and -; xị ¼ ð0; 0Þ are shown in Table We observe from Tables and that at almost every pair of points ðt; xÞ, we have E ðt; xÞ < , which confirms the stability of the method with respect to a perturbation of the source data Graphical presentations are given in Fig for exact and approximate solutions at different values of t that is ðt ¼ 0:1; t ¼ 0:25; t ¼ 0:5; t ¼ 0:75Þ in the case -; x; kị ẳ 0; 0; 4ị Similarly in Fig 2, the exact and approximate solutions at different values of t that is ðt ¼ 0:1; t ¼ 0:25; t ẳ 0:5; t ẳ 0:75ị in the case Table Absolute errors in the case ð-; x; k; Þ ¼ ð0; 0; 4; 0:05Þ u ðt; xÞ E ðt; xÞ 0:00390000 0:01550000 0:03490000 0:06170000 0:01550000 0:06280000 0:14130000 0:24990000 0:03490000 0:14130000 0:31850000 0:56480000 0:06170000 0:24990000 0:56480000 1:00540000 0:00000490 0:00001900 0:00008200 0:00004500 0:00002000 0:00024000 0:00050000 0:00047000 0:00008000 0:00050000 0:00117000 0:00178000 0:00000000 0:00050000 0:00180000 0:00430000 (0.25, (0.25, (0.25, (0.25, (0.5, (0.50, (0.50, (0.50, (0.75, (0.75, (0.75, (0.75, (1.00, (1.00, (1.00, (1.00, ðt; xÞ uà ðt; xÞ 0.25) 0.50) 0.75) 1.00) 0.25) 0.50) 0.75) 1.00) 0.25) 0.50) 0.75) 1.00) 0.25) 0.50) 0.75) 1.00) 0:00385100 0:01548100 0:03481800 0:06174500 0:01548000 0:06256000 0:14080000 0:24943000 0:03482000 0:14080000 0:31733000 0:56302000 0:06170000 0:24940000 0:56300000 1:00110000 u ðt; xÞ E ðt; xÞ 0:00400000 0:01570000 0:03500000 0:06180000 0:01570000 0:06390000 0:14320000 0:25170000 0:03500000 0:14320000 0:32310000 0:57200000 0:06180000 0:25170000 0:57200000 1:02260000 0:00014900 0:00021900 0:00018200 0:00005500 0:00022000 0:00134000 0:00240000 0:00227000 0:00018000 0:00240000 0:00577000 0:00898000 0:00010000 0:00230000 0:00900000 0:02150000 K Shah et al / Journal of Advanced Research 25 (2020) 39–48 47 Fig Comparison between exact and approximate solutions over the square ðt; xÞ ½0; 1  ½0; 1 and taking ð-; x; kị ẳ 0; 0:1; 4ị -; x; kị ẳ 0; 0:1; 4Þ are presented In both cases the effect of time and the parameters values have testified At taking ð-; xị ẳ 0; 0ị for parameters, we get the solution more precise as compare to -; xị ẳ 0; 0:1ị at same scale k ¼ Further for more explanation, we give comparison between exact and approximate solution in Fig by using -; xị ẳ 0; 0:1ị at same scale k ¼ 4, to the given problem We see that both surfaces coincide very well which illustrate the accuracy of the considered method Conclusion The suggested method provides an easy way to solve numerically the class of fractional partial differential Eqs (1)–(3) Using shifted Jacobi polynomial basis, the considered problem is reduced to a system of linear algebraic equations which has been solved by Matlab using Gauss elimination method for the unknown coefficient matrix which then used to obtained the required numerical solution of the considered problem Moreover, from numerical experiments, we observed that the method is stable with respect to a perturbation of the source data In future, the method can be easily extended to solve other types of fractional partial differential equations from physics and other fields of science Compliance with Ethics Requirements Our research work does not contain any studies with human or animal subjects Declaration of Competing Interest The authors declare that there are no conflicts of interest regarding the publication of this paper Acknowledgments We are thankful to the reviewer for their nice suggestions which improved this paper very well References [1] Akbarzade M, Langari J Application of Homotopy perturbation method and variational iteration method to three dimensional diffusion problem Int J Math Anal 2011;5:871–80 [2] Borhanifar A, Sadri Kh Numerical solution for systems of two dimensional integral equations by using Jacobi operational collocation method Sohag J Math 2014;1:15–26 [3] Debanth L Recents applications of fractional calculus to science and engineering Int J Math Sci 2003;54:3413–42 [4] Doha EH, Bhrawy AH, Baleanu D, Ezz-Eldien SS The operational matrix formulation of the Jacobi Tau approximation for space fractional diffusion equation Adv Differ Equns 2014;231:1687–847 [5] Deghan M, Yousefi YA, Lotfi A The use of Hes variational iteration method for solving the telegraph and fractional telegraph equations Comm Numer Method Eng 2011;27:219–31 [6] Hilfer R Applications of fractional calculus in physics Singapore: World Scientifc Publishing Company; 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