Numerical study of time-fractional hyperbolic partial differential equations

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1 Available online at wwwisr-publicationscom/jmcs J Math Computer Sci, 7 7, Research Article Journal Homepage: wwwtjmcscom - wwwisr-publicationscom/jmcs Numerical study of time-fractional hyperbolic partial differential equations Saima Arshed Department of Mathematics, University of the Punjab, Lahore 5459, Pakistan Abstract In this study, a numerical scheme is developed for the solution of time-fractional hyperbolic partial differential equation In the proposed scheme, cubic B-spline collocation is used for space discretization and time discretization is obtained by using central difference formula Caputo fractional derivative is used for time-fractional derivative The stability and convergence of the developed scheme, have also been proved The numerical examples support the theoretical results c 7 All rights reserved Keywords: Time-fractional hyperbolic equation, cubic B-spline, collocation method, convergence analysis, stability analysis MSC: 35R, 74H5 Introduction The idea of differentiation and integration to non-integer order has been studied in the fractional calculus The subject fractional calculus is the generalized form of classical calculus This subject is as old as conventional calculus, but researchers did not pay attention towards this subject, for so many years In recent decades, it has been observed by many mathematicians, scientists and researchers that fractional derivatives and integrals play a significant role in describing the properties of physical phenomena Fractional derivatives and integrals efficiently describe the memory and hereditary properties of different processes and materials It has further been observed that fractional models are more efficient and accurate than already developed classical models The mathematical modeling of real world problems such as earthquake, traffic flow, fluid flow, signal processing, viscoelastic problems give rise to fractional partial differential equations FPDEs Sometimes it is hard to calculate the exact solutions of fractional partial differential equations by using analytical methods Due to this reason, mathematicians are motivated to develop numerical methods for the numerical solution of fractional PDEs The numerical methods for solving fractional partial differential equations have become popular among the researchers in the last ten years A variety of numerical methods have been used for finding the solution of FPDEs, such as the finite difference method, finite element method, finite volume method and the collocation method etc In this article, a numerical solution for time-fractional hyperbolic partial differential equation has been proposed address: saimamath@puedupk Saima Arshed doi:436/jmcs75 Received 6-4-7

2 S Arshed, J Math Computer Sci, 7 7, The considered equation has the following form with initial conditions and boundary conditions t α a u t ru u fx, t, x Ω [, L], < t T, ux, g x, x Ω, ux, t g x, u, t, ul, t, t [, T], where < α < is a parameter describing the order of the fractional time and a, r are constants g x and g x are given smooth functions and fx, t is the source term The time-fractional derivative of order α has been considered in the Caputo form, defined as { t α t Γ α ux,s ds, < α <, t s α s u t, α For a >, r >, is a fractional telegraph equation The study of flow of electric signals in transmission line and wave propagation phenomena gives rise to telegraph equation It is also used for modeling reaction diffusion equation For a >, r, takes the following form t α a u t u fx, t, which represents a fractional damped wave equation For a, r, becomes t α u fx, t, which represents a fractional wave equation For a, r >, becomes t α ru u fx, t, which represents a fractional linear Klein-Gordon equation The equation occurs in relativistic physics and is used to describe dispersive wave phenomena The numerical solution of partial differential equations with sub and super diffusion is proposed by Li et al [6] Wei et al [8] proposed the numerical solution of fractional telegraph equation Chen et al [3] presented the approximate and exact solutions of the fractional diffusion-wave equation Srivastava et al [4] developed the method for solving telegraph equation Sweilam and Assiri [5] presented finite difference scheme for the solution of nonlinear wave equation with fractional derivative The numerical solution of fractional convection-diffusion equations is proposed by Zhang [9] by using finite difference method Ding and Li [4] proposed a numerical algorithm for the fractional diffusion-wave equation Siddiqi and Arshed [] presented a numerical solution of convection-diffusion integro-differential equations with a weakly singular kernel Tadjeran et al [6] developed the solution of fractional diffusion equation Liu et al [9] provided the stability and convergence of the difference schemes for the fractional advection-diffusion equation Sousa [3] proposed an approximation of the Caputo fractional derivative of order < α Liu et al [8] developed a numerical method for solving the time fractional advectiondispersion equation Zhang and Han [3] proposed a quasi-wavelet method for time-dependent fractional partial differential equation Lynch et al [] proposed a numerical methods for the solution of partial differential equations of fractional order In [8], Baskonus et al used active control method for fractional

3 S Arshed, J Math Computer Sci, 7 7, order economic system Lin and Xu [7] developed the approximate solution of time-fractional diffusion equation Baskonus and Bulut [6] developed the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton Method Bulut et al [] proposed some new analytical solutions for the nonlinear time-fractional KdV-Burgers-Kuramoto equation Baskonus et al [5] used variational iteration method for the solutions of nonlinear fractional differential equations systems Atangana [ 3] proposed the stability and a convergence analysis of fractional differential equations Baskonus and Bulut [7] developed the complex analytical solutions for the fractional nonlinear double sinh-poisson Equation Bulut et al [] proposed modified trial equation method for fractional wave equation and time-fractional generalized Burgers equation In [9], Bulut et al developed the analytical solutions of some fractional ordinary differential equations by Sumudu Transform Method In [4], Atangana et al used the results from fractional calculus for the modeling of nonlinear wave motion In [7], Demiray et al used generalized Kudryashov method for solving time fractional differential equations In [], Bulut et al developed exact solutions of time fractional KdV equations by using generalized Kudryashov method In [] finite difference approximations are used for fractional advection-dispersion flow equations In [5] finite central difference/finite element approximations are applied for parabolic integro-differential equations The paper is divided into six sections The cubic B-spline basis function is presented in Section Section 3, presents the finite difference approximation for time discretization of the given problem The stability and convergence of temporal discretization are established Section 4, presents cubic B-spline collocation method for the space discretization Approximate solutions that are discussed in Section 5, support the theoretical results The concluding remarks are given in Section 6 Cubic B-spline The domain interval [, ] has been partitioned into N elements by having uniform step size h with knots x i, i,,,, N such that { x < x < x < < x N }, h x i x i, i,,, N The cubic B-spline basis function B i x, i,,, N are defined as B i x h 3 x x i 3, x [x i, x i ], h 3 3h x x i 3hx x i 3x x i 3, x [x i, x i ], h 3 3h x i x 3hx i x 3x i x 3, x [x i, x i ], x i x 3, x [x i, x i ],, otherwise The values of successive derivatives B r i x, i,, N, r,, at nodes, are listed in Table Table : Coefficients of cubic B-spline and its derivatives at knots x i x i x i x i x i x i else B i x 4 B i x 3 3 h h B i x 6 h h 6 h The approximate solution can be written as a linear combination of cubic B-spline basis functions B i x as U n x N i c i tb i x, where c i are time dependent unknown quantities to be computed from the boundary conditions and collocation form of the fractional differential equation

4 S Arshed, J Math Computer Sci, 7 7, Discretization in time The time-fractional derivative in is defined in Caputo form Let t n n t, n,,,, K, where t T K is the temporal step size and T is the final time Caputo fractional derivative is discretized by using the central difference approximation as x, t n t α Γ α n Γ α Γ α Γ α Γ α Γ3 α Γ3 α tn j n j n j n j n j n j ux, s ds s t n s α, tj ux, s t j s ds t n s α, ux, t j ux, t j ux, t j t ux, t j ux, t j ux, t j t tj t j tn j t n j ux, t n j ux, t n j ux, t n j t ds rn t n s α t, dτ rn τα t, tj t j dτ rn τα t, ux, t n j ux, t n j ux, t n j t α j α j α r n t, b j ux, t n j ux, t n j ux, t n j t α r n t, where b j j α j α and τ t n s In particular, for j, denote u ux, tu x, g x tg x Rewrite 3 in operator form as Then 3 leads to L α t ux, t n : t α Γ α Γ3 α n j tj n j ux, s t j s b j ux, t n j ux, t n j ux, t n j t α 3 ds t n s α Lα t ux, t n r n t 3 The second order approximation of Caputo derivative as discussed in [3] is given as r n c t, 33 t n s α, the finite differ- where c is a constant By using L α t ux, t n as an approximation of α u t α Γ α ence scheme for has the form L α t ux, t n a ux, t n t t tn ux,s s ds rux, t n ux, t n fx, t n 34 By applying the θ-weighted scheme to the space derivative at two adjacent time levels, the above equation takes the following form ux, L α tn ux, t n t ux, t n a θ {rux, t n } θ {rux, t n } t

5 S Arshed, J Math Computer Sci, 7 7, By taking θ, we get { } ux, t n θ θ { ux, t n ux, L α tn ux, t n t ux, t n a t uxx x, t n u xx x, t n r fx, t n } fx, t n ux, tn ux, t n By inserting the value of L α t ux, t n, we have Γ3 α n ux, t n j ux, t n j ux, t n j ux, tn ux, t n b j t α a t j ux, tn ux, t n r uxx x, t n u xx x, t n fx, t n After simplification, the above scheme takes the following form aα t rα u n α un xx b n u b n b n u n j bj b j b j u n j b b u n rα un α un xx aα t un α f n, n, where u n x ux, t n and α Γ3 α t α For n, the above scheme 35 becomes aα t rα u α u xx b u b b u b b u aα t u α u xx rα u α f For n, the above scheme can be rewritten as aα t α r u α u xx Equations 37 and 36, with the boundary conditions aα t u α u xx α f α r u u n u n L, n, and initial conditions represent the time-discrete problem ux, g x, x Ω, ux, t g x,

6 S Arshed, J Math Computer Sci, 7 7, The error term r n is defined as r n : α ux, t n t α L α t ux, t n By using 3, r n r n c t 38 t To check the stability and convergence of the time-discrete problem, the inner product and norms are defined as H Ω { f H Ω, f Ω }, f f, g f, g, g, f f, f /, f f, f / In this paper, it is preferred to apply instead of H -norm, defined as f f A f /, 39 where A α β rα, β aα t The weak formulation of 37 and 36 has the following form, ie, by finding u n H Ω, such that for all v H Ω, u, v A u xx, v B C u, v 4B ra u, v and where A u xx, v A f, v, u n, v A u n xx, v Bb n u, v B b n b n u, v A Discrete Gronwall Inequality B α β rα, B n j bj b j b j u n j, v B b u n, v ra u n, v A u n xx, v C u n, v A f n, v, β rα, C If the sequences {a j } and {z j }, j,,, n, satisfy inequality j z j a i z i b, j,,, n, where a j, b, then the inequality i z j b exp j β β rα 3 3 a i, j,,, n 3 i

7 S Arshed, J Math Computer Sci, 7 7, Stability and convergence analysis The stability of time-discrete problem is discussed in the following theorem Theorem 4 The proposed time-discrete scheme 36 is stable, if for all t >, the following relation holds n D g t g, n,,,, K, 4 where is defined in 39 Proof For n, let v u in 3, it can be written as u, u A u xx, u B C u, u 4B ra u, u A u xx, u By using integration by parts and applying the boundary conditions, the above equation becomes u, u u A, u B C u, u 4B ra u, u A u x, u x By using v v, v and Schwarz inequality, the above equation takes the following form B u C u 4B u u ra u, B C B C, A v, 4 since B ± C, the above equation can be rewritten as 3 43 By using the inequality u g t g, 43 becomes u D g t g 44 Assume that the above relation holds for v u j, ie, j D g t g, j, 3,, n 45 For v u n in 3, we have u n, u n u n A, u n Bb n u, u n B b n b n u, u n B n j bj b j b j u n j, u n B b u n, u n ra u n, u n u n A, un C u n, u n By using integration by parts and applying boundary conditions, the above equation takes the following

8 S Arshed, J Math Computer Sci, 7 7, form u n, u n A u n, un Bb n u, u n B b n b n u, u n B By using Schwartz inequality, it can be written as n j bj b j b j u n j, u n B b u n, u n C u n, u n u n A, un n Bb n u u n B b n b n u n B n j ra u n, u n bj b j b j u n j u n B b u n u n C n u n u n u n ra u n u n By using induction hypothesis and discrete Gronwall inequality 3, we have n g t g exp B b n b n C ra D g t g Theorem 4 The numerical solution obtained by the proposed method converges to the exact solution, if the following relation holds ut n u n E t, n,,, K 46 Proof Let the error term is defined as e n ux, t n u n x, for n, the error equation can be written as e, v e A, v B C e, v 4B ra e, v e A, v r, v, for all v H Ω Let v e, noting e yields e, e e A, e B C e, e r, e, v H Ω, e B C e e r e, v H Ω, e B C e r, v H Ω Since B C < and e t, this together with 38, gives Hence 46 is proved for the case n For inductive part, suppose 46 holds for n,, 3,, s, ie, t u E t E t 47 ut n u n E t 48

9 S Arshed, J Math Computer Sci, 7 7, To prove the lemma for n s, equations and 3 are used and the error equation can be written, for all v H Ω, as e n, e n e n A, en Bb n e, e n B b n b n e, e n B n j bj b j b j e n j, e n B b e n, e n e n A C e n, e n r n, e n,, en ra e n, e n e n Bb n e B n j bj b j b j e n j B b e n e n ra e n C e n r n By using induction hypothesis and discrete Gronwall inequality 3, we have e n Bb n e r n where E is a constant exp B n j bj b j b j B b ra C Bb n t t exp B b n b n C ra, tn u n E t, 5 Space discretization using collocation method Cubic B-spline collocation method is used for spatial discretization Substituting into 36, we have the following equation aα t rα b n c i 4c i n j rα c n i 4cn i c n α i 6 c n h i cn i c n i c i bn b n c i 4c i c i c n j i bj b j b j c n j i 4cn j i c n i 4c n i cn i α b b c n i 4cn i cn i 6 c n h i c n i aα cn i c n i t 4cn i c n i α f n i After simplification, the above equation represents the linear system of N equations in N 3 unknowns c n, cn, c n,, cn n, cn N aα t rα α 6 h c n i 4 4aα t 4rα 6α h c n i aα t rα α 6 h c n i F i, n, i,,,, N,

10 S Arshed, J Math Computer Sci, 7 7, where F i b n c i 4c i n j rα c i bn b n c i 4c i c i c n j i bj b j b j c n j i 4cn j i c n i 4c n i cn i α b b c n i 4cn i cn i 6 c n h i c n i aα cn i c n i t 4cn i To find unique solution of this system, eliminate c n and c n N using boundary conditions u, t c 4c c, ul, t c N 4c N c N, c 4c c, c N 4c N c N c n i α f n i After eliminating c and c N, the system is reduced to a tri-diagonal system of N linear equations in N unknowns This system can be rewritten in matrix form as AC n F, n,, 3,, K, 5 where C n [ c n, c n,, cn n ] T, n,, 3,, K The coefficient matrix A is given as below where A x y x x y x x y x x aα t rα 6α h y 4 aα t rα, 6α h, To calculate C [ c, c,, T N] c, first we need to calculate C [ c, c,, T N] c C is evaluated, solving 37, as aα t rα α 6 h c i 4 4aα t 4rα 6α h c i where F i aα t α aα t rα α c i 4c i c i 6 h c i c i c i 6 h c i F i, α r c i 4c i i c α f i

11 S Arshed, J Math Computer Sci, 7 7, Numerical results Following three problems have been considered to check the accuracy and efficiency of the developed method The main purpose of these examples is to verify the convergence rate of the numerical results with respect to the fractional order α From the following Tables, 3 and 4, it is clear that the numerical results are highly consistent with the theoretical results It can be shown that the convergence rate of numerical solutions is of order as t approaches zero Example 6 Following is the time-fractional telegraph equation with the initial conditions and the boundary conditions The exact solution of the problem is 5 u u t5 t u u fx, t, x [, ], < t T, 6 ux, sin πx, x [, ], u, t u, t, < t T ux, t t sin πx The maximum norm errors for different time steps t with N 5 and the corresponding rates of convergence at T are presented in Table From Table, it is easy to see that the temporal convergence rate of the numerical results obtained by the proposed method is consistent with the theoretical results Table : Temporal convergence rate at T t e K Rate Example 6 Following is the time-fractional wave equation with the initial condition and boundary conditions The exact solution of the problem is t α u fx, t, x [, L], < t T, 6 ux, sin πx, x [, ], u, t u, t, < t T ux, t t sin πx The Maximum norm errors for different time steps t with N 5 and the corresponding rates of convergence at T are presented in Table 3 From Table 3, it is easy to see that the temporal convergence rate of the numerical results obtained by the proposed method is in good agreement with the theoretical estimation

12 S Arshed, J Math Computer Sci, 7 7, Table 3: Temporal convergence rate at T t e K Rate Example 63 Following is the time-fractional linear Klein Gordon equation t α u u fx, t, x [, L], < t T, 63 with the initial condition ux, π 5 sin πx π 5 cos πx cos 3πx, x [, ], π5 and boundary conditions u, t u, t, < t T The exact solution of the problem is ux, t t π 5 sin πx π 5 cos πx π 5 cos 3πx The Maximum norm errors for different time steps t with N 5 and the corresponding rates of convergence at T are presented in Table 4 From Table 4, it is easy to see that the temporal convergence rate of the numerical results obtained by the proposed method is in good agreement with the theoretical estimation Table 4: Temporal convergence rate at T t e K Rate Conclusion The solution of time-fractional hyperbolic partial differential equation has been developed Central difference formula and cubic B-spline are used for discretizing time and space variables respectively The proposed method is stable and the approximate results approach the analytical results with order O t The approximate results obtained by the suggested method support the theoretical estimates References [] A Atangana, Convergence and stability analysis of a novel iteration method for fractional biological population equation, Neural Comput Appl, 5 4, 3 [] A Atangana, On the stability and convergence of the time-fractional variable order telegraph equation, J Comput Phys, 93 5, 4 4 [3] A Atangana, On the stability of iteration methods for special solution of time-fractional generalized nonlinear ZK-BBM equation, J Vib Control, 6,

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