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Applied Sciences
, Volume 8 (6) – Jun 12, 2018

/lp/multidisciplinary-digital-publishing-institute/a-finite-difference-method-on-non-uniform-meshes-for-time-fractional-gpwgpuvhw8

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- Multidisciplinary Digital Publishing Institute
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- © 1996-2019 MDPI (Basel, Switzerland) unless otherwise stated
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- 2076-3417
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- 10.3390/app8060960
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Article A Finite Difference Method on Non-Uniform Meshes for Time-Fractional Advection–Diffusion Equations with a Source Term ID Riccardo Fazio, Alessandra Jannelli * and Santa Agreste Department of Mathematics and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, 98166 Messina, Italy; rfazio@unime.it (R.F.); sagreste@unime.it (S.A.) * Correspondence: ajannelli@unime.it; Tel.: +39-090-676-5030 Received: 8 May 2018; Accepted: 7 June 2018; Published: 12 June 2018 Featured Application: In this paper, the authors develop an unconditionally-stable, finite difference method on non-uniform grids for solving time-fractional advection–diffusion equations with a source term involving the Caputo fractional derivative. In previous years, the interest in these equations has come from their mathematical structures and from their applications. Fractional partial differential equations (FPDEs) are applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and so on. In regard to its simplicity, the method is applicable to a wide class of fractional advection–diffusion equations in many fields of the applied sciences. Abstract: The present paper deals with the numerical solution of time-fractional advection–diffusion equations involving the Caputo derivative with a source term by means of an unconditionally-stable, implicit, ﬁnite difference method on non-uniform grids. We use a special non-uniform mesh in order to improve the numerical accuracy of the classical discrete fractional formula for the Caputo derivative. The stability and the convergence of the method are discussed. The error estimates established for a non-uniform grid and a uniform one are reported, to support the theoretical results. Numerical experiments are carried out to demonstrate the effectiveness of the method. Keywords: time-fractional advection–diffusion–reaction equation; Caputo fractional derivative; implicit ﬁnite difference method; non-uniform grid; stability; convergence 1. Introduction The fractional partial differential equations (FPDEs) have become increasingly popular in recent years. The interest in these equations comes from their mathematical structure and from their applications. FPDEs are applied in various areas of engineering, science, ﬁnance, applied mathematics, bio-engineering and so on. Several ways to solve them theoretically have been proposed [1–3], including the Green function method and Laplace and Fourier transform methods [4,5], the Adomian decomposition [6,7], and the Homotopy Perturbation methods [8,9]. Generally, numerical solution techniques are preferred when dealing with fractional models since the analytical solutions are available for a few simple cases, but also the memory effect of the problem under consideration may lead to arbitrarily wild solutions, as shown in [10]. In recent years, efﬁcient numerical methods have been developed to solve fractional differential equations, including ﬁnite difference methods [11–15], the ﬁnite volume method [16], the ﬁnite element method [17–19], and the spectral method [20,21]. Generally, Riemann–Liouville derivatives and Caputo derivatives are used for the formulation of fractional differential problems, and often, the Riemann–Liouville formula is approximated by Appl. Sci. 2018, 8, 960; doi:10.3390/app8060960 www.mdpi.com/journal/applsci Appl. Sci. 2018, 8, 960 2 of 16 the Caputo one. As for the non-fractional differential equations involving the Caputo derivative, ﬁnite difference methods are one of the most important classes of numerical methods for solving FPDEs. Zhuang and Liu [22] obtained an implicit difference approximation to solve time-fractional diffusion equations. Lin and Xu [21,23] proposed the numerical solution by ﬁnite/difference approximations for a time-fractional diffusion equation. Liu et al. [24] developed an explicit difference method and an implicit difference method for solving a space–time fractional advection dispersion equation on a ﬁnite domain. In [25–27], high order numerical difference schemes were constructed in order to solve the Caputo-type advection–diffusion equations. Zhang et al. [28] obtained a ﬁnite difference method for FPDEs involving the Caputo derivative on a non-uniform mesh. Recently Jannelli et al. [29–32] determined exact and numerical solutions for the time and space fractional advection–diffusion differential equations involving the Riemann–Liouville derivative with a non-linear source term by means of the Lie symmetries. The authors transformed the fractional partial differential equations into a fractional ordinary differential equations, written in terms of Caputo derivatives, which were then solved using implicit ﬁnite difference methods. The main goal of this paper was to construct an unconditionally-stable, implicit finite difference method for solving the time-fractional advection–diffusion equations (TFADEs) with a non-homogeneous source term involving the Caputo fractional derivative on non-uniform grids. We chose to use a special non-uniform mesh in order to improve the numerical accuracy of the classical discrete fractional formula for the Caputo derivative, since the fractional derivatives are integrals with weakly singular kernels, and the discretization on the uniform mesh may lead to poor accuracy. The consistency, stability, and convergence of the proposed difference method were investigated. Three numerical examples are given to show the reliability and efficiency of the derived difference method. 2. The Mathematical Model We considered the following linear time-fractional advection–diffusion equation a 2 ¶ ¶ ¶ u(x, t) + K u(x, t) K u(x, t) = f (x, t), a < x < b, 0 < t T, (1) 1 2 a 2 ¶t ¶x ¶x with the initial and boundary conditions given by u(x, 0) = f(x), a x b, u(a, t) = j(t), u(b, t) = y(t), 0 < t T, where u is the ﬁeld variable that can represent, for example, the solute concentration, and K and K are 1 2 the constant ﬂuid velocity and the dispersion coefﬁcient, respectively. The time fractional derivative, u(x, t) is the a order Caputo fractional derivative deﬁned by ¶t ¶ 1 ¶ u(x, t) = u(x, s)(t s) ds 0 < a < 1. (2) ¶t G(1 a) ¶s The function f (x, t) can be used to represent sources and sinks. f(x), j(t) and y(t) are known smooth functions. We took K 6= 0 and K > 0 and we assumed problem (1) had a unique and 1 2 sufﬁciently smooth solution under the above initial and boundary conditions. The fractional Equation (1) represents a suitable mathematical model for describing anomalous physical processes that exhibit fractional order behavior that varies with time or space and that cannot be modeled accurately by normal integer order equations. In fact, it has been treated by a number of authors as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns [33]. The TFADE is also used in groundwater hydrology research to model the transport of passive tracers carried by ﬂuid ﬂow Appl. Sci. 2018, 8, 960 3 of 16 in porous mediums [34] and in neurology [35]. It is notable that, when a = 1, the model (1) reduces to the classical advection–diffusion–reaction equation ¶ ¶ ¶ u(x, t) + K u(x, t) K u(x, t) = f (x, t), a x b, 0 < t T, 1 2 ¶t ¶x ¶x used in order to describe several phenomena of relevant interest in many ﬁelds of applied sciences. It is a mathematical model that describes how the concentration of the substance distributed in a medium changes under the inﬂuence of three processes: advection, diffusion and reaction. In [36–38], a fractional step approach with variable time step was used in order to solve numerically mathematical models that describe evolution problems on computational domains of one or three space dimensions. 3. Discretization in Time on a Non-Uniform Mesh The main goal of this work was to construct an unconditionally-stable implicit ﬁnite difference method deﬁned on a non-uniform mesh. In general, the existence of a weakly singular kernel (t s) , 0 < a < 1, in fractional derivatives makes it more difﬁcult to obtain a higher-order scheme. In particular, when the solutions are not suitably smooth, numerical methods on uniform meshes seem to have a poor convergent rate. For these reasons, numerical schemes on non-uniform meshes have been developed in the last years. In this section, ﬁrst, we construct the non-uniform mesh and then, in order to approximate the Caputo derivative on the non-uniform grid, we deﬁne a suitable discrete fractional derivative formula. n1 n 0 1 We divide the interval, [0, T], into N subintervals, [t , t ], for n = 1, , N and with 0 = t < t N n < t = T. We denote the time step Dt as n n n1 Dt = t t , 1 n N, and let max n min n Dt = max Dt Dt = min Dt . 1nN 1nN A sequence of mesh is described as quasi-uniform if there a ﬁnite constant, C, exists such that max min Dt /Dt C. max 1 n 1 In this case, it holds that Dt bT N . When C = 1, it holds that Dt = T N for all n = 1, , N, and the mesh is reduced to a uniform mesh. A sequence of meshes is not quasi-uniform if max min Dt /Dt ! +¥ as N ! +¥. The non-uniform mesh and quasi-uniform mesh methods have been used for solving differential equations by several authors. In [39–42], quasi-uniform meshes were used for solving a class of boundary values problems on inﬁnite domains. In this work, we are interested on the non-uniform mesh [28], which is deﬁned as follows Dt = (N + 1 n)s, 1 n N, (3) 2T n N where s = . Note that the time steps, fDt g , are a monotonically decreasing sequence, n=1 N(N + 1) 1 1 N 2 with Dt = O(N ) and Dt = O(N ). Figure 1 shows a sample of the non-uniform grid (3) obtained for N = 10. The fractional derivatives are integrals with weakly singular kernels, and it is well known that the discretization on the uniform mesh may lead to poor accuracy. We decided to use a non-uniform mesh in order to improve the accuracy of the numerical solution. Another interesting Appl. Sci. 2018, 8, 960 4 of 16 choice could be to construct non-uniform meshes that reﬁne the computational domain according to the fractional order of a or to use adaptive procedures to dynamically choose the size of the time-steps according to the local behavior of the solution. 0.5 −0.5 −1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 1. Non-uniform grid for N = 10. To motivate the choice of the non-uniform mesh (3), we discretized the Caputo derivative (2) of a function v(t) by means of the following classical approximate formula Z n d 1 n 0 n a v(t ) = v (s)(t s) ds (4) dt G(1 a) Z k n k k1 1 v(t ) v(t ) n a n = (t s) ds + R , k k1 k1 G(1 a) t t t k=1 which is the well-known, so-called L1 formula deﬁned in [2] where R is the local truncation error. The L1 formula has been used to solve the fractional differential equations with Caputo derivatives (see [22,43]). Moreover, using the relationship between the Caputo derivative and the Riemann–Liouville fractional derivative, the L1 formula was also applied to the time fractional diffusion equation with the Riemann–Liouville fractional derivative (see [44,45]). High-order approximations, such as the compact difference scheme [43,45–47] and spectral method [21,23,48] were applied to improve the spatial accuracy of fractional diffusion equations. It is important to note that it is rather difﬁcult to get a high-order time approximation due to the singularity of fractional derivatives. For any temporal mesh, for 0 < a < 1 and v(t) 2 C [0, T], it can be veriﬁed that [28] Z n Z k n k k1 t t v(t ) v(t ) 0 n a n a n v (s)(t s) ds = (t s) ds + r , k k1 k1 0 t t t k=1 where n 2 2 (Dt ) (Dt ) max n n a 00 jr j + (Dt ) max jv (t)j 1 n N. 2(1 a) 8 0tt Since " # Z n Z k n k k1 t t 1 v(t ) v(t ) n 0 n a n a R = v (s)(t s) ds (t s) ds k k1 k1 G(1 a) 0 t t t k=1 = r , G(1 a) we obtain n 2 2 1 (Dt ) (Dt ) max n n a 00 jR j + (Dt ) max jv (t)j 1 n N. G(1 a) 2(1 a) 8 0tt Appl. Sci. 2018, 8, 960 5 of 16 n n 2a For the uniform mesh, i.e., Dt = Dt for all n = 1, 2, , N, then R = O(Dt ) (see [21,49]). From the truncation error estimate of the L1 formula, it is clear that the accuracy is dependent on the fractional order, a. This is justiﬁed since a weakly singular kernel, (t s) , is contained in the integral. In order to improve the accuracy of the L1 numerical approximation of the derivative of fractional order, it is possible to consider non-uniform meshes. Numerical methods developed with the non-uniform meshes have been developed to solve weakly singular integro-differential equations. Mustapha [50], Yuste and Quintana-Murillo [51,52] proposed an implicit ﬁnite-difference time-stepping method for the discretization of the time diffusion equation. Zhang et al. [28] obtained a numerical integration formula for any a 2 (0, 1) by employing the special non-uniform grid (3) and obtained the following results for sufﬁciently smooth solutions: for the non-uniform mesh (3), for 0 < a < 1 and v(t) 2 C [0, T], it held that Z n Z k k k1 t t v(t ) v(t ) 0 n a n a n v (s)(t s) ds = (t s) ds + r 1 n N, k k1 k1 t t 0 t k=1 where 1a n 00 2a a2 jr j 1 + a + max jv (t)jT (N + 1) , 1 n N 1, 1 a 0tt and Z Z N k N k k1 t t v(t ) v(t ) 0 N a N a N v (s)(t s) ds = (t s) ds + r , k k1 k1 t t 0 t k=1 where 1 + a N 1a 00 2a 2 jr j 2 max jv (t)jT N . 1 a 0tT Then, we obtain 1a 1 2 n 00 2a a2 jR j 1 + a + max jv (t)jT (N + 1) , 1 n N 1, G(1 a) 1 a 0tt (5) 1 1 + a N 1a 00 2a 2 jR j 2 max jv (t)jT N . G(1 a) 1 a 0tT We use these estimates to study the consistency, stability and convergence of the method presented in the following sections. 4. An Implicit Finite Difference Method For the derivation of the implicit difference method, first, we constructed a computational uniform grid in the x direction, that is, the spatial size of the mesh, Dx = x x , is constant, for 1 j J, j j1 and it is non-uniform in the time direction. We deﬁned the mesh points (x , t ) with x = a + jDx, j j n n1 n n j = 0, , J and t = t + Dt , for n = 1, , N, with Dt deﬁned by (3). J and N are positive integers. We denoted the numerical approximation provided by the difference method of the exact n n n solution u(x , t ) by U at the mesh points (x , t ), for j = 0, , J and n = 0, , N. j j As usual, we supposed that the solution was sufﬁciently smooth and discretized its ﬁrst ¶/¶x and 2 2 second order ¶ /¶x spatial derivatives by means of the second order, three-point central difference formula so that Appl. Sci. 2018, 8, 960 6 of 16 n n u(x , t ) u(x , t ) j+1 j1 n 2 u(x , t ) = + O(Dx ) (6) ¶x 2Dx n n n u(x , t ) 2u(x , t ) + u(x , t ) ¶ j+1 j j1 n 2 u(x , t ) = + O(Dx ). (7) 2 2 ¶x Dx According to the discretization for the Caputo derivative (4), we approximated the time fractional derivative in (1) as follows u(x , t ) = (8) ¶t k k1 n h i u(x , t ) u(x , t ) j j n k1 1a n k 1a n (t t ) (t t ) + R , k k1 G(2 a) t t k=1 where Z k 1 1 n a n k1 1a n k 1a (t s) ds = [(t t ) (t t ) ]. k1 G(1 a) G(2 a) n n Replacing u(x , t ) with its numerical approximation, U , and neglecting the local truncation errors, the time-fractional advection–diffusion Equation (1) was discretized as follows n n n n n U U U 2U + U j+1 j1 j+1 j j1 k k1 n T (U U ) + K K = f , (9) å n,k 1 2 j j j 2 G(2 a) 2Dx Dx k=1 n n for 1 n N and 1 j J 1, where f = f (x , t ) is the source term and n k1 1a n k 1a (t t ) (t t ) T = 1 k n, 1 n N. n,k k k1 t t The initial and boundary conditions can be rewritten as U = f(x ) 0 j J n n n n U = j(t ) U = y(t ) 0 n N. (10) For any temporal meshes on [0, T] and for any n = 1, 2, , N, it holds that [22] T > 0 T > T , 1 k n. n,k n,k n,k1 n a Taking into account that T = (Dt ) , we can write n,n n n1 k k1 k k1 n a n n1 T (U U ) = T (U U ) + (Dt ) (U U ). å n,k å n,k j j j j j j k=1 k=1 Then, the following implicit ﬁnite difference scheme was obtained: n n n (K1 K2) U + (1 + 2 K2) U + (K1 K2) U (11) j1 j j+1 n1 n1 n a k k1 n = U (Dt ) T (U U ) + F , 1 n N, 1 j J 1, å n,k j j j j k=1 where we set n a n a K (Dt ) G(2 a) K (Dt ) G(2 a) 1 2 n n a n K1 = , K2 = , F = (Dt ) G(2 a) f . j j 2Dx Dx Appl. Sci. 2018, 8, 960 7 of 16 The Equation (11) can be written in vectorial form as n n1 n KU = LU + F , (12) where K is a tridiagonal matrix and where L denotes the following difference operator n1 n1 n1 n a k k1 L U = U (Dt ) T (U U ). (13) å n,k j j k=1 Here, and in the following text, we assume the convention that the summation is equal to zero if the lower bound is larger that the upper bound. The obtained method (12) is implicit. In order to compute the numerical solution, U , a system with the tridiagonal coefﬁcients matrix, K, has to be solved. It is interesting to note that the operator, L, is a kind operator with memory, due to the non-local n n character of the fractional derivative. This means that the effect on U at time t , U depends on all the 0 1 n1 0 1 n1 previous values, U , U , , U , evaluated at all the previous time, t , t , , t . The main difference compared to the non-fractional case is that, in order to evaluate L, 0 1 n1 the numerical solutions for all the n previous time values t , t , , t are required, while for n1 non-fractional equations, only the solution to the previous value t is used. The computational cost n n1 to compute the solution at the time t from the solution at the time t grows as n. That is, it grows as the number of terms in the summation that compares in the second term of the (13). This implies that 0 n 2 the computational cost goes from t to t , thus growing by n . 5. Consistency, Stability, and Convergence In this section, we discuss the consistency, the stability, and the convergence of the implicit ﬁnite difference scheme. Consistency. According to Equations (6)–(8), the local truncation error of the difference scheme (9) is n k k1 R = T (u(x , t ) u(x , t )) å n,k j j G(2 a) k=1 n n n n n u(x , t ) u(x , t ) u(x , t ) 2u(x , t ) + u(x , t ) j+1 j1 j+1 j j1 +K K f (x , t ) 1 2 2Dx Dx 1 ¶ k k1 n = T (u(x , t ) u(x , t )) u(x , t ) å n,k j j j G(2 a) ¶t k=1 n n u(x , t ) u(x , t ) j+1 j1 +K u(x , t ) 1 j 2Dx ¶x n n n u(x , t ) 2u(x , t ) + u(x , t ) j+1 j j1 K u(x , t ) 2 j 2 2 Dx ¶x a2 2 2 a2 2 = O(N ) + K O(Dx ) + K O(Dx ) = O(N + Dx ). (14) The implicit ﬁnite difference scheme deﬁned by (9) or (11) is consistent with model (1) of order a2 2 O(N + Dx ). Stability. For the stability analysis of the implicit finite difference scheme Equation (11) was rewritten as n n n (K1 K2) U + (1 + 2 K2)U + (K1 K2) U (15) j1 j j+1 n1 n a k n = (Dt ) (T T )U + F , 1 n N, 1 j J 1. å n,k+1 n,k j j k=0 Appl. Sci. 2018, 8, 960 8 of 16 Let U be another approximate solution of the difference scheme (11), and let n n n r = U U , 0 j J, 1 n N, j j j be the corresponding round-off error. We let n n n n T r = (r , r , , r ) , 0 1 J and we considered the inﬁnity norm n n n jjr jj = max jr j = jr j. j i 0j J The round-off error satisﬁed the following round-off equations n n n (K1 K2)r + (1 + 2 K2)r + (K1 K2)r (16) j1 j j+1 n1 n a k = (Dt ) (T T )r . å n,k+1 n,k k=0 In order to check whether the ﬁnite difference scheme was stable, we studied how the size of the round-off error, r , evolved over time. We deﬁned n n n n L r = (K1 K2)r + (1 + 2 K2)r + (K1 K2)r , j j1 j j+1 and n1 n1 n a k L r = (Dt ) (T T )r . (17) 2 å n,k+1 n,k j k=0 Equation (16) can be written as n n1 L r = L r . j j From (17), and taking into account that T T > 0, we obtained n,k+1 n,k n1 n1 n1 n a k n1 n a jL r j = j(Dt ) (T T )r j jr j(Dt ) (T T ), (18) 2 å n,k+1 n,k å n,k+1 n,k j j k=0 k=0 where we deﬁned n1 k jr j = max jr j. 0kn1 n1 n a n a But, since (T T ) = (Dt ) and recalling that T = (Dt ) and T = 0, n,k+1 n,k n,n n,0 k=0 n1 n1 n a k n1 jL r j = j(Dt ) (T T )r j jr j, å n,k+1 n,k j j j k=0 Thus, we concluded that n n n n n jjr jj = jr j = j(K1 K2)r + (1 + 2 K2)r + (K1 K2)r j i i i i n n1 n1 n1 = jL r j = jL r j jr j = jjr jj , 1 2 ¥ i i Appl. Sci. 2018, 8, 960 9 of 16 i.e., n 0 jjr jj jjr jj , ¥ ¥ for 1 n N. This means that the present method is unconditionally-stable. Convergence. We let u(x , t ) be the exact solution of Equation (1) at mesh point (x , t ) for j j n n n j = 0, 1, , J and n = 0, 1, , N. Denoting e = u(x , t ) U , we obtained the error equations j j n n1 n L e = L e + R , j = 0, 1, , J, n = 0, 1, , N, (19) 1 2 j j 0 n n with e = 0, for j = 0, 1, , J and e = 0, for n = 1, 2, , N. R is the local truncation error. j j We introduced the following norm n n n jje jj = max je j = je j, j i 0j J then, we obtained n n n n1 n jje jj = je j = jL e j = jL e + R j 1 2 i i i i n1 n n1 n n1 jL e j + jR j je j + jR j jje jj + R , 2 ¥ max i i i i where R = max jR j. For n = 1, we obtained max n,i 1 0 jje jj jje jj + R = R . ¥ ¥ max max Then, jje jj R , 1 n N. ¥ max To obtain the error estimates of the numerical solutions, we needed the uniform error bounds on all time levels. Then, applying the ﬁrst of the estimates (5), we were able to consider a2 2 R C (N + Dx ), 1 n N, (20) max R where C is a positive constant that is dependent on T, a and the exact solution u(x, t), but is independent of N and Dx. Thus, we proved that the solution of the difference method (11), with initial and boundary conditions given by (10), is convergent. 6. Numerical Experiments In this section, we report some numerical examples of the TFPDEs to demonstrate the accuracy and efﬁciency of the numerical method. At ﬁrst, we wanted to show that the approach proposed in this paper properly works, so we chose two examples in such a way that the exact solutions of FPDE could be evaluated analytically. This allowed us to check the accuracy and the order of convergence of the numerical solution. In both the examples, we assumed suitable smooth solutions, and we used the proposed method on the non-uniform grid and on a uniform grid. We compared the numerical results, observing that the ﬁnite difference scheme generated more accurate numerical solutions on the non-uniform grid than on the uniform one. In the third test, we solved a TFADE of physical interest with the source term chosen as a linear function of the ﬁeld variable in order to show that the method is applicable to a wide class of TFADEs. The presented problem is one of the most used mathematical models in the applied sciences. It describes how the concentration of one or more substances, chemical or biological species, distributed in a medium changes under the inﬂuence of three processes, namely, advection, diffusion and reaction. With this last test, we illustrate how the changes in the solution behavior arise when the fractional order is varied. Appl. Sci. 2018, 8, 960 10 of 16 Example 1. We consider the following TFADE a 2 ¶ ¶ ¶ u(x, t) + u(x, t) u(x, t) = f (x, t), 0 < x < 1, 0 < t T, a 2 ¶t ¶x ¶x u(x, 0) = 0, 0 x 1, (21) b b u(0, t) = t , u(1, t) = et , 0 < t T, where the source term is given by G(b + 1) x ba f (x, t) = e t . G(b + 1 a) The exact solution is x b u(x, t) = e t . In this example, we took K = K = 1 with a = 0.5 and b = 5. Figure 2 shows the comparison 1 2 n n between the numerical solution, U , and the exact solution, u(x , t ), at different times computed with N = J = 20. From Figure 2, it can be seen that the numerical solution, U , is in good agreement with the exact solution u(x , t ). The exact solution is reported with the solid line. We report only the numerical results obtained for the value parameter a = 0.5. Analogous results were obtained for 0.1 a 0.9. t= 0.5667 t=0.9 t=1 2.5 1.5 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 2. Comparison of the exact and the numerical solutions of the fractional partial differential equations (FPDE) (21) for a = 0.5 with N = J = 20 at different time steps. Solid lines: exact solution; star, circle and square points: numerical solutions. In order to investigate the temporal error and the convergence order of the numerical difference method, we deﬁned the maximum error between the exact solution, u(x , t ), and the numerical N N solution, U , at the ﬁnal time, t N N e (N, J) = max ju(x , t ) U j, (22) ¥ j 1j J n n U ,u(x ,t ) j j Appl. Sci. 2018, 8, 960 11 of 16 and the convergence order, as follows e (N, J) Order = log . (23) e (2N, J) In order to show the efﬁciency of the method, we deﬁned U as the numerical solution obtained by the implicit difference method on a uniform grid deﬁned by Dt = T/N, for n = 1, , N. We used notations similar to the (22) and (23) to deﬁne the maximum error, e ¯ (N, J), between the exact solution, N N N u(x , t ), and the numerical solution, U , at the ﬁnal time, t , and the convergence Order . In this test, we ﬁxed J = 100, a value large enough such that the spatial error is negligible as compared with the temporal error. Table 1 shows the values of e and e ¯ and the corresponding numerical convergence ¥ ¥ orders for a = 0.1, 0.5 and 0.9. It can be seen that the method is stable and convergent when solving the problem (21) on both the computational grids. By using the L1 formula on non-uniform mesh, we improvde the order of accuracy in time of the proposed method. In fact, we observe that the solutions are more accurate on the non-uniform mesh and the convergence order on the non-uniform mesh is greater than the convergence order obtained with the uniform mesh. The numerical results agree well with the theoretical results. n n Table 1. e , e ¯ and convergence orders, related to the numerical solutions U and U respectively, ¥ ¥ j j for different values of N and a, with J = 100. a N e Order e ¯ Order ¥ ¥ ¯ 4 3 10 3.6363 10 1.3741 10 5 4 20 9.1021 10 1.9982 4.3575 10 1.6570 0.1 5 4 40 2.2054 10 2.0452 1.3222 10 1.7206 6 5 80 4.4649 10 2.3043 3.8355 10 1.7854 3 2 10 5.5793 10 1.9875 10 3 3 20 1.7121 10 1.7044 7.7106 10 1.3660 0.5 4 3 40 5.4236 10 1.6584 2.8895 10 1.4160 4 3 80 1.7544 10 1.6283 1.0602 10 1.4465 2 1 10 4.6556 10 1.0115 10 2 2 20 2.1358 10 1.1242 4.9512 10 1.0307 0.9 3 2 40 9.8735 10 1.1131 2.3691 10 1.0634 3 2 80 4.5790 10 1.1085 1.1201 10 1.0807 Example 2. We consider the following TFADE a 2 ¶ ¶ ¶ u(x, t) + u(x, t) u(x, t) = f (x, t), 0 < x < 1, 0 < t T, a 2 ¶t ¶x ¶x u(x, 0) = 0, 0 x 1, (24) u(0, t) = 0, u(1, t) = t , 0 < t T, where the source term is given by 2 3a 3 f (x, t) = x t + 2t (x 1) G(4 a) and the exact solution is 2 3 u(x, t) = x t . We took K = K = 1 and set a = 0.1. Figure 3 shows the comparison between the numerical 1 2 n n solution, U , and the exact solution, u(x , t ), at different times computed with N = J = 20. The exact j Appl. Sci. 2018, 8, 960 12 of 16 solution is reported with the solid line. The numerical solution U is in good agreement with the exact solution u(x , t ). We report only the numerical results obtained for the value of the parameter, a = 0.1. Analogous results were obtained for 0.1 < a 0.9. t= 0.5667 t=0.9 0.9 t=1 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 3. Comparison of the exact and the numerical solutions of the FPDE (24) for a = 0.1 with N = J = 20 at different time steps. Solid lines: exact solution; star, circle and square points: numerical solutions. The values of e and e ¯ and the corresponding numerical convergence orders obtained for ¥ ¥ a = 0.1, 0.5 and 0.9 are reported in Table 2. The results conﬁrm that the method is stable and convergent when solving problems (24) on both the computational grids. The numerical solutions computed on non-uniform mesh were more accurate and the convergence order on the non-uniform mesh was greater than the convergence order obtained with the uniform mesh. The errors e and e ¯ satisﬁed the ¥ ¥ relationships (22). The convergence orders, Order and Order , satisﬁed the relationships (23). n n Table 2. e , e ¯ and convergence orders, related to the numerical solutions, U and U , respectively, ¥ ¥ j j for different values of N and a, with J = 100. a N e Order e ¯ Order ¥ ¥ ¯ 5 5 10 3.7836 10 9.4723 10 6 5 20 9.6542 10 1.9705 2.8745 10 1.7204 0.1 6 6 40 2.4674 10 1.9682 8.5307 10 1.7526 7 6 80 6.2843 10 1.9732 2.4911 10 1.7759 4 3 10 4.4597 10 1.3182 10 4 4 20 1.3436 10 1.7309 4.8998 10 1.4278 0.5 5 4 40 4.1920 10 1.6803 1.7898 10 1.4529 5 5 80 1.3435 10 1.6416 6.4671 10 1.4686 3 3 10 3.3229 10 6.9875 10 3 3 20 1.4844 10 1.1625 3.3290 10 1.0682 0.9 4 3 40 6.7617 10 1.1345 1.5524 10 1.0837 4 4 80 3.1119 10 1.1196 7.2847 10 1.0915 Example 3. In this test, in order to show the efﬁciency of the method, we solve the following TFADE a 2 ¶ ¶ ¶ u(x, t) + K u(x, t) K u(x, t) = f (u(x, t)), a 1 2 ¶t ¶x ¶x (25) 2 2 u(x, 0) = x (5 x) , a x b, u(a, t) = u(b, t) = 0, 0 < t T, n n U ,u(x ,t ) j j Appl. Sci. 2018, 8, 960 13 of 16 where the source term is chosen as a linear function of the ﬁeld variable f (u) = bu(x, t). The model describes the one-dimensional transport problem of a concentration, u(x, t), of a chemical or biological species in a ﬂowing medium, such as air or water. The species concentration is assumed to be horizontally and vertically well mixed, such that it varies only in the longitudinal or downstream direction. Moreover, a steady and uniform ﬂow ﬁeld is imposed and the effects of the dispersion are constant in time and space. A reaction where the transformation rate b is proportional to the species concentration was considered; according to the sign of rate, decay effects may or may not have occurred. This model has been used by several authors, because it is able to model a wide variety of physical and biological phenomena, see, for example, [53–55]. In this example, we used a = 0, b = 5, K = 1, K = 1 and b = 0.2. Figure 4 shows the solution 1 2 behavior at different times obtained for a = 0.1 and a = 0.9 and with N = J = 100. The height of the solution proﬁle decreased as the time increased. Analogous results were obtained for other values of a. Lower proﬁles were obtained for a = 0.1, and higher ones were obtained for a = 0.9 at the ﬁnal time, t = 1, that the behavior of solutions were comparable. Figure 5 shows the solution behavior with different values of a between 0 and 1 at the ﬁnal time, t = 1. Figure 5 also shows that the solution exhibited an anomalous diffusion behavior. The height of the solution proﬁle decreased when 0.1 a 0.5 and increased when 0.6 a 0.9. Thus, the solution continuously depends on the a-order of the time-fractional derivative. 40 40 t=0 t=0 t=0.3424 t= 0.3424 35 35 t=0.7374 t=0.7374 t=1 t=1 30 30 25 25 20 20 15 15 10 10 5 5 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x x Figure 4. Numerical solutions of the FPDE (25) with N = J = 100 at different time steps. Left frame: a = 0.1. Right frame a = 0.9. 30 30 = 0.1 = 0.5 = 0.2 = 0.6 25 = 0.3 25 = 0.7 = 0.4 = 0.8 = 0.5 = 0.9 20 20 15 15 10 10 5 5 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x x Figure 5. Numerical solutions of the FPDE (25) for different values of a with N = J = 100. n n U U j j n n U U j j Appl. Sci. 2018, 8, 960 14 of 16 7. Conclusions and Final Remarks In this paper, we developed an unconditionally-stable, ﬁnite difference method on a non-uniform grid for solving time-fractional advection–diffusion equations involving the Caputo fractional derivative. The Caputo time derivative was discretized by means of a direct generalization of the well-known fractional L1 formula (4) to the case of non-uniform meshes. The L1 formula takes into account the non-local character of the time-fractional operator and allows the order of accuracy in time of the proposed method to be improved by using of the non-uniform time discretization obtained by the mesh (3). We proved the stability and convergence of the proposed method. Numerical experiments were carried out to support the theoretical results. The reported numerical experiments pointed out that the difference method is more accurate on the non-uniform grid than on the uniform mesh, and the convergence order is greater on the non-uniform mesh than on the uniform mesh. Moreover, it is important to note that, in regard to its simplicity, the method is applicable to a wide class of TFADEs occurring in applied sciences. Recent convergence results [56] have shown how the grading of the mesh and the regularity of the solution affect the order of convergence of the ﬁnite difference schemes for time-fractional diffusion equations. 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**Published: ** Jun 12, 2018

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