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A finite difference scheme based on cubic trigonometric Bsplines for a time fractional diffusionwave equation
Advances in Difference Equations volumeÂ 2017, ArticleÂ number:Â 274 (2017)
Abstract
In this paper, we propose an efficient numerical scheme for the approximate solution of a time fractional diffusionwave equation with reaction term based on cubic trigonometric basis functions. The time fractional derivative is approximated by the usual finite difference formulation, and the derivative in space is discretized using cubic trigonometric Bspline functions. A stability analysis of the scheme is conducted to confirm that the scheme does not amplify errors. Computational experiments are also performed to further establish the accuracy and validity of the proposed scheme. The results obtained are compared with finite difference schemes based on the Hermite formula and radial basis functions. It is found that our numerical approach performs superior to the existing methods due to its simple implementation, straightforward interpolation and very low computational cost. A convergence analysis of the scheme is also discussed.
1 Introduction
1.1 Problem description
For \(T>0\) and \(\Omega=[a,b]\), we consider the following model of the time fractional diffusionwave equation with reaction term:
with initial conditions
and the following boundary conditions:
where a, b, \(\phi_{1}(x)\), \(\phi_{2}(x)\), \(\psi_{1}(t)\) and \(\psi_{2}(t)\) are given, \(\alpha>0\) is the reaction coefficient and \(\frac{\partial ^{\gamma}}{\partial t^{\gamma}}u(x,t)\) represents the Caputo fractional derivative of order Î³ given by [1]
To obtain the time fractional diffusionwave equation from the standard diffusion or wave equation, we replace the ordinary first or second time derivative by a fractional derivative of order Î³, where \(0<\gamma<1\) or \(1<\gamma<2\). As Î³ changes from 0 to 2, the process transforms from slow diffusion to classical diffusion and from diffusionwave to classical wave phenomenon. We consider, in this paper, the case of diffusionwave, i.e., \(1 < \gamma< 2\). It can be used to deal with viscoelastic problems and disordered media to examine structures, semiconductors and dielectrics.
1.2 Applications and literature review
The subject of fractional calculus [1â€“4] in its modern form has a history of at least three decades and has developed rapidly due to its wide range of applications in fluid mechanics, plasma physics, biology, chemistry, mechanics of material science and so on [3, 5]. Other applications include system control [1], viscoelastic flow [6], hydrology [7, 8], tumor development [9] and finance [10â€“12]. Since the fractional models in certain situations tend to behave more appropriately than the conventional integer order models, several techniques have been developed to study these models. These techniques have been continuously improved and modified to achieve more and more accuracy.
Since exact analytical solutions of only a few fractional differential equations exist, the search for approximate solutions is a concern of many recently published articles. Many research publications have been devoted to numerical techniques for solving time fractional diffusionwave equations. Zeng [13] proposed two second order stable and one conditionally stable finite difference schemes for the time fractional diffusionwave model. Using a class of finite difference methods based on the Hermite formula, Khader and Adel [14] obtained numerical solutions of a fractional diffusionwave equation. Avazzadeh et al. [15] obtained numerical solutions of a fractional diffusionwave equation by using the radial basis function method. Pskhu [16] obtained fundamental solutions of a fractional order diffusionwave equation. It has been shown that this fundamental solution gives the corresponding solutions for diffusion and wave equations when the fractional order is equal to one or approaches two. Povstenko [17] discussed Neumann boundaryvalue problems for a timefractional diffusionwave equation in a halfplane. Numerical solutions to the fractional diffusionwave equation under Dirichlet and Neumann boundary conditions were obtained by Povstenko. Liemert and Kienle [18] discussed a time fractional wavediffusion equation in an inhomogeneous halfspace. Ren and Sun [19] obtained efficient numerical solutions of the multiterm time fractional diffusionwave equation by using a compact finite difference scheme with fourthorder accuracy. Jin et al. [20] utilized a Galerkin finite element method to find approximate solution for a multiterm timefractional diffusion equation. Jianfei et al. [21] presented two efficient finite difference schemes to approximate solutions of time fractional diffusion equations. A second order BDF alternating direction implicit difference scheme for the twodimensional fractional evolution equation was discussed in [22].
An efficient numerical scheme based on trigonometric cubic B spline functions is presented in this paper to find the approximate solutions of a time fractional diffusionwave equation with reaction term. First, we discretize the Caputo time fractional derivative by the usual finite difference formula and then use trigonometric cubic Bspline basis to approximate derivatives in space. Trigonometric cubic Bspline functions provide better accuracy than the usual finite difference schemes due to their minimal support and \(C^{2}\) continuity. Numerical experiments are carried out, and the obtained results are compared with those of [14] and [15]. The comparison shows that the presented scheme has accuracy up to 10^{âˆ’11}, whereas the scheme discussed in [14] has accuracy of 10^{âˆ’5}. The scheme is shown to be unconditionally stable using a procedure similar to VonNeumann stability analysis, whereas the scheme of [14] is conditionally stable. Convergence analysis of the presented scheme is also discussed. Numerical experiments confirm the validity and efficiency of the algorithm.
The outline of this paper is as follows. In SectionÂ 2, we give temporal discretization using a forward finite difference scheme of Eq. (1). In SectionÂ 3, we present the derivation of the scheme for the fractional diffusionwave equation using the trigonometric cubic Bspline functions. The stability analysis of the proposed scheme is given in SectionÂ 4. SectionÂ 5 discusses convergence analysis of the scheme. Computational experiments are conducted to check the efficiency and validity of the scheme, and the numerical results are reported in SectionÂ 6. The last section is devoted to the concluding remarks of the study.
2 Temporal discretization
To find time discretization of Eq. (1), we discretize the Caputo time fractional derivative \(\frac{\partial^{\gamma }u(x,t)}{\partial t^{\gamma}}\) appearing in the equation using the usual finite difference method. Following the standard notations, we let \(t_{n}=n \Delta t\), \(n=0,1,\ldots M\), where \(\Delta t=\frac{T}{M}\) is the time step. First we approximate the second order differential operators using a forward finite difference method as follows:
where \(s \in[t_{n},t_{n+1}]\). Using (5), we can obtain an efficient approximation to the fractional derivative \(\frac{\partial^{\gamma}u(x,t)}{\partial t^{\gamma}}\) as follows:
where \(e_{\Delta t}^{n+1}\) is the truncation error, \(r=(t_{n+1}s)\) and \(b_{j}=(j+1)^{2\gamma}j^{2\gamma}\). The reader may verify that

\(b_{j} >0\), \(j=0,1,2,\ldots,n\),

\(1=b_{0}>b_{1}>b_{2}>\cdots>b_{n}\) and \(b_{n}\rightarrow0\) as \(n \rightarrow\infty\),

\(\sum_{j=0}^{n} (b_{j}b_{j+1})=(1b_{1})+\sum_{j=1}^{n1} (b_{j}b_{j+1})+b_{n}=1\).
Substituting (6) into (1), we obtain the following temporal discretization:
Letting \(\alpha_{0}=\frac{1}{(\Delta t)^{\gamma}\Gamma(3\gamma)}\), \(u^{n+1}=u(x,t_{n+1})\), the last equation can be rewritten as
where \(n=0,1,\ldots M\). It is observed that the term \(u^{1}\) will appear when \(n=0\) or \(j=n\). To eliminate \(u^{1}\), we utilize the initial condition to obtain
It follows then that \(u^{1}=u^{1}2 \Delta t u_{t}^{0}\) or \(u^{1}=u^{1}2 \Delta t \phi_{2}(x)\).
3 Description of the numerical scheme
In this section, we derive the cubic trigonometric Bspline collocation method (CuTBSM) for finding the numerical solution of time fractional diffusionwave equation problem (1). The solution domain \(a\leq x\leq b\) is uniformly partitioned by knots \(x_{i}\) into N subintervals \([x_{i}, x_{i+1}]\) of equal length \(h=\frac{ba}{N}\), \(i=0,1,2,\ldots,N1\), where \(a=x_{0}< x_{1}<\cdots<x_{n1}<x_{N}=b\). Our numerical approach for solving (1) using trigonometric cubic Bsplines is to seek an approximate solution \(U(x,t)\) to the exact solution \(u(x,t)\) in the following form [23, 24]:
where \(c_{i}(t)\) are to be required for the approximate solution \(U(x,t)\) to the exact solution \(u(x,t)\). The twice differentiable trigonometric basis functions \(TB^{4}_{i} (x)\) [25] at the knots \(x_{i}\) are given by
where
Since there are three nonzero terms at each knot, notably \(TB^{4}_{j1}(x)\), \(TB^{4}_{j}(x)\) and \(TB^{4}_{j+1}(x)\), therefore the approximation \(u_{j}^{n}\) at the grid point \((x_{j},t_{n})\) to the exact solution at nth time level is given as
The time dependent unknowns \(c_{j}^{n}(t)\) are to be determined by making use of the initial and boundary conditions, and the collocation conditions on \(TB^{4}_{i} (x)\). As a result, we obtain the approximations \(u_{j}^{n}\) together with their necessary derivatives as given below:
where
To obtain full discretization which relates the successive time levels and the unknowns \(c_{j}^{n+1}\), we plug in the approximations \(u_{j}^{n}\) and their derivatives (13) into Eq. (8). After some simplifications, we arrive at the following recurrence relation:
System (14) contains \((N+1)\) linear equations in \((N+3)\) unknowns. To obtain two additional equations, the boundary conditions (3) are utilized to obtain a unique solution of the problem. Consequently, a matrix system of dimension \((N + 3)\times(N + 3)\), which is a tridiagonal system, is obtained. The Thomas algorithm [26] is then used to uniquely solve this system.
3.1 Initial vector \(c^{0}\)
In order to commence the iteration process, it is required to find the initial solution vector \(c^{0}=[c_{1}^{0},c_{0}^{0}, \ldots, c_{N+1}^{0}]^{T}\). The process of finding the initial vector involves the computation of initial condition and its derivatives at the two boundaries as explained below [25]:

(i)
\((u_{j}^{0})_{x}=\frac{d}{dx}\phi_{1}(x_{j})\), \(j=0\),

(ii)
\(u_{j}^{0}=\phi_{1}(x_{j})\), \(j=0,1,\ldots N\),

(iii)
\((u_{j}^{0})_{x}=\frac{d}{dx}\phi_{1}(x_{j})\), \(j=N\).
The above tridiagonal system consists of \((N+3)\) linear equations in \((N+3)\) unknowns whose matrix form is given as
where
and \(b=[\phi_{1}'(x_{0}), \phi_{1}(x_{0}), \ldots, \phi_{1}(x_{N}), \phi_{1}'(x_{N})]^{T}\).
4 Stability analysis
By Duhamelâ€™s principle [27], it follows that the solution to an inhomogeneous problem is the superposition of the solutions to homogeneous problems. As a consequence, a scheme is stable for the inhomogeneous problem if it is stable for the homogeneous one. It is sufficient to present the stability analysis for scheme (14) for the forcefree case (\(f=0\)) only. The growth factor of a Fourier mode is assumed to be \(\rho_{j}^{n}\), and let \(\tilde{\rho}_{j}^{n}\) be its approximation. Define \(E_{j}^{n}=\rho _{j}^{n}\tilde{\rho}_{j}^{n}\) which on substitution in (14) gives the following roundoff error equation:
The error equation satisfies the boundary conditions
and the initial conditions
Define the grid function
Note that the Fourier expansion of \(E^{K}(x)\) is
where \(a^{k}(m)=\frac{1}{(ba)}\int_{a}^{b} E^{k}(x)e^{\frac{i 2 \pi m x}{(ba)}} \,dx\). Let
and introduce the norm
By Parsevalâ€™s equality, it is observed that
so that the following relation is obtained:
Suppose that Eqs. (16)(18) have a solution of the form \(E_{j}^{n}=\xi_{n} e^{i \beta j h}\), where \(i=\sqrt{1}\) and Î² is real. Substituting this expression in Eq. (16) and dividing by \(e^{i \beta j h}\), we obtain
Using the relation \(e^{i \beta h}+e^{i \beta h}=2\cos(\beta h)\) and grouping like terms, we obtain the following relation:
where \(\nu=1+\frac{2(a_{1}\alphaa_{4})\cos(\beta h)+(a_{2}\alpha a_{5})}{\alpha_{0}(2 a_{1} \cos(\beta h)+a_{5})}\). Obviously, \(\nu\geq1\).
Proposition 1
If \(\xi_{n}\) is the solution of Eq. (21), then \(\vert \xi _{n} \vert \leq2 \vert \xi_{0} \vert \), \(n=0,1,\ldots, T \times M\).
Proof
Mathematical induction is used to prove the result. For \(n=0\), we have from Eq. (21) that \(\xi_{1}=\frac{2}{\nu} \xi_{0}\). Since \(\nu\geq1\), we have
Now suppose that \(\vert \xi_{n} \vert \leq2 \vert \xi _{0} \vert \), \(n=1,\ldots,T \times M1 \), so that from (21) we obtain
â€ƒâ–¡
Theorem 1
The collocation scheme (14) is unconditionally stable.
Proof
Using formula (19) and PropositionÂ 1, we obtain
which establishes that the scheme is unconditionally stable.â€ƒâ–¡
5 Convergence analysis
First we introduce some usual notations and a lemma due to LopezMarcos [28] that play a crucial role in convergence analysis of the scheme.
Let \(\Omega_{h}=\{x_{j}\vert0 \leq i \leq N\}\) and \(\Omega_{\tau}=\{ t_{n}\vert0 \leq n \leq M\}\) be uniform partitions of the intervals \([a,b]\) and \([0,L]\), respectively, where \(x_{i}=ih\) and \(t_{n}=n \tau\) with \(\tau=\frac {T}{M}\). Let \(u_{j}^{n}\) be approximation to exact solution at the point \((x_{j},t_{n})\) and \(V=\{v_{j}\vert0\leq j\leq M\}\) and \(W=\{w_{j}\vert0\leq j\leq M\}\) be two grid functions defined on \(\Omega_{h} \). Introduce
From [28], we have the following important lemma regarding the nonnegative nature of some real quadratic forms possessing a convolution structure.
Lemma 5.1
Let \(\{w_{n}\}_{n=0}^{\infty}\) be a monotonically decreasing sequence of nonnegative real numbers with the property \(a_{n+1}+a_{n1}\geq2 a_{n}\) (\(n\geq1\)), then for any positive integer K and real vector \((V_{1},v_{2},\ldots, V_{K})\in R^{K}\), we have
Let C be a positive number which assumes different values at different locations and is independent of i, n, h and Ï„ such that
Then, for scheme (7), we have
and
where \(u(x_{j},t_{n})\) is exact and \(u_{j}^{n}\) is approximate solution at the point \((x_{j},t_{n})\) and \(f_{j}^{n+1}=f(x_{j},t_{n})\).
Theorem 2
Let \(u(x,t)\) and \(u_{i}^{n}\) be solutions of (1) and (24), respectively, and \(u(x,t)\) satisfies the smoothness condition (23), then for sufficiently small h and Ï„, it holds that
where \(e_{i}^{n+1}=u(x_{i},t^{n+1})u_{i}^{n+1}\).
Proof
To obtain the error equation, we subtract (24) from (25) to get
where \(r_{j}^{n+1}=O(\tau^{2}+\tau h^{2})\).
Multiplying both sides of (26) by \(he_{j}^{n+1}\) and summing up for j from 1 to M, we obtain
Rearranging terms, we obtain
Since \(\frac{1}{\alpha} \Vert (e^{n+1})_{x} \Vert ^{2} \geq 0\), therefore
Then
Adding up all the above inequalities gives
Using Lemma (5.1), it follows that \(\sum_{p=0}^{n}\sum_{k=0}^{p} b_{k} ( \delta^{2}e^{p+1k}, e^{p+1})\geq0\) so that we obtain from the last inequality
So
By the CauchySchwarz inequality, we obtain
Then
from where (26) can be very easily deduced.â€ƒâ–¡
6 Numerical results and discussion
In this section, numerical experiments are carried out for the time fractional diffusionwave equation (1) with initial (2) and boundary conditions (3). The efficiency and accuracy of the method are checked by calculating the error norms \(L_{2}\) and \(L_{\infty}\) given by
and
respectively. We compare the numerical solutions obtained by CuTBSM for onedimensional fractional diffusion equations (1) with known exact solutions. Numerical calculations are carried out by using Mathematica 9 on an IntelÂ®Coreâ„¢ i52410M CPU @2.30Â GHz with 8Â GB RAM and 64bit operating system (Windows 7).
Example 1
As a first experiment, we consider the following fractional diffusionwave equation for \(\alpha=0\):
where the source term is \(f(x,t)=\frac{2 t^{2\gamma}\sin(\pi x)}{\Gamma[3\gamma]}+(t^{2}t)\sin(\pi x)\pi^{2}\). The exact solution of the problem is \(u(x,t)=\sin(\pi x)(t^{2}t)\) [14].
FigureÂ 1 compares the graphs of the exact and approximate solutions with different values of Î³, h and Î”t at different time levels. The graphs show excellent agrement between the solutions. In FigureÂ 2, we exhibit the absolute error profiles at different time levels from where high accuracy of the method can be observed. FigureÂ 3 compares the graphs of the exact and approximate solutions using our scheme with those obtained in [14] at time \(t=2\). It is observed that our scheme gives much better accuracy. FigureÂ 4 shows a very close comparison of 3D plots of the exact and approximate solutions at time \(t=0.1\). In TablesÂ 12, the maximum errors obtained are compared with those of Hermite formula (HF) [14] for different values of Î³ to demonstrate that our scheme is more accurate and gives accuracy of 10^{âˆ’7}. In TablesÂ 34, error norms are computed for various values of parameters to further confirm the accuracy and efficiency of the presented scheme.
Example 2
As a second experiment, consider the time fractional diffusionwave equation
where the forcing term \(f(x,t)\) is supposed to be
The exact solution of problem (29) is \(u(x,t)=t^{2} x (1x)\).
The proposed scheme is applied to solve this problem. FigureÂ 5 shows the graphs of the exact and approximate solutions at different time levels with \(\gamma=1.5\), \(\Delta t=0.01\) and \(t=1\) for \(N=80\). An excellent agrement between the exact and approximate solutions can be observed. To exhibit the accuracy of the scheme, the absolute error profile is plotted at different time levels in FigureÂ 6 (with \(N=60\), \(\Delta t=0.001\)). The 3D plot approximate and exact solutions are shown in FigureÂ 7 by fixing values of different parameters. A tremendous similarity can be seen between the solutions. The absolute errors at different points \((x_{j},t_{j})\in[0,1]\times [0,1]\) for different values of Î³ chosen in the range \(1<\gamma \leq2\) are tabulated in TableÂ 5. From FigureÂ 7 and TableÂ 5, it is clear that the proposed scheme is very accurate and efficient. It is worthwhile to note that the numerical solutions are in excellent agreement with the exact solutions for many values of Î³.
Example 3
As the last example, consider the time fractional diffusionwave equation
where the source term is
The exact solution of problem (30) is \(u(x,t)=t^{2} \sinh(x)\) [15].
The above problem is solved by using the proposed scheme. FigureÂ 8 exhibits the graphs of the exact and approximate solutions at different time levels with \(\gamma=1.5\), \(\Delta t=0.01\) and \(t=1\) for \(N=80\). By taking \(N=40\), \(\Delta t=0.001\), the absolute error profile is plotted at different time levels in FigureÂ 9. FiguresÂ 8 and 9 show a very close comparison between the exact and approximate solutions. The 3D exact and numerical solutions are shown in FigureÂ 10 by fixing values of different parameters. In TableÂ 6, we present the absolute errors at different points \((x_{j},t_{j})\in[0,1]\times[0,1]\) for different values of Î³ chosen in the range \(1<\gamma\leq2\). The comparison between the obtained results and those of radial basis functions (RBF) [15] is given in TablesÂ 79. From FigureÂ 10 and TablesÂ 69, it is clear that the proposed scheme is very accurate and efficient. It is noticed that the numerical solutions are in close agreement with the exact solutions for many values of Î³.
7 Concluding remarks
This study presents a finite difference scheme with a combination of cubic trigonometric Bspline basis for the time fractional fractional diffusionwave equation with reaction term. This algorithm is based on a discretization using finite difference formulation for the Caputo sense. The cubic trigonometric Bspline basis functions have been used to approximate derivatives in space. The scheme provides accuracy of 10^{âˆ’11}, and the obtained numerical results are in superconformity with the exact solutions. A special attention has been given to study the stability of the scheme by using a procedure similar to Von Neumann stability analysis. The scheme is shown to be unconditionally stable, whereas the scheme of [14] is conditionally stable. A convergence analysis of the scheme is also presented.
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Yaseen, M., Abbas, M., Nazir, T. et al. A finite difference scheme based on cubic trigonometric Bsplines for a time fractional diffusionwave equation. Adv Differ Equ 2017, 274 (2017). https://doi.org/10.1186/s136620171330z
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DOI: https://doi.org/10.1186/s136620171330z