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A finite difference scheme based on cubic trigonometric Bsplines for a time fractional diffusionwave equation
 Muhammad Yaseen^{1},
 Muhammad Abbas^{1}Email authorView ORCID ID profile,
 Tahir Nazir^{1} and
 Dumitru Baleanu^{2}
https://doi.org/10.1186/s136620171330z
© The Author(s) 2017
 Received: 11 June 2017
 Accepted: 26 August 2017
 Published: 8 September 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.
Keywords
 time fractional diffusionwave equation
 trigonometric basis functions
 cubic trigonometric Bsplines method
 stability
1 Introduction
1.1 Problem description
1.2 Applications and literature review
The subject of fractional calculus [1–4] 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 [10–12]. 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^{−11}, whereas the scheme discussed in [14] has accuracy of 10^{−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 Section 2, we give temporal discretization using a forward finite difference scheme of Eq. (1). In Section 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 Section 4. Section 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 Section 6. The last section is devoted to the concluding remarks of the study.
2 Temporal discretization

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

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

\(\sum_{j=0}^{n} (b_{j}b_{j+1})=(1b_{1})+\sum_{j=1}^{n1} (b_{j}b_{j+1})+b_{n}=1\).
3 Description of the numerical scheme
3.1 Initial vector \(c^{0}\)
 (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)
\((u_{j}^{0})_{x}=\frac{d}{dx}\phi_{1}(x_{j})\), \(j=N\).
4 Stability analysis
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
Theorem 1
The collocation scheme (14) is unconditionally stable.
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.
Lemma 5.1
Theorem 2
Proof
6 Numerical results and discussion
Example 1
The maximum error for Example 1 at different values of h , Δ t for \(\pmb{\gamma=1.5}\) and \(\pmb{t=0.2}\)
h  \(\boldsymbol{\frac{1}{5}}\)  \(\boldsymbol{\frac{1}{10}}\)  \(\boldsymbol{\frac{1}{20}}\)  \(\boldsymbol{\frac{1}{30}}\)  \(\boldsymbol{\frac{1}{30}}\)  \(\boldsymbol{\frac{1}{40}}\)  \(\boldsymbol{\frac{1}{40}}\)  \(\boldsymbol{\frac{1}{45}}\) 

Δ t  \(\boldsymbol{\frac{1}{50}}\)  \(\boldsymbol{\frac{1}{100}}\)  \(\boldsymbol{\frac{1}{150}}\)  \(\boldsymbol{\frac{1}{150}}\)  \(\boldsymbol{\frac{1}{200}}\)  \(\boldsymbol{\frac{1}{200}}\)  \(\boldsymbol{\frac{1}{210}}\)  \(\boldsymbol{\frac{1}{220}}\) 
HF [14]  0.01149  0.00361  0.00120  0.00115  0.00021  0.00019  0.00006  0.00004 
Present method  6.410E − 04  8.203E − 05  1.027E − 05  3.049E − 06  3.035E − 06  1.281E − 06  1.280E − 06  8.989E − 07 
The maximum error for Example 1 at different values of h , Δ t for \(\pmb{\gamma=1.7}\) and \(\pmb{t=0.4}\)
h  \(\boldsymbol{\frac{1}{10}}\)  \(\boldsymbol{\frac{1}{20}}\)  \(\boldsymbol{\frac{1}{30}}\)  \(\boldsymbol{\frac{1}{50}}\)  \(\boldsymbol{\frac{1}{50}}\)  \(\boldsymbol{\frac{1}{60}}\)  \(\boldsymbol{\frac{1}{60}}\)  \(\boldsymbol{\frac{1}{70}}\) 

Δ t  \(\boldsymbol{\frac{1}{50}}\)  \(\boldsymbol{\frac{1}{100}}\)  \(\boldsymbol{\frac{1}{200}}\)  \(\boldsymbol{\frac{1}{250}}\)  \(\boldsymbol{\frac{1}{300}}\)  \(\boldsymbol{\frac{1}{400}}\)  \(\boldsymbol{\frac{1}{450}}\)  \(\boldsymbol{\frac{1}{480}}\) 
HF [14]  0.01396  0.01064  0.00736  0.00653  0.00586  0.00494  0.00460  0.00443 
Present method  2.490E − 04  3.113E − 05  9.178E − 06  1.982E − 06  1.980E − 06  1.149E − 06  1.443E − 06  7.202E − 07 
Error norms and order of convergence for Example 1 when \(\pmb{\Delta t=\frac{1}{120}}\) , \(\pmb{\gamma=1.5}\) for different N
N  \(\boldsymbol{L_{2}}\) Norm  \(\boldsymbol{L_{\infty}}\) Norm  Order 

10  5.8885E − 04  1.1194E − 04  … 
20  1.4794E − 04  2.9588E − 04  1.4023 
40  3.7034E − 05  7.4070E − 05  1.9981 
80  9.2619E − 06  1.8524E − 05  1.9995 
160  2.3157E − 06  4.6314E − 06  1.9999 
Error norms for Example 1 when \(\pmb{h= \frac{1}{60}}\) , \(\pmb{\Delta t=\frac{1}{120}}\) , \(\pmb{\gamma=1.5}\) at different time levels
t  \(\boldsymbol{L_{2}}\) Norm  \(\boldsymbol{L_{\infty}}\) Norm 

0.2  3.6622E − 06  7.3244E − 06 
0.4  1.3860E − 05  2.77120E − 05 
1  1.6464E − 05  6.2983E − 05 
2  1.4790E − 04  2.9581E − 04 
Example 2
Absolute errors of Example 2 for many values of γ at different points
( x , t )  γ = 1.1  γ = 1.3  γ = 1.5  γ = 1.7  γ = 1.9 

(0.1,0.1)  9.5133E − 09  6.6004E − 09  4.4920E − 09  2.9885E − 09  1.9326E − 09 
(0.2,0.2)  1.0530E − 07  7.9127E − 08  5.7844E − 08  4.1404E − 08  2.8903E − 08 
(0.3,0.3)  9.6665E − 07  3.3461E − 07  2.5678E − 07  1.9243E − 07  1.4105E − 07 
(0.4,0.4)  1.0813E − 06  9.1574E − 07  7.3594E − 07  5.7117E − 07  4.3402E − 07 
(0.5,0.5)  2.2190E − 06  1.6516E − 06  1.6516E − 06  2.2190E − 06  1.0367E − 06 
(0.6,0.6)  3.9341E − 06  3.1570E − 06  3.1570E − 06  3.9341E − 06  2.1091E − 06 
(0.7,0.7)  6.3114E − 06  5.3801E − 06  5.3809E − 06  6.3114E − 06  3.8408E − 06 
(0.8,0.8)  9.4176E − 06  8.4176E − 06  8.4176E − 06  9.4176E − 06  6.4421E − 06 
(0.9,0.9)  1.3302E − 05  1.2877E − 05  1.2324E − 05  1.3302E − 05  1.3302E − 05 
Example 3
Absolute errors of Example 3 for several values of γ at different points
( x , t )  γ = 1.1  γ = 1.3  γ = 1.5  γ = 1.7  γ = 1.9 

(0.1,0.1)  1.2498E − 09  6.0892E − 10  2.9307E − 10  1.3992E − 10  6.6457E − 11 
(0.2,0.2)  2.0339E − 08  1.1879E − 08  6.5709E − 09  3.5960E − 09  1.9547E − 09 
(0.3,0.3)  9.7339E − 08  6.6352E − 08  4.0629E − 08  2.4101E − 08  1.4186E − 08 
(0.4,0.4)  2.8201E − 07  2.1618E − 07  1.4732E − 07  9.3061E − 08  1.7977E − 08 
(0.5,0.5)  6.2459E − 07  5.1997E − 07  3.9209E − 07  2.6549E − 07  1.7292E − 07 
(0.6,0.6)  1.1728E − 06  1.0335E − 06  8.4665E − 07  6.2213E − 07  4.2256E − 07 
(0.7,0.7)  1.9709E − 06  1.8065E − 06  1.5758E − 06  1.2559E − 06  8.9991E − 07 
(0.8,0.8)  3.0599E − 06  2.8817E − 06  2.6324E − 06  2.2492E − 06  1.7266E − 06 
(0.9,0.9)  4.4761E − 06  4.4761E − 09  4.0559E − 06  3.6618E − 06  3.0216E − 06 
The comparison of results for Example 3 when \(\pmb{N=50}\) and \(\pmb{\gamma=1.5}\) at \(\pmb{t=1}\)
x  N = 10  N = 20  N = 50  Exact  

Present  RBF [ 15 ]  Present  RBF [ 15 ]  Present  RBF [ 15 ]  
0.1  0.10016466  0.09950933  0.10016467  0.09951043  0.10016467  0.09951107  0.10016675 
0.2  0.20133195  0.20079972  0.20133195  0.20080174  0.20133195  0.20080291  0.20133600 
0.3  0.30451450  0.30402685  0.30451450  0.30402960  0.30451450  0.30403116  0.30452029 
0.4  0.41074514  0.41029949  0.41074514  0.41030268  0.41074515  0.41030448  0.41075232 
0.5  0.52108721  0.52065615  0.52108721  0.52065950  0.52108722  0.52066138  0.52109530 
0.6  0.63664520  0.63621617  0.63664520  0.63621936  0.63664520  0.63622115  0.63665358 
0.7  0.75857580  0.75812244  0.75857580  0.75812517  0.75857580  0.75812672  0.75858370 
0.8  0.88809951  0.88761746  0.88809951  0.88761947  0.88809951  0.88762062  0.88810598 
0.9  1.02651281  1.02593218  1.02651281  1.02593326  1.02651281  1.02593389  1.02651672 
1.0  1.17520119  1.17520119  1.17520119  1.17520119  1.17520119  1.17520119  1.17520119 
The comparison of results for Example 3 when \(\pmb{N=50}\) and \(\pmb{\gamma=1.25}\) at \(\pmb{t=1}\)
x  N = 10  N = 50  Exact  

Present  RBF [ 15 ]  Present  RBF [ 15 ]  
0.1  0.10016466  0.09950226  0.10016465  0.09950368  0.10016675 
0.2  0.20133193  0.20078691  0.20133193  0.20078954  0.20133600 
0.3  0.30451447  0.30400968  0.30451448  0.30401323  0.30452029 
0.4  0.41074511  0.41027962  0.41074512  0.41028373  0.41075232 
0.5  0.52108717  0.52063540  0.52108718  0.52063970  0.52109530 
0.6  0.63664516  0.63619639  0.63664517  0.63620048  0.63665358 
0.7  0.75857577  0.75810541  0.75857578  0.75810893  0.75858370 
0.8  0.88809948  0.88760481  0.88809949  0.88760740  0.88810598 
0.9  1.02651280  1.02592522  1.02651280  1.02592662  1.02651672 
1.0  1.17520119  1.17520119  1.17520119  1.17520119  1.17520119 
The comparison of results for Example 3 when \(\pmb{N=50}\) and \(\pmb{\gamma=1.75}\) at \(\pmb{t=1}\)
x  N = 10  N = 50  Exact  

Present  RBF [ 15 ]  Present  RBF [ 15 ]  
0.1  0.10016466  0.09951844  0.10016464  0.09952297  0.10016675 
0.2  0.20133194  0.20081683  0.20133192  0.09952297  0.20133600 
0.3  0.30451449  0.30405039  0.30451445  0.30406191  0.30452029 
0.4  0.41074513  0.41032718  0.41074509  0.41034071  0.41075232 
0.5  0.52108720  0.52068530  0.52108715  0.52069953  0.52109530 
0.6  0.63664519  0.63624390  0.63664514  0.63625745  0.63665358 
0.7  0.75857579  0.75814606  0.75857575  0.75815761  0.75858370 
0.8  0.88809950  0.88763465  0.88809947  0.88764309  0.88810598 
0.9  1.02651281  1.02594135  1.02651279  1.02594589  1.02651672 
1.0  1.17520119  1.17520119  1.17520119  1.17520119  1.17520119 
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^{−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.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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