A Finite Difference Scheme based on Cubic Trigonometric B-splines for Time Fractional Diffusion-wave Equation

In this paper, we propose an efficient numerical scheme for the approximate solution of the time fractional diffusion-wave 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 B-spline 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 a 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, straight forward interpolation and very less computational cost.

We consider, in this paper, the case of diffusion-wave i.e. 1 < γ < 2. It can be used to deal with viscoelastic problems and disordered media to examine structures, semiconductors and dielectrics.

Applications and Literature Review
The subject of fractional calculus [1][2][3][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][11][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 diffusion-wave equations. Zeng [13] proposed two second order stable and one condition- An efficient numerical scheme based on trigonometric cubic B spline functions is presented in this paper to find the approximate solutions of the time fractional diffusion-wave equation with reaction term. First, we discretize the Caputo time fractional derivative by the usual finite difference formula and then use trigonometric cubic B-spline basis to approximate derivatives in space. Trigonometric cubic B-spline functions provide better accuracy than the usual finite difference schemes due to its minimal support and C 2 continuity. Numerical experiments are carried out and the obtained results of a special case are compared with those of [14]. The comparison shows that the presented scheme has an 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 Von-Neumann stability analysis, whereas the scheme of [14] is conditionally stable. 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

Temporal Discretization
To find time discretization of Eq. (1.1) we discretize the Caputo time fractional derivative ∂ γ u(x,t) ∂t γ appearing in the equation using the usual finite difference method. Following the standard notations we let t n = n∆t, n = 0, 1, · · · M, where ∆t = T M is the time step. First we approximate the second order differential operators using a forward finite difference method as follows Using (2.1), we can obtain an efficient approximation to the fractional derivative ∂ γ u(x,t) ∂t 1−γ as follows: where e n+1 ∆t is the truncation error, r = (t n+1 − s) and b j = (j + 1) 2−γ − j 2−γ . The reader may verify that • b j > 0, j = 0, 1, 2, · · · , n, Substituting (2.2) into (1.1), we obtain the following temporal discretization , the last equation can be rewritten as where n = 0, 1, · · · 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

Description of the Numerical Scheme
In this section we derive the cubic trigonometric B-spline collocation method (CuTBSM) for finding the numerical solution of time fractional diffusion-wave equation problem (1.1).
The solution domain a ≤ x ≤ b is uniformly partitioned by knots Our numerical approach for solving (1.1) using trigonometric cubic B-splines is to seek an approximate solution U(x, t) to the exact solution u(x, t) in the following form [21,22] U(x, t) = 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 T B 4 i (x) [23] at the knots x i are given by Since there are three non zero terms at each knot notably T B 4 j−1 (x), T B 4 j (x) and T B 4 j+1 (x), therefore the approximation u n j at the grid point (x j , t n ) to the exact solution at n th time level is given as: The time dependent unknowns c n j (t) are to be determined by making use of the initial and boundary conditions, and the collocation conditions on T B 4 i (x). As a result we obtain the approximations u n j together with its necessary derivatives as given below:

+ 4 cos (h) .
To obtain full discretization which relates the successive time levels and the unknowns c n+1 j , we plug in the approximations u n j and its derivatives (3.4) into the equation (2.4). After some simplifications we arrive at the following recurrence relation: obtained which is a tri-diagonal system. Thomas Algorithm [24] is then used to uniquely solve this system.

Initial Vector c 0
In order the commence the iteration process, it is required to find the initial solution The process of finding the initial vector involves the computation of initial condition and its derivatives at the two boundaries as explained below [23]: The above tri-diagonal system consists of (N + 3) linear equations in (N + 3) unknowns whose matrix form is given as: −a 3 0 a 3 · · · · · · · · · · · · 0 a 1 a 2 a 1 . 0 · · · · · · · · · · · · −a 3 0 a 3

Stability Analysis
By Duhamels principle [25], 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 inhomogeneous problem if it is stable for the homogeneous one. It is sufficient to present the stability analysis for the scheme (3.5) for the force free case (f = 0) only.
The growth factor of a Fourier mode is assumed to be ρ n j and letρ n j be its approximation. Define E n j = ρ n j −ρ n j which on substitution in (3.5), gives the following roundoff error equation: ]. (4.1) 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 and introduce the norm: so that the following relation is obtained (4.5) Using the relation e −iβh +e iβh = 2 cos(βh) and grouping like terms, we obtain the following relation where ν = 1 + 2(a 1 α−a 4 ) cos(βh))+(a 2 α−a 5 ) α 0 (2a 1 cos(βh)+a 5 ) . Obviously ν ≥ 1.
Proof. Using formula (4.4) and Proposition 1, we obtain which establishes that the scheme is unconditionally stable.

Numerical Results and Discussion
In respectively. We compare the numerical solutions obtained by CuTBSM for one dimensional fractional diffusion equations (1.1) with known exact solutions. Numerical calculations are carried out by using Mathematica 9 on an Intel Core T M i5-2410M CPU @ 2.30 GHz with 8GB RAM and 64-bit operating system (Windows 7).

Example 1.
As a first experiment we consider the following fractional diffusion-wave The exact solution of the problem is u(x, t) = sin(πx)(t 2 − t) [14]. Figure 1 compares the graphs of the exact and approximate solutions with different values of γ, h and ∆t at different time levels. In Figure 2, we exhibit the absolute error profiles at different time levels. Figure 3 compares the graphs of 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. Figure   4 shows 3D plots of exact and approximates solutions at time t = 0.1. In Tables 1-2, the maximum errors obtained are compared with those of Hermite Formula (HF) [14] for different values of γ to demonstrate that our scheme is more accurate and gives accuracy of 10 −7 . In tables 3-4, 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 diffusion-wave equation The exact solution of the problem The proposed scheme is applied to solve this problem. Figure 5 shows  Table ??. From Figure 7 and Table ??, 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 γ.   Example 3. As a last example consider the time fractional diffusion-wave equation The above problem is solved by using the proposed scheme. Figure 8

Concluding Remarks
This study presents a finite difference scheme with a combination of cubic trigonometric B-spline basis for the time fractional fractional diffusion-wave equation with reaction term. This algorithm is based on a discretization using finite difference formulation for the Caputo sense. The cubic trigonometric B-spline 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.