In this paper, we propose an efficient numerical scheme for the approximate solution of a 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 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.

This paper presents a linear computational technique based on cubic trigonometric cubic B-splines for time fractional burgers' equation. The nonlinear advection term is approximated by a new linearization technique which is very efficient and significantly reduces the computational cost. The usual finite difference formulation is used to approximate the Caputo time fractional derivative while the derivative in space is discretized using cubic trigonometric B-spline functions. The method is proved to be globally unconditionally stable. To measure the accuracy of the solution, a convergence analysis is also provided. A convergence analysis is Computational experiments are performed to further establish the accuracy and stability of the method. Numerical results are compared with those obtained by a scheme based on parametric spline functions. The comparison reveal that the proposed scheme is quite accurate and effective.

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This paper presents a new approach and methodology to solve the second order one dimensional hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions using the cubic trigonometric B-spline collocation method. The usual finite difference scheme is used to discretize the time derivative. The cubic trigonometric B-spline basis functions are utilized as an interpolating function in the space dimension, with a weighted scheme. The scheme is shown to be unconditionally stable for a range of  values using the von Neumann (Fourier) method. Several test problems are presented to confirm the accuracy of the new scheme and to show the performance of trigonometric basis functions. The proposed scheme is also computationally economical and can be used to solve complex problems. The numerical results are found to be in good agreement with known exact solutions and also with earlier studies.Keywords: Second order one dimensional telegraph equation, Cubic trigonometric B-spline basis functions, Cubic trigonometric B-spline collocation method, Stability * Corresponding Email address: m.abbas@uos.edu.pk 2 1.3.Literature review Several numerical methods have been developed to solve the telegraph equation subject to Dirichlet boundary conditions and the references are in [2,[5][6][7][8]. In [9], two semi-discretization methods based on quartic splines function have been developed to solve the telegraph equations. A class of unconditionally stable finite difference schemes constructed with the help of quartic splines functions has been developed by Liu and Liu [10] for the solution of the telegraph equation. Further several numerical methods have been developed by Dehghan [11][12] in collaboration with different authors. These include the thin plate splines radial basis functions (RBF) for the numerical solution of the telegraph equation [11] and high-order compact finite difference method to solve the telegraph equation [12]. Further details on other numerical methods including interpolating scaling functions [13], radial basis functions [14], quartic B-spline collocation method (QuBSM) [15], cubic B-spline collocation method (CuBSM) [16][17] for the solution of the telegraph equation subject to Dirichlet boundary conditions are in the literature. Thus many numerical methods have been developed to solve the telegraph equation (1) with Dirichlet boundary conditions. Some numerical methods have been developed for numerical solution of the telegraph equation with Neumann boundary conditions. These include methods by Dehghan and Ghesmati [1] who constructed a dual reciprocity boundary integral equation (DRBIE) method in which cubic-radial basis function (C-RBF), thin plate-spline-radial basis function (TPS-RBF) and linear radial basis functions (L-RBF) are utilized for the numerical solution of the telegraph equation with Neumann boundary conditions. Liu and Liu [18] have developed a compact difference unconditionally stable scheme (CDS) to solve the telegraph equation with Neumann boundary conditions. Further, Mi...

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