Numerical solution of second-order hyperbolic telegraph equation via new cubic trigonometric B-splines approach

Abstract

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...

An efficient algorithm for solving the conformable time-space fractional telegraph equations

“…The L 2 and L ∞ errors and CPU time in seconds is shown in Table 1. Numerical results are compared with the obtained results in [23] and [28]. It can be concluded that the numerical solutions obtained by our algorithm are good.…”

“…Using the linear system (6.6) for m = 5 and N = 1000 with the initial data V 0 = (0.2373, 0.2273, −0.0124, −0.0023, 0.0001, 0) T , V0 = V0 = (0, 0, 0, 0, 0, 0) T , we compute the approximate solution u 5 (x, t). The efficiency of Algorithm 1 is measured using L 2 , L ∞ and root mean square errors with ∆t = 0.0001, h = 0.01 which are shown in Table 3.The numerical results are compared with the obtained results in [28,23]. It can be concluded that the numerical solutions obtained by our algorithm are good.…”

“…The exact solution of this example is u e (x, t) = x − x 2 t 1+α e −t . If α = 1, this example was studied in [23,28] by different numerical methods. We apply Algorithm 1 for m = 5 and approximate to the solution u (x, t) as follows:…”

Parabolic Methods for One Dimensional Advection-Diffusion Type Equation and Application to Burger Equation

“…The Fourier analysis (Von-Neumann) stability [32] analysis technique is applied to investigate the stability of the proposed method. Such an approach has been used by many researchers like [23][24][25][26]32].…”

“…The Fourier analysis (Von-Neumann) stability [32] analysis technique is applied to investigate the stability of the proposed method. Such an approach has been used by many researchers like [23][24][25][26]32]. Now consider that the make nonlinearity in the difference scheme is linear by taking ] = max ( such that Thus this shows that the scheme is stable.…”