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 θ weighted scheme. The scheme is shown to be unconditionally stable for a range of θ 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.
Highlights
Further details on other numerical methods including interpolating scaling functions (Lakestani & Saray, 2010), radial basis functions (RBF) (Esmaeilbeigi, Hosseini, & Mohyud-Din, 2011), quartic B-spline collocation method (QuBSM) (Dosti & Nazemi, 2012), cubic B-spline collocation method (CuBSM) (Mittal & Bhatia, 2013; Rashidinia, Jamalzadeh, & Esfahani, 2014) for the solution of the telegraph equation subject to Dirichlet boundary conditions are in the literature
We found that our numerical results are comparable to that of QuBSM (Dosti & Nazemi, 2012) and CuBSM (Mittal & Bhatia, 2013) in terms of L2, L∞ errors
This paper has investigated the application of cubic trigonometric B-spline collocation method to find the numerical solution of the telegraph equation with initial condition and Dirichlet as well as Neumann’s type boundary conditions
Summary
Further details on other numerical methods including interpolating scaling functions (Lakestani & Saray, 2010), RBFs (Esmaeilbeigi, Hosseini, & Mohyud-Din, 2011), quartic B-spline collocation method (QuBSM) (Dosti & Nazemi, 2012), cubic B-spline collocation method (CuBSM) (Mittal & Bhatia, 2013; Rashidinia, Jamalzadeh, & Esfahani, 2014) for the solution of the telegraph equation subject to Dirichlet boundary conditions are in the literature. W. Liu (2013) have developed a compact difference unconditionally stable scheme (CDS) to solve the telegraph equation with Neumann boundary conditions. Mittal and Bhatia (2014) have developed a technique based on collocation of cubic B-spline collocation method (CuBSM) for solving the telegraph equation with Neumann boundary conditions. A numerical collocation finite difference technique based on cubic trigonometric B-spline is presented for the solution of telegraph Equation (1) with initial conditions in Equation (2) and different two types of boundary conditions in Equations (3) and (4).
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