Special Functions and Its Application in Solving Two Dimensional Hyperbolic Partial Differential Equation of Telegraph Type

• Published in: Symmetry (Basel) Vol. 15; no. 4; p. 847
• Main Authors: Ali, Ishtiaq, Saleem, Maliha Tehseen, Din, Azhar ul
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.04.2023
• Subjects: Algorithms; Approximation; Boundary conditions; Chebyshev approximation; Collocation; Computational efficiency; Computing costs; Computing time; Derivatives; Differential equations; Differentiation; Dirichlet problem; Discretization; Exact solutions; Fast Fourier transformations; Finite difference method; Mathematical analysis; Mathematical models; Methods; Numerical analysis; Ordinary differential equations; Partial differential equations; Polynomials; Propagation; Transmission lines
• ISSN: 2073-8994; 2073-8994
• DOI: 10.3390/sym15040847

In this article, we use the applications of special functions in the form of Chebyshev polynomials to find the approximate solution of hyperbolic partial differential equations (PDEs) arising in the mathematical modeling of transmission line subject to appropriate symmetric Dirichlet and Neumann boundary conditions. The special part of the model equation is discretized using a Chebyshev differentiation matrix, which is centro-asymmetric using the symmetric collocation points as grid points, while the time derivative is discretized using the standard central finite difference scheme. One of the disadvantages of the Chebyshev differentiation matrix is that the resultant matrix, which is obtained after replacing the special coordinates with the derivative of Chebyshev polynomials, is dense and, therefore, needs more computational time to evaluate the resultant algebraic equation. To overcome this difficulty, an algorithm consisting of fast Fourier transformation is used. The main advantage of this transformation is that it significantly reduces the computational cost needed for N collocation points. It is shown that the proposed scheme converges exponentially, provided the data are smooth in the given equations. A number of numerical experiments are performed for different time steps and compared with the analytical solution, which further validates the accuracy of our proposed scheme.