Solutions of Hyperbolic Telegraph Equation using Radial Basis Function-Finite Difference Method Abstract

Dulashini Karunarathna

Abstract


In this study, we present a Radial Basis Function-Finite Difference (RBF-FD) based numerical scheme for solving the hyperbolic telegraph equation, a model widely used in reaction–diffusion processes and known to generalize the conventional diffusion equation. This method is a revolutionary meshless approach that has several advantages over conventional numerical techniques, especially in terms of accuracy and flexibility. Four exemplary test instances covering telegraph equation parameter settings are used to validate the suggested scheme. The computed numerical solutions are illustrated graphically compared with the corresponding analytical solutions. Comprehensive error analysis further shown the robustness and superior accuracy of the method, with consistently negligible errors observed across all test scenarios.

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References


Bayona, V., (2019). An insight into RBF-FD approximations augmented with polynomials, Computers and Mathematics with Applications, 77:2337- 2353.

C ́ecile, P., Dissanayake, N., Gierke, J., Fornberg, B., (2019). The Radial Basis Functions Method for improved Numerical Approximations of Geological Processes in Heterogeneous Systems, Mathematical Geosciences, 52:08.

Dehghan, M., & Shokri, A. (2007). A numerical method for solving the hyperbolic telegraph equation. Numerical Methods for Partial Differential Equations: An International Journal, 24(4), 1080-1093.

Du Toit, W. (2008). Radial basis function interpolation (Doctoral dissertation, Stellenbosch: Stellenbosch University).

El-Azab, M. S., & El-Gamel, M. (2007). A numerical algorithm for the solution of telegraph equations. Applied Mathematics and Computation, 190(1), 757-764.

Estruch, O., Lehmkuhl, O., Borrell, R., Segarra, C. P., & Oliva, A. (2013). A parallel radial basis function interpolation method for unstructured dynamic meshes. Computers & Fluids, 80, 44-54.

Fornberg, B., and Flyer, N., (2015). A Primer on Radial Basis Functions with Applications to the Geosciences. Society for Industrial and Applied Mathematics, Philadelphia, PA.

Gao, F., & Chi, C. (2007). Unconditionally stable difference schemes for a one-space-dimensional linear hyperbolic equation. Applied Mathematics and Computation, 187(2), 1272-1276.

Hashemi, M. S., Karatas, E., & Darvishi, E. (2019). Numerical treatment on one-dimensional hyperbolic telegraph equation by the method of line-group preserving scheme. The European Physical Journal Plus, 134(4), 153.

Li, J., Zhai, S., Weng, Z., Feng, X., (2017). H-adaptive RBF-FD method for the high-dimensional convection-diffusion equation, International Communications in Heat and Mass Transfer, 89:139-146.

Martin, B., Fornberg, B., (2017). Using radial basis function-generated finite differences (rbf-fd) to solve heat transfer equilibrium problems in domains with interfaces, Engineering Analysis with Boundary Elements, 79:38–48.

Mudiyanselage, N. D. K., Blazejewski, J., Ong, B., Piret, C., (2022). A Radial Basis Function-Finite Difference and Parareal Framework for Solving Time Dependent Partial Differential Equations Dolomites Research Notes on Approximation, 15(5),8-23.

Momani, S., (2005). Analytic and approximate solutions of the space- and time-fractional telegraph equations, Applied Mathematics and Computation 170, 1126–1134.

Mongillo, M. (2011). Choosing basis functions and shape parameters for radial basis function methods. SIAM undergraduate research online, 4(190-209), 2-6.

Ouédraogo, P. F., Sawadogo, W. O., & So, O. (2019). Numerical resolution of Richards equation by the RBF-MQ method. Annals of the University of Craiova-Mathematics and Computer Science Series, 46(1), 109-124.

Usta, F. (2016). Fractional type Poisson equations by radial basis functions Kansa approach. Journal of Inequalities and Special Functions, 7(4), 143-149.


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