Numerical study of integer-order hyperbolic telegraph model arising in physical and related sciences
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Numerical study of integer-order hyperbolic telegraph model arising in physical and related sciences Imtiaz Ahmad1, Hijaz Ahmad2,a M. Abdel-Aty5
, Ahmed E. Abouelregal3 , Phatiphat Thounthong4 ,
1 2 3 4
Department of Mathematics, University of Swabi, Swabi 23430, Khyber Pakhtunkhwa, Pakistan Department of Basic Sciences, University of Engineering and Technology Peshawar, Peshawar, Pakistan Department of Mathematics, College of Science and Arts, Jouf University, Al-Qurayyat, Saudi Arabia Renewable Energy Research Centre, Department of Teacher Training in Electrical Engineering, Faculty of Technical Education, King Mongkut’s University of Technology North Bangkok, 1518 Pracharat 1 Road, Bangsue, Bangkok 10800, Thailand 5 Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt Received: 22 July 2020 / Accepted: 16 September 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract More recently, it is discovered in the field of applied sciences and engineering that the telegraph equation is better suited to model reaction-diffusion than the ordinary diffusion equation. In this article, the second-order hyperbolic telegraph equations are analyzed numerically by means of an efficient local differential quadrature method utilizing the radial basis functions. The explicit time integration technique is used to semi-discretize the model in the time direction, while the space derivatives are discretized by the proposed meshless procedure. To test the accuracy and capabilities of the method, five test problems are considered utilizing both rectangular and non-rectangular domains, which show that the proposed scheme solutions are converging extremely quick in comparison with the different existing numerical techniques in the recent literature.
1 Introduction The telegraph equation, which has been used to describe phenomena in various fields, belongs to the hyperbolic partial differential equation (PDEs) scope. For instance, the two-dimensional (2D) second-order hyperbolic telegraph equations can model different real-world phenomena in sciences and engineering and furthermore has many applications in different fields [1]. The generalized 2D second-order hyperbolic telegraph equation have the following form 2 ∂ 2 W (z, t) ∂ W (z, t) ∂ W (z, t) ∂ 2 W (z, t) 2 + β + 2α W (z, t) − δ + ∂t 2 ∂t ∂x2 ∂ y2 (1) = F(z, t), z ∈ , t > 0, with initial-boundary conditions
Electronic supplementary material The online version of this article (https://doi.org/10.1140/epjp/ s13360-020-00784-z) contains supplementary material, which is available to authorized users. a e-mail: [email protected] (corresponding author)
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W (z, 0) = g1 (z),
Eur. Phys. J. Plus
(2020) 135:759
∂ W (z, 0) = g2 (z), C (z, t) = g3 (z, t), z ∈ ∂, ∂t
(2)
where α > 0, β and δ are known coefficients and F(z, t) is the source function. It is well-known that it is difficult to get the analytical solutions for relatively complex problems [2–4]. Thu
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