New unconditionally stable scheme for telegraph equation based on weighted Laguerre polynomials

Abstract

This article proposes a new unconditionally stable scheme to solve one‐dimensional telegraph equation using weighted Laguerre polynomials. Unlike other numerical schemes, the time derivatives in the equation can be expanded analytically based on the Laguerre polynomials and basis functions. By applying a Galerkin temporal testing procedure and using the orthogonal property of weighted Laguerre polynomials, the time variable can be eliminated from computations, which results in an implicit equation. After solving the equation recursively one can obtain the numerical results of telegraph equation by using the expanded coefficients. Some numerical examples are considered to validate the accuracy and stability of this proposed scheme, and the results are compared with some existing numerical schemes.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1603–1615, 2017

A fundamental sequences method with time-reduction for one-dimensional lateral Cauchy problems

“…As explained in the nal paragraph of Section 3, the element ũp depends on the parameters β j and thus ũp diers in (4.3) and (4.4). We mention that numerical methods for direct problems for the wave and telegraph equations related to the above are [11,14,29,34]. Of course, for the direct problem for the wave equation (4.1), any of the standard numerical methods such as the nite element method (with or without time-discretisation), nite dierences or the boundary element method can be applied although with more eort for mesh generation, see the overview [33].…”

Optimal Approximation of Fractional Order Brain Tumor Model Using Generalized Laguerre Polynomials

“…( 2021 ) used a novel Laguerre neural network with three layers of neurons for solving Black–Scholes equations and proved its high accuracy and superiority over other existing algorithms. Zhang and Miao ( 2017 ) applied weighted LPs to an unconditionally stable scheme for solving one-dimensional telegraph equation.…”

High order accurate method for the numerical solution of the second order linear hyperbolic telegraph equation

“…Among these equations, the hyperbolic telegraph equation is an important type of hyperbolic partial differential equation commonly used in modeling of mathematical problems such as signal analysis, random walking theory and wave propagation [1]. Many methods have been applied to solve numerically the second order linear hyperbolic telegraph equation, such as the implicit three-level difference scheme [2], the collocation methods based on the thin plate splines, the quartic B-spline, the septic B-spline, the cubic B-spline, the modified cubic B-spline, the extended cubic B-spline and the cubic trigonometric B-spline [3][4][5][6][7][8][9][10][11], the Rothe-Wavelet method [12], the Legendre multiwavelet Galerkin method [13], the Chebyshev tau method [14], the finite difference methods based on the quartic spline functions, the non-polynomial splines and the cubic Hermite interpolation polynomial function [15][16][17], the unconditionally stable parallel difference scheme [18], the boundary integral equation method and the dual reciprocity technique [19], the differential quadrature method [20,21], the Rothe-wavelet-Galerkin method [22], the reduced differential transform method [23], the reproducing Kernel Hilbert space method [24], the sinc-collocation method [25], the Chebyshev wavelets method [26], the meshfree method based on the radial basis functions [1], the polynomial scaling functions method [27], the high-order shifted Gegenbauer pseudospectral method [28], the uniformly convergent Euler matrix method [29], the Euler method [30], the Bessel collocation method [31], the Galerkin method [32] and the adaptive Monte Carlo method …”