A reliable method for voltage of telegraph equation in one and two space variables in electrical transmission: approximate and analytical approach

Abstract

In this paper we investigate approximate analytical solution so called voltage in one and two space variables for linear and nonlinear telegraph equations by a reliable method namely Modified Laplace Decomposition Method (MLDM) using MATLAB and MATHEMATICA software tools. The MLDM is a mixture of Laplace transform with modified Adomian decomposition method based on Newton Raphson method. The nonlinearity of the problem is tackled by Adomian decomposition and approximate analytical solution to the partial differential equation handled by using the Laplace and inverse Laplace transform technique without differentiation in time domain. We use Newton Raphson method in the domain of Adomian polynomial to modify it. Theoretical concepts for the approximate analytical solution of present scheme are well behaved through stability and convergence analysis. Five numerical examples are carried out in order to check the effectiveness and applicability of the proposed scheme. The telegraph equation with one space variable is solved numerically whereas the approximate analytical solution obtained for two space variables. Employing MLDM, it is possible to obtain the approximate analytical solution (i.e., voltage) of a telegraph equation and found to be in good agreement with exact solutions and also compared with earlier studies for one space variable.

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“…For instances, recently LADT has been successfully implemented in a variety of differential model like; SDIQR model of Covid-19 [18], Lane emden differential equation [19], model of lassa diseases [20], numerical solution of large deflection beam [21], simulation of unsteady MHD flow in incompressible fluid [22], approximation of time fractional advection dispersion equation [23], approximate solution of fractional order sterile insect technology model [24] etc. Similarly various integral transforms [25,26] are implemented to solve higher order partial differential equations. Also various new soliton solution approaches are investigated; like Sardar sub-equation technique [27], multi soliton solution of Vakhnenko-Parkes equation [28], Coke price prediction approach [29], multivariate stochastic volatility models by using optimization mechanisms [30], Galilean transformation of Schrodinger equation [31], bifurcation analysis of Hindmarsh-Rose model [32] etc B-spline collocation [33][34][35] and the block method [36,37] are additional techniques that improve the current article and are advantageous to the existing system.…”