On the Approximate Solutions of the Constant Forced (Un)Damping Helmholtz Equation for Arbitrary Initial Conditions

Alvaro H Salas,Castillo H Jairo E,M R Alharthi

doi:10.1155/2021/8887566

Abstract

This paper presents some novel solutions to the family of the Helmholtz equations (including the constant forced undamping Helmholtz equation (equation (1)) and the constant forced damping Helmholtz equation (equation (2))) which have been reported. In the beginning, equation (1) is solved analytically using two different techniques (direct and indirect solutions): in the first technique (direct solution), a new assumption is introduced to find the analytical solution of equation (1) in the form of the Weierstrass elliptic function with arbitrary initial conditions. In the second case (indirect solution), the solution of the undamping (standard) Duffing equation is devoted to determine the analytical solution to equation (1) in the form of Jacobian elliptic function with arbitrary initial conditions. Moreover, equation (2) is solved using a new ansatz and with the help of equation (1) solutions. Also, the evolution equations (equations (1) and (2)) are solved numerically via the Adomian decomposition method (ADM). Furthermore, a comparison between the approximate analytical solution and approximate numerical solutions using the fourth-order Runge–Kutta method (RK4) and ADM is reported. Furthermore, the maximum distance error for the obtained solutions is estimated. As a practical application, the Helmholtz-type equation will be derived from the fluid governing equations of quantum plasma particles with(out) taking the ionic kinematic viscosity into account for investigating the characteristics of (un)damping oscillations in a degenerate quantum plasma model.

Highlights

Academic Editor: Viet- anh Pham is paper presents some novel solutions to the family of the Helmholtz equations (including the constant forced undamping Helmholtz equation (equation (1)) and the constant forced damping Helmholtz equation (equation (2))) which have been reported
Equation (1) is solved analytically using two different techniques: in the first technique, a new assumption is introduced to find the analytical solution of equation (1) in the form of the Weierstrass elliptic function with arbitrary initial conditions
In the second case, the solution of the undamping Duffing equation is devoted to determine the analytical solution to equation (1) in the form of Jacobian elliptic function with arbitrary initial conditions

Summary

Introduction

Academic Editor: Viet- anh Pham is paper presents some novel solutions to the family of the Helmholtz equations (including the constant forced undamping Helmholtz equation (equation (1)) and the constant forced damping Helmholtz equation (equation (2))) which have been reported. E exact analytic solutions to the unforced and undamping Helmholtz equation have been derived in detail in the literature in terms of the Weierstrass elliptic function [34,35,36,37,38] and Jacobi elliptic functions [38,39,40,41]. It is known that the unforced and undamping Helmholtz equations are completely integrable, so they have exact solutions, but if the friction force (damping term) is included, the unforced damping Helmholtz equation becomes nonintegrable and cannot support an exact solution for arbitrary values of its coefficients (c, α, β). Us, under certain condition, the unforced damping Helmholtz equation has been solved analytically in terms of the Jacobi elliptic functions by Johannessen [24]. Almendral and Sanjuan [27] derived an exact solution to the unforced damping Helmholtz equation using the Lie theory under certain conditions for the coefficients (c, α, β)

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