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

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.