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Analytical solution of wave system in ${ℝ}^{n}$ with coupling controllers. (English) Zbl 1231.65248
Summary: The purpose of this paper is to consider analytical solution of wave system in ${R}^{n}$ with coupling controllers by using the homotopy perturbation method (HPM). HPM is applied to the system of linear partial differential equations, i.e. the system of waves in the two-dimensional version of system equations (1) and (2). This problem is motivated by an analogous problem in ordinary differential equations for coupled oscillators and has potential application in isolating a vibrating object from the outside disturbances. For example, rubber or rubber-like materials can be used to either absorb or shield a structure from vibration. As an approximation, these materials can be modeled as distributed springs. In this paper, HPM was used to obtain analytical solution of wave system in with coupling controllers. The method provides the solutions in the form of a series with easily computable terms. Unlike other common methods for solving any physical problem, linear or nonlinear, that requires linearization, discretization, perturbation, or unjustified assumptions that may slightly change the physics of the problem, the HPM finds approximate analytical solutions by using the initial conditions only. The method proposed in this paper is very reliable and efficient and is being used quite extensively for diversified nonlinear problems of a physical nature. The algorithm is being used for the first time on such problems.
##### MSC:
 65N99 Numerical methods for BVP of PDE 35L53 Second-order hyperbolic systems, initial-boundary value problems 35C10 Series solutions of PDE 35C15 Integral representations of solutions of PDE