:Homotopy perturbation is one of the newest methods for numerical analysis of deferential equations. We have used for solving wave equation around a black hole. Our conclusions have this method far reaching consequences for comparison of theoritical physics and experimental physics.

We intend to propose an experimental sketch to detect gravitational waves (GW) directly, using an cold electronic plasma in a long pipe. By considering an cold electronic plasma in a long pipe, the Maxwell equations in 3+1 formalism will be invoked to relate gravitational waves to the perturbations of plasma particles. It will be shown that the impact of GW on cold electronic plasma causes disturbances on the pathes of the electrons. Those electrons that absorb energy from GW will pass through the potential barrier at the end of the pipe. Therefore, crossing of some electrons over the barrier will imply the existence of the GW.

Considering the dynamics of geometry and the matter fields, dynamical equations of geometry and the matter fields are re-derived. The solutions of these equations are studied. We focus on a charged particle and explain the axiomatic approach to drive the electromagnetic self-force on its motion, then the energy conservation is considered. A new mathematical concept, which is introduced in axiomatic approach in general, is discussed.

In the previous paper, we introduced a new method of gravitational waves (GW) detection [1]. In our proposal, we replaced usual Weber's metallic bar with a cold electronic plasma. We obtained a nonhomogenous differential equation for tangential electric field, E φ , that on it GW is known as nonhomogenous term. In this paper we estimate, the dimension of pipe, the electron density and some other associated parameters for obtaining the best detection.

In this paper, first, we will try to introduce the gravitational domain wall as a physical system. In the second step, we also introduce the Hun differential equation as a mathematical tools. We factorize the known Heun’s equation as form of operators [Formula: see text], [Formula: see text] and [Formula: see text]. Then we compare the differential equation of gravitational domain wall with corresponding Hun equation. In that case the above-mentioned operators can be obtained for the gravitational system by the comparing process. Finally, we employ such operators and achieve the corresponding symmetry algebra with the usual commutation relation of operators to each other. Here, by having such operators, we investigate the stability of system.