GL method for solving equations in math-physics and engineering

Abstract

In this paper, we propose a GL method for solving the ordinary and the partial differential equation in mathematical physics and chemics and engineering. These equations govern the acustic, heat, electromagnetic, elastic, plastic, flow, and quantum etc. macro and micro wave field in time domain and frequency domain. The space domain of the differential equation is infinite domain which includes a finite inhomogeneous domain. The inhomogeneous domain is divided into finite sub domains. We present the solution of the differential equation as an explicit recursive sum of the integrals in the inhomogeneous sub domains. Actualy, we propose an explicit representation of the inhomogeneous parameter nonlinear inversion. The analytical solution of the equation in the infinite homogeneous domain is called as an initial global field. The global field is updated by local scattering field successively subdomain by subdomain. Once all subdomains are scattered and the updating process is finished in all the sub domains, the solution of the equation is obtained. We call our method as Global and Local field method, in short, GL method. It is different from FEM method, the GL method directly assemble inverse matrix and gets solution. There is no big matrix equation needs to solve in the GL method. There is no needed artificial boundary and no absorption boundary condition for infinite domain in the GL method. We proved several theorems on relationships between the field solution and Green’s function that is the theoretical base of our GL method. The numerical discretization of the GL method is presented. We proved that the numerical solution of the GL method convergence to the exact solution when the size of the sub domain is going to zero. The error estimation of the GL method for solving wave equation is presented. The simulations show that the GL method is accurate, fast, and stable for solving elliptic, parabolic, and hyperbolic equations. The GL method has advantages and wide applications in the 3D electromagnetic (EM) field, 3D elastic and plastic etc seismic field, acoustic field, flow field, and quantum field. The GL method software for the above 3D EM etc field are developed.