A New GL Method for Solving Differential Equation in Electromagnetic and Phys-Chemical and Financial Mathematics

Abstract

In this paper, we propose a new GL method for solving the ordinary and the partial difierential equation. These equations govern the electromagnetic fleld etc. macro and micro physical, chemical, flnancial sciences and engineering. The difierential equation is held on an inflnite domain which includes a flnite inhomogeneous domain. The inhomogeneous domain is divided into flnite sub domains. We present the solution of the difierential equation as an explicit recursive sum of the integrals in the inhomogeneous sub domains. The analytical solution of the equation in the inflnite homogeneous domain is called as an initial global fleld. The global fleld is updated by local scattering fleld successively subdomain by subdomain. Once all subdomains are scattered and the updating process is flnished in all the sub domains, the solution of the equation is obtained. We call our method as Global and Local fleld method, in short GL method. It is difierent from FEM method, the GL method directly assemble inverse matrix and solution. There is no big matrix equation needs to solve in the GL method. There is no needed artiflcial boundary and no absorption boundary condition for inflnite domain in the GL method. We proved several theorems and proposed a triangle formula of the Greens functions 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 1-D 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 1-D 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) fleld, 3D elastic and plastic etc seismic fleld, acoustic fleld, ∞ow fleld, and quantum fleld. The GL method software for the above 3D EM etc fleld are developed.