A Numerical Method For Computing Nonlinear Eigenpairs Of The Semi-variational P-Laplacian Systems

Eigenpairs problem of the partial differential equation in the field of physics, me-chanics, quantum chemistry and so on, is very important in theoretical reasearch and has widespread application value. In the free vibration of bounded strings and heat conduction process is the key to solve eigenpairs problem. And p-Laplacian equation in many fluid mechanics, for example nonlinear diffusion, glaciology, non-newtonian fluid model (non-Darcian) and climatology, turbulence, porous media and power-law material flow, has very widespread application.This paper mainly studies a numerical method for computing nonlinear eigenpairs of the semi-variational p-Laplacian systems, discusses the application of the prom-toted local minimum orthogonal algorithm from Hilbert space to Banach space, and applies this algorithm solving nonlinear eigenpairs of the semi-variational p-Laplacian systems. First applying Rayleigh quotient, converting eigenpairs problem to "critical point" problem of energy functional, and proving that the "critical point" and "critical value" of Rayleigh quotient are the solutions of the eigenpairs problem. Second, due to the "critical point" on the whole space B is not easy to find, so introducing a L-(?) select function, constructing a submanifolds M, then translating the original eigenpairs problem into some relatively stable point on manifold M; Then, using pseudo gradient in Banach space replacing the gradient in Hilbert space, and project mapping pseudo gradient to appropriate space, and applying gradient descent method, combining with the update step lemma to search the direction, searching the the relatively stable point on M. Finally, combining with multiple numerical example, successfully get multiple eigenparis on square and circular area.