A Unique Study Of Free Boundary Problems

The free boundary problem is a class of partial differential equations with unknown boundary,whose boundary is partly related to each other and must be determined simultaneously.In the problem of free boundary,not only the condition of definite solution but also the condition of free boundary must be set at least.Therefore,the problem of free boundary is a nonlinear problem.The problem of free boundary in the direction of partial differential equations has attracted much attention in the field of mathematics.The free boundary problem has been in the American option pricing,tumor growth,dam seepage,crystal growth,metallurgy industry in metal melting solidification,image processing,population migration and wound healing have a deep and detailed research.In this paper,we study the problem of free boundary into a homogeneous solution of a kind of free boundary problem.The study of a class of free boundary problems will help us to explain more and more nonlinear problems in nature.In this paper,we first study the uniqueness of a homogeneous solution for a class of free boundary problems in three-dimensional space.Secondly,through the Mathematica software programming of a free boundary problem with generalized Legendre function theorem is deduced and proved by the different boundary conditions the obtained results and draw the function and derivative image analysis,obtained under different boundary conditions the number of generalized Legendre function of a homogeneous solution is existent and unique.