Research On Solution Of High Dimensional Wave Equation In Finite Region

The study of wave propagation in underground media has a very wide research value and very important application value in geological exploration.The study of solution of the wave equation is an important part of the study of direct problem and has important applications in many fields.As we all know,the problem of solving the wave equation is to solve the definite solution problem based on its initial value and the boundary value.In practice,the solution region is generally high-dimensional.In the paper,two-dimensional wave equation in the finite region is adoped to study the problem of solving the wave equation.The paper first introduces the initial boundary value problem of the wave equation in finite region,and then introduces two methods which would be used in solving the problem,including the separation variable method of one-dimensional wave equation and the changeable paratemer method of the initial value problem of first-order ordinary differential equation.The introduction lays a solid foundation for the later research.Secondly,based on the one-dimensional wave equation of finite region,the two-dimensional wave equation of finite region is taken as the mathematical model,and the six kinds of homogeneous boundary condition model are given.The six models are deeply studied,and the solution formula are given,respectively.The unknown multivariate function is first decomposed into the product of the unary function using the separated variable method,and then the corresponding homogeneous boundary condition is used to form the corresponding eigenvalue problem,and the eigenvalue and the eigenfunction are obtained to form a general solution in the form of series.Then,using the initial conditions and the Fourier transform,the formula for calculating the coefficients in the general solution is designed.When there is a free term in the equation,the solution is considered as the form of the variation coefficient,and the problem of the ordinary differential equation that the variable coefficient should satisfy is constructed by Fourier transform,and then calculated.The research in this chapter has certain research value and significance,and also lays a foundation for the smooth research of the non-homogeneous boundary conditions of the high-dimensional wave equations in the following limited regions.Thirdly,the two-dimensional wave equation of finite region is taken as the mathematical model,and the six-class non-homogeneous boundary condition model are given.The six models are studied in depth,and the solution formula of each case are given,respectively.The paper will construct the appropriate auxiliary function to successively homogenize the non-homogeneous boundary conditions,thus transforming the original problem into a homogeneous boundary problem,and then using the method of the previous chapter to solve.Finally,the paper gives conclusions and prospects.