Reproducing Kernel Numerical Algorithm For Solving Hyperbolic Equations

Partial differential equation is an important research field in mathematics,and many phenomena in physics and biology use partial differential equation to build mathematical models.Hyperbolic partial differential equation is a very classic equation,and with the deepening of its understanding,many experts and scholars have paid close attention to it.A very important type of equation in hyperbolic equation is wave equation,which provides valuable modeling tools for simulating complex natural phenomena in various scientific fields.Spline is a function composed of smooth functions bonded together in a smooth way.The simplest form is piecewise polynomial.The polynomial on each adjacent segment has the property of continuity,so it has the simplicity of polynomial and relatively independent properties among segments.Spline is simple in structure,easy to use,with good structural properties and excellent approximation ability.In recent years,spline functions have been rapidly developed and widely used.Spline function plays an important role in function approximation,signal processing,data analysis,computational geometry,finite element and wavelet.This paper mainly studies a kind of important hyperbolic equation: wave equation.This paper mainly studies from both theoretical and numerical aspects.Theoretically,this paper defines the weak solution of the initial-boundary value problem of the wave equation by using Galerkin method,and proves the existence and uniqueness of the weak solution by estimating its energy.Numerically,this paper combines the finite difference method and reproducing kernel method to solve the approximate solution of the wave equation.At the same time,the convergence analysis is also given,and it is strictly proved theoretically.Finally,the corresponding numerical examples are given to prove the effectiveness of this algorithm.