Reproducing Kernel Method For Differential Equations

Many natural phenomena are described by linear or nonlinear equations, whichhave been widely used as important mathematical models in numerous field such asphysics, biology and control. And the analysis of these phenomena usually boilsdown to the solution of differential equation, it is difficult to acquire analyticexpression of the accurate solution for this class of equations so that thedevelopment of suitable numerical methods has theoretical significance andpractical value. So it is more and more important how to solve these meaningfulequations.In this paper, Burgers equation, reproducing kernel method and extrapolation ofthe basic theory are given. reproducing kernel method for Burgers equations isproposed. An improved reproducing kernel method for solving second-ordernonlinear ordinary differential equations with initial value problem is presented.The first chapter, describes the research status of initial value problems forordinary differential equations and Burgers equation; application background anddevelopment history for the reproducing kernel space are described in detail. Thenwe study the extrapolation, give the basic theory of extrapolation and analyze theadvantages and disadvantages of extrapolation in solving differential equations.The second chapter, proposes a reproducing kernel method for Burgersequations, and gives the theoretical proof and error estimates. Two numericalexamples illustrate the feasibility of this method.The third chapter, we combine the reproducing kernel method with theextrapolation method, then obtain an improved reproducing kernel method to solvesecond-order nonlinear ordinary differential equation with initial value condition.Calculation formula and error estimate of the method are presented, and the errorestimate is proved. According to the analysis of convergence and comparison of theresults of numerical experiment, we summarize the advantages and disadvantages of this method.