Numerical Analysis Of Boundary Value Methods For Several Differential Equations

Boundary value methods are a class of effective numerical methods for solving ordinary differential equations.The methods are able to achieve high accuracy while maintaining good stability.The core idea of boundary value methods is to transform initial value problem into discrete boundary value problem.This idea can be used to deal with different types of differential equations,and the corresponding numerical methods can keep the good characteristics of boundary value methods.This dissertation is devoted to the numerical analysis of boundary value methods,and the corresponding research objects include diffusion equations,fractional ordinary differential equations,fractional delay differential equations and delay differential algebraic equations.In this dissertation,the numerical schemes of these methods are constructed,and the corresponding numerical properties such as convergence and stability are analyzed.The main work of this dissertation include the following four parts:By using full discretizations with multi-time-level and multi-space-level,fully discretized methods based on boundary value methods are established for solving the initial boundary value problem of diffusion equations.By using the characteristic equation to analyze the corresponding full discretizations,the equivalent condition of the order of local truncation error is given.For the general fully discretized methods based on boundary value methods,considering that the methods have the structure similar to boundary value methods,the corresponding results of stability and convergence are obtained.By approximating the fractional derivative with local technique,fractional generalized backward differentiation formulae are constructed for solving the initial value problem of fractional ordinary differential equations.For the Toeplitz matrix related to the methods,the estimate of the inverse of the corresponding matrix is established.Using this estimate,the convergence and stability of the method are analyzed,and it is shown that the methods can have high accuracy and maintain good stability.Fractional generalized Adams methods are developed for solving the initial value problem of fractional delay differential equations.By using the corresponding matrix structure,the convergence of the methods is analyzed in detail.The stability results of fractional generalized Adams methods for fractional ordinary differential equations are generalized.Based on this result,the numerical stability for fractional delay differential equations is studied,and the relationship of numerical stability of the methods in the two cases is established.For the initial value problem of nonlinear delay differential algebraic equations with index-1 and index-2,the general forms of the corresponding block boundary value methods are obtained,and the convergence analysis of the methods is given.It is proved that if the delay term is properly dealt with,the convergence order of the block boundary value methods is consistent with the underlying one in the case of ordinary differential equations.