The Research Of Algorithms For Solving Several Differential Equations With Multi-point Boundary Value Problems

Multi-point boundary value problems arise in many fields of applied mathematicsand physics. The issues in non-uniform electromagnetic field theory, water and wet soildynamics and elastic stability theory can come down to the diferential equations withmulti-point boundary value problems. It is very important to solve the diferential equa-tions with multi-point boundary value problems because of the wide application to phys-ical fields. However, some data on the discrete points could be obtained frequently inthe practical problems and their analytical solutions are difcult to be obtained usually.Therefore, efective numerical algorithms become main tools to solve these problems.In this thesis, five chapters are included and numerical algorithms are emphaticallyproposed for several diferential equations with multi-point boundary value problems.The main research contents are as follows:Firstly, using the idea of weighted functions, we construct two kinds of weightedreproducing kernel spaces：one kind is on an finite interval and another kind is on aninfinite interval. The former spaces expand the scope of the functions in the reproducingkernel spaces and the latter spaces make their functions unbounded and satisfy the specialconditions in the infinity points. Meanwhile, we present the computational methods andexplicit expressions of the two kinds of weighted reproducing kernel functions.Secondly, the numerical algorithms of three-point boundary value problems with alimit condition on an infinite interval are obtained by constructing the projection iterativesequences in the three-point weighted reproducing kernel space. Moreover, we prove thatthe projection iterative sequences converge uniformly to the exact solution of equations.Furthermore, we obtain the calculation formulas for multi-point weighed reproducingkernel functions and extend the algorithms to nonlinear diferential equations with multi-point boundary value problems on an infinite interval. In addition, the error estimationand complexity analysis of the algorithms are presented.Thirdly, the reproducing kernel methods are applied to solve nonlinear Riemann-Liouville fractional diferential equations with three-point boundary value problems andCaputo fractional integro-diferential equations with multi-point boundary value prob-lems. The reproducing kernel spaces for the fractional diferential models are established by the relations between two fractional derivative definitions. Moreover, the exact solu-tions for the nonlinear fractional diferential equations with multi-point boundary valueconditions are obtained by the good properties of the reproducing kernel spaces. Fur-thermore, the approximate solutions of the equations are given using the least squareprinciple.Finally, by constructing the reproducing kernel spaces with space-time three-pointboundary value conditions, we give the Fourier series approximation algorithms for aclass of partial diferential equations with three-point boundary value problems. Thenthe convergence of the algorithms are proved. Moreover, we extend the algorithms tomulti-point boundary value problems of parabolic partial diferential equations with morecomplex integral conditions. Numerical simulations are carried out to verify the efective-ness.