The Studies Of Solutions About Several Types Of Differential Equations And Integral Equations

In the many fields of applied mathematics and applied physics and so on, since integral equations and differential equations have extensive applications, and many practical problems can be transformed into integral equations or differential equation boundary value problems to study, hence, in this thesis, we have done more thorough researches on the existence of solutions of several classes of differential equations and integral equations. The main works of this thesis are arranged as follows:In Chapter one, we briefly introduce the general situation both at home and abroad of integral equations and boundary value problems of differential equations.In Chapter two, we mainly study the solutions of a class of Volterra integral equations. We first prove the existence and uniqueness of the solution of Volterra integral equation by using Banach fixed point theorem, and then give several concrete solving methods. For some integral equations without analytical solution, we give a solving method of numerical solution (numerical integral method) to solve this kind of integral equations.In Chapter three, we mainly study the existence of positive solutions for a class of three-point boundary value problems of the third order singular nonlinear differential equations. By using the cone expansion and compression fixed point theorem, we obtain some existence results of single and multiple positive solutions for these problems.In Chapter four, we mainly study the numerical solutions of a class of obstacle boundary value problems of the third order ordinary differential equations. Firstly, we introduce the basic theoretical knowledge for the problems; secondly, we give several classical solving methods; thirdly, via our analysis, we construct a new algorithm of non-polynomial spline functions with base functions consisted of exponential function, trigonometric function and polynomial function; finally, we prove that the proposed algorithm has the second-order convergence in theory, and the local truncation error is o(h6) by using Taylor expansion formula, and the error analysis is done by a concrete example, and consequently, the used algorithm is effective and feasible.