The Research Of Two Classes Of Multi-Point Boundary Value Problems For Nonlinear Differential Equations

In the practical engineering applications,many mathematical models of the nonlinear vibration problems can be simulated by using Duffing equation.For some of the more complex vibrations,the nonlinear term is significant.Here we consider the generalized Duffing equation.Fractional calculus is an important extension of the typical calculus.The generalized fractional Bagley-Torvik equation with variable coefficient is with more profound physical background and good application.In this thesis,by considering the two kinds of important nonlinear differential equation with multi-point boundary value conditions,the existence,uniqueness and approximation of solutions are addressed.The main contents are given as follows:(1)The generalized Duffing equation with multi-point boundary value conditions is transformed into the nonlinear Hammerstein integral equation of second kind by applying the integration method.According to the fixed point principle on the Banach space,the uniqueness of the solutions is studied in the square integrable space.And the integral mean value theorem and piecewise approximation thought is proposed to construct the approximate solution about obtained nonlinear Hammerstein integral equation of the second kind.The convergence and error estimate of the approximate solution are analyzed.Numerical results are carried out to show the feasibility and effectiveness of the proposed method by comprising with the existing methods.(2)The nonlinear Bagley-Torvik equation with variable coefficients and four-point boundary value conditions is considered.The nonlinear Fredholm-Hammerstein integral equations of the second kind with weakly singular kernel or continuous kernel are determined.The uniqueness of solutions is further studied by using the fixed point theory in the continuous function space.The integral-type piecewise Taylor series expansion method is proposed to obtain the approximate solution of the obtained nonlinear Fredholm-Hammerstein integral equation with weakly singular kernel.The obtained theorems are verified by analyzing the convergence,error estimate and numerical computing of the approximate solution.The obtained results enhance the theory and numerical methods of boundary value problems of ordinary differential equations.The observations give some theoretical foundations for simulating the physics and mechanical procedures.