Several Three-Point Boundary Value Problems For Secong-Order Ordinary Differential Equations In Mechanics

In order to accurately simulate the physical procedures, it is necessary to study systematically the second-order ordinary differential equations with non-local boundary value conditions in mechanics. In this dissertation, several linear, nonlinear and fractional-order ODEs are considered. The existence, uniqueness and approximation of solutions are addressed under three-point boundary value conditions. The main contents are given as follows:In Chapter 1, the background and the research progress of damped vibration equations, Duffing equations and fractional-order Bagley-Torvik equations are introduced briefly.Chapter 2 focuses on the relative preliminaries used throughout this dissertation including the definitions of fractional calculus, integral equations and some fixed point theorems.In Chapter 3, the generalized damped vibration equation with three-point boundary value conditions is transformed into the Fredholm integral equation of the second kind by applying the integration method. The uniqueness of the solutions is studied by using the contraction operator principle in the square integrable space. The differential-type piecewise Taylor series expansion method is proposed to construct the approximate solution of the obtained Fredholm integral equation. 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.Chapter 4 discusses the generalized Duffing equation with three-point boundary value conditions. The Hammerstein integral equation of the second kind is given. The existence and uniqueness of solution are dealt with by using the Schauder fixed point theorem and the Banach contraction mapping principle, and some sufficient conditions are given. The piecewise Taylor-series expansion method is extended to obtain the approximate solution of the nonlinear integral equation. The convergence and error estimate of the approximate solution are made, and the proposed method is verified through numerical examples.Chapter 5 considers the fractional Bagley-Torvik equation with variable coefficients and three-point boundary value conditions. The Fredholm integral equations of the second kind with weakly singular kernel or continuous kernel are determined. The uniqueness of solution 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 Fredholm integral equation with weakly singular kernel. The obtained theorems are verified by analyzing the convergence, error estimate and numerical computing of the approximate solution.Finally, the conclusions are shown, and the problems studied in the future are proposed.