Solving the fourth-order nonlinear boundary value problem by a boundary shape function method

Abstract

A method to construct the boundary shape function (BSF) and then two novel methods are developed to obtain the solutions of fourth-order singularly perturbed beam equation and nonlinear boundary value problem (BVP). In the first-type algorithm, the free function is a series of complete basis functions, while the corresponding BSFs are new bases. The trial functions with fractional powers exponential are suitable for the singularly perturbed beam equation under fixed-end and simply supported boundary conditions. With the aid of the BSF, we can improve the asymptotic and uniform approximations to exactly satisfy the prescribed boundary conditions. In the second-type algorithm, the solution of a nonlinear BVP is viewed as a boundary shape function, while the free function is regarded as a new variable. With this means, the fourth-order nonlinear BVP is exactly converted to an initial value problem with a new variable, the terminal value of which is unknown, when the initial conditions are given. The computed order of convergence and an error estimation are given. Numerical illustrations, including the singularly perturbed examples, show that the present methods, based on the new idea of the BSF, are highly effective, accurate, and fast convergent.