Abstract
Nonlinear boundary conditions are difficult to be fulfilled exactly, when one employs numerical methods to treat a highly nonlinear boundary value problem (NBVP). In this paper, a novel iterative algorithm to solve NBVP involved with two coupled nonlinear boundary conditions at two-end of a unit interval is developed, of which the solution can satisfy the nonlinear boundary conditions automatically. By letting the free function in the boundary shape function (BSF) be a new variable, an initial value problem (IVP) is created from the second-order NBVP. While the initial values of the new variable are given, the terminal values are viewed as unknown parameters to be determined iteratively. Therefore, a very accurate solution for the NBVP with nonlinear boundary conditions can be quickly determined through a few iterations. Some numerical examples confirm the efficiency and accuracy of the proposed iterative scheme, wherein the examples with multiple solutions and unique solution are worked out.
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