The paper considers the second-order nonlinear boundary value problem (NBVP), which is equipped with nonlinear integral boundary conditions (BCs). Two novel iterative algorithms are developed to overcome the difficulty of NBVP with double nonlinearities involved. In the first iterative algorithm, two nonlocal shape functions incorporating the linear integral terms are derived, and a nonlocal boundary shape function (NBSF) is formulated to assist the solution. Let the solution be the NBSF so that the NBVP can be exactly transformed into an initial value problem. The new variable is a free function in the NBSF, and its initial values are given. For the NBVP with linear integral BCs, three unknown constants are to be determined, while for the nonlinear integral BCs, five unknown constants are to be determined. Two-point local shape functions and local boundary shape functions are derived for the second iterative algorithm, wherein the integral terms in the boundary conditions are viewed as unknown constants. By a few iterations, four unknown constants can be determined quickly. Through numerical experiments, these two iterative algorithms are found to be powerful for seeking quite accurate solutions. The second algorithm is slightly better than the first, with fewer iterations and a more accurate solution.
Liu, Chein-Shan; Chen, Yung-Wei; and Shen, Jian-Hung
"Solving Nonlinear Boundary Value Problems with Nonlinear Integral Boundary Conditions by Local and Nonlocal Boundary Shape Functions Methods,"
Journal of Marine Science and Technology: Vol. 30:
6, Article 1.
Available at: https://jmstt.ntou.edu.tw/journal/vol30/iss6/1