Solving higher-order nonlocal boundary value problems with high precision by the fixed quasi Newton methods

The paper presents a numerical solution of a higher-order nonlocal and nonlinear boundary value problem (NNBVP); it exhibits triple nonlinearities: nonlinear ordinary differential equation (ODE), nonlinear local boundary values and nonlinear integral boundary conditions (BCs). We consider one nonlocal, two nonlocal and three nonlocal BCs. New variables are incorporated, whose values at right-end present the integral parts in the specified nonlocal BCs. For an extremely precise solution of NNBVP, it is crucial to acquire highly accurate initial values of independent variables by solving target equations very accurately. Because of the implicit form of the target equations, they are solved by a fixed quasi-Newton method (FQNM) without calculating the differentials, which together with the shooting technique would offer nearly exact initial values. Higher-order numerical examples are examined to assure that the proposed methods are efficient to achieve some highly accurate solutions with 14- and 15-digit precisions, whose computational costs are very saving as reflected in the small number of iterations. The computed order of convergence (COC) reveals that the iterative methods converge faster than linear convergence. The novelty lies on the introduction of new variables to present the nonlocal boundary conditions, and develop FQNM to solve the target equations. The difficult part of nonlinear and nonlocal BC is simply transformed to find an ended value of the new variable governed by an ODE. When the constant iteration matrix is properly established, the numerical examples of NNBVP reveal that FQNM is convergent fast and is not sensitive to the initial guesses of unknown initial values.