Diagonal Block Method for Solving Two-Point Boundary Value Problems with Robin Boundary Conditions

Abstract

This numerical study presents the diagonal block method of order four for solving the second-order boundary value problems (BVPs) with Robin boundary conditions at two-point concurrently using constant step size. The solution is obtained directly without reducing to a system of first-order differential equations using a combination of predictor-corrector mode via shooting technique. The shooting method was adapted with the Newton divided difference interpolation approach as the strategy of seeking for the new initial estimate. Five numerical examples are included to examine and illustrate the practical usefulness of the proposed method. Numerical tested problem is also highlighted on the diffusion of heat generated application that imposed the Robin boundary conditions. The present findings revealed that the proposed method gives an efficient performance in terms of accuracy, total function calls, and execution time as compared with the existing method.

Highly Accurate Method for Boundary Value Problems with Robin Boundary Conditions

“…For N = 0, 1, 2, 3, 4, 5 , we get: Problem 7.2 Consider the linear boundary value problem, [9,10,34] subject to the non-homogeneous Robin conditions The exact solution is y(x) = cos x and the computed approximate solution y 11 (x) has the form: This solution agrees perfectly with the exact solution of accuracy 10 −15 as shown in Table 1. ‖E N ‖ ∞ Problem 7.3 Consider the nonlinear boundary value problem, [9,10] subject to the non-homogeneous Robin conditions…”

“…Journal of Nonlinear Mathematical Physics (2023) 30:1239-1263 Problem 7. 4 Consider the nonlinear boundary value problem, [9,10] subject to the non-homogeneous Robin conditions The exact solution is y(x) = ln((x + 2)∕2) and the computed approximate solution y 13 (x) has the form:…”

“…Problem 7.5 Consider the nonlinear boundary value problem, [9,10,35] subject to the non-homogeneous Robin conditions…”

“…Several numerical approaches have been developed and implemented to solve BVP (1.1)- (1.3). In this regard, these approaches include the Adomian decomposition method [4,5], the Laplace transform-homotopy perturbation [6], the homotopy analysis method [7], the finite difference method [8,9], Diagonal block method [10,11], B-spline collocation method [12,13], Cubic Hermite collocation method [14], Gegenbaur integration matrices [15] and the spectral method [16,17].…”

Finite difference spectral collocation schemes for the solutions of boundary value problems

“…Here we slightly modify Scheme (15) and extend its application to the problem [36] subject to the Robin conditions The exact solution is known to be .…”