Robustness of convergence demonstrated byparametric-guiding andcomplex-root-tunneling algorithms for Bratu’s problem

Purpose This study aims to propose the parametric-guiding algorithm, the complex-root (CR) tunneling algorithm and the method that integrates both algorithms for the heat and fluid flow (HFF) community, and apply them to nonlinear Bratu’s boundary-value problem (BVP) and Blasius BVP. Design/methodology/approach In the first algorithm, iterations are primarily guided by a diminishing parameter that is introduced to reduce magnitudes of fictitious source terms. In the second algorithm, when iteration-related barriers are encountered, CRs are generated to tunnel through the barrier. At the exit of the tunnel, imaginary parts of CRs are trimmed. Findings In terms of the robustness of convergence, the proposed method outperforms the traditional Newton–Raphson (NR) method. For most pulsed initial guesses that resemble pulsed initial conditions for the transient Bratu BVP, the proposed method has not failed to converge. Originality/value To the best of the authors’ knowledge, the parametric-guiding algorithm, the CR tunneling algorithm and the method that integrates both have not been reported in the computational-HFF-related literature.

Subdivision Collocation Method for One-Dimensional Bratu’s Problem

The purpose of this article is to employ the subdivision collocation method to resolve Bratu’s boundary value problem by using approximating subdivision scheme. The main purpose of this researcher is to explore the application of subdivision schemes in the field of physical sciences. Our approach converts the problem into a set of algebraic equations. Numerical approximations of the solution of the problem and absolute errors are compared with existing methods. The comparison shows that the proposed method gives a more accurate solution than the existing methods.

Discrete cubic spline technique for solving one-dimensional Bratu’s problem

In this paper, discrete cubic spline method based on central differences is developed to solve one-dimensional (1D) Bratu’s and Bratu’s type highly nonlinear boundary value problems (BVPs). Convergence analysis is briefly discussed. Four examples are given to justify the presented method and comparisons are made to confirm the advantage of the proposed technique.