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

Abstract

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.

Numerical simulation of Bratu’s problem using a new form of the Adomian decomposition technique

“…Saravi et al (2013) and Das et al (2016) used the variational iteration method (VIM) to analyse the numerical solution of Bratu’s problem. Similarly, the Adomian decomposition method (ADM) (Deeba et al , 2000; Wazwaz, 2005), finite difference method (Temimi and Ben-Romdhane, 2016), Laplace transform decomposition (LTM) (Khuri, 2004), neural network method (Kumar and Yadav, 2015), Laguerre wavelet method (Shiralashetti and Kumbinarasaiah, 2018), restarted Adomian method (Vahidi and Hasanzade, 2012), Bernstein polynomial approximation (Ghomanjani and Shateyi, 2019), Bessel collocation method (BCM) (Roul and Prasad Goura, 2020), Picard’s method (Tomar and Pandey, 2019), parametric-guiding and the complex-root tunnelling algorithm (Liu et al , 2022), the hybrid method (Raja et al , 2016), etc., have been used to analyse the solution of Bratu’s problem.…”

“…algorithm (Liu et al, 2022), the hybrid method (Raja et al, 2016), etc., have been used to analyse the solution of Bratu's problem.…”