A hybrid reptile search algorithm and Levenberg–Marquardt algorithm based Haar wavelets to solve regular and singular boundary value problems

Various engineering applications lead to the appearance of partial differential equations resulting in boundary value problems (BVPs). Orthogonal collocation method based Haar wavelets has gained significant attention in solving these problems. The Haar wavelets have several properties like vanishing moment, compact effect, and orthogonality. These properties prioritize being used as base functions for solving BVPs. However, the approximation leads to a relatively high number of unknown coefficients, which needs an efficient and reliable nonlinear solver to reach their values. The Levenberg–Marquardt algorithm (LM) is one of the most efficient nonlinear solvers. However, it may diverge in case of too far initial guesses, especially in many unknowns. The reptile search algorithm (RSA) is a recent reliable, nature-inspired optimization technique that has a noticeable capability in dealing with high-dimensional issues. Therefore, this paper proposes a hybrid optimization algorithm that integrates the RSA and LM algorithms using Haar wavelets as bases, named RSA–LM–Haar algorithm, to solve regular and singular natures of the BVPs more efficiently. To evaluate the performance of the proposed RSA–LM–Haar algorithm, it is tested on twelve case studies of BVPs including the singular and regular problems. The results are compared with those based on the LM alone, named LM–Haar algorithm. Finally, the applicability of the proposed algorithm is verified using two practical chemical applications to prove its ability to solve real-time applications effectively. All the results affirmed the capability of the proposed algorithm in solving both regular and singular BVPs. All results and comparisons illustrated that the proposed hybridization algorithm provides remarkable performance.