A matrix technique‐based numerical treatment of a nonlocal singular boundary value problems

• Published in: Mathematical methods in the applied sciences Vol. 48; no. 7; pp. 8322 - 8341
• Main Authors: Sriwastav, Nikhil, Barnwal, Amit K., Srivastav, Avinash Kumar, Chandra, Harish
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 15.05.2025
• Subjects: Astrophysics; Bernoulli polynomials; Boundary conditions; Boundary value problems; collocation method; Collocation methods; Differential equations; Error analysis; Exact solutions; Initial conditions; Lipschitz condition; Mathematical analysis; Operators (mathematics); Polynomials; Series expansion; singular boundary value problems; three‐point boundary value problems
• DOI: 10.1002/mma.9809
• ISSN: 0170-4214

The mathematical modeling of the decisive event of astrophysics, physiology, and many other areas of science and technology witness the involvement of singular boundary value problems. The nonlocal boundary conditions are more informative than local boundary conditions (initial conditions and two‐point boundary conditions) to evaluate some mathematical models. This article presents a collocation approach‐based matrix technique to approximate the solution of the fusion of a class of singular differential equations subject to nonlocal three‐point boundary conditions. The proposed strategy utilizes the truncation of the series expansion of a function belonging to L2[0,1]$$ {L}_2\left[0,1\right] $$ in terms of Bernoulli polynomials. It transforms the singular boundary value problems into a set of nonlinear algebraic equations, which can be dealt with by any mathematical software. The Lipschitz condition on an equivalent completely continuous nonlinear operator has been used to prove the convergence analysis of the scheme. Some extremely nonlinear test examples are solved and provided in contrast with the exact solution. These numerical results are also examined against some existing numerical techniques to verify the applicability and significance of the proposed methodology. There are a few numerical examples that are application based but do not have exact solutions. In such cases, residual error norm is employed to measure the accuracy of the numerical strategies. The computational data demonstrate the superiority and validity of the proposed technique over existing numerical approaches.