10.1007/s40096-022-00505-w

A high-order numerical method for solving nonlinear derivative-dependent singular boundary value problems using trigonometric B-spline basis function

  • Mohammad Prawesh Alam1,
  • Arshad Khan*1,
  1. Department of Mathematics, Jamia Millia Islamia, New Delhi, IN

Published in Issue 2023-01-31

How to Cite

Alam, M. P., & Khan, A. (2023). A high-order numerical method for solving nonlinear derivative-dependent singular boundary value problems using trigonometric B-spline basis function. Mathematical Sciences, 18(3 (September 2024). https://doi.org/10.1007/s40096-022-00505-w
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Abstract

Abstract In the present work, we have developed high-order computationally reliable numerical method based on trigonometric quintic B-spline basis functions for solving a class of nonlinear derivative-dependent singular boundary value problems, which arises in the study of the charge densities and potential in many scientific models like atoms, molecules, metals, and crystals are presented. To develop the method, we first apply the quasilinearization technique to the original problem. This method annihilates the singular behavior at the point z=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${z}=0$$\end{document} and handles this problem very easily. The error and convergence analysis of the proposed method is substantiated through the matrix approach. It is proved that the developed method preserves O(h4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {O}(h^4)$$\end{document} convergent approximation to the solution of the underlying problem. We have solved three numerical examples to prove the efficiency and robustness of the method and to validate the theoretical results. It is displayed that the rate of convergence of the present method is higher when compared to the cubic B-spline collocation method (Roul and Goura in Appl Math Comput 341:428–450, 2019) and the Bernstein collocation method (Shahni and Singh in Eng Comput:1–10, 2020), respectively. It is shown that the developed method executes superior to the continuing methods due to its easy accomplishment, and it takes very less computational cost.

Keywords

  • Nonlinear derivative-dependent boundary value problems,
  • Trigonometric quintic B-spline functions,
  • Quasilinearization technique,
  • Convergence analysis with matrix approach

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