Abstract <p>The authors used an effective approximation technique to obtain accurate approximate solutions for a class of second-order non-local, non-linear ordinary differential equations with various non-local terms and boundary conditions that often appear in applied sciences. The underlying mathematical ingredients of the proposed scheme is the finite Whittaker Cardinal function approximation of functions in the basis generating Shannon–Kotelnikov multi-resolution analysis of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2296_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="213" /> </InlineMediaObject> <EquationSource Format="TEX">\({{L}^{2}}(\Omega ),\;(\Omega = [a,\,b] \subset \mathbb{R}\;{\text{or}}\;{{\mathbb{R}}^{ + }})\)</EquationSource>  </InlineEquation>. Formulae relating the exponent <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2296_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\)</EquationSource>  </InlineEquation> in the desired order <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2296_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">\((O{{(10}^{{ - n}}}))\)</EquationSource>  </InlineEquation> of accuracy, the resolution <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2296_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(J\)</EquationSource>  </InlineEquation> of the bandwidth of the approximation space, the dependences of the lower and upper limits in the finite sum in the approximation and a formula for <i>a posteriori</i> error in the approximate solution are provided. The efficiency and elegance of the scheme have been examined for various second-order, non-local, non-linear ordinary differential equations of physical interest and found efficient.</p>

An Efficient DE Sinc Function-Based Approximation Scheme for Non-Local Elliptic Boundary Value Problems

期刊论文
发表日期：

2025年8月1日

查看全文

Sharda Kumari,
Sudam Bin,
Sourav Roy,
M. M. Panja

摘要

Abstract

The authors used an effective approximation technique to obtain accurate approximate solutions for a class of second-order non-local, non-linear ordinary differential equations with various non-local terms and boundary conditions that often appear in applied sciences. The underlying mathematical ingredients of the proposed scheme is the finite Whittaker Cardinal function approximation of functions in the basis generating Shannon–Kotelnikov multi-resolution analysis of \({{L}^{2}}(\Omega ),\;(\Omega = [a,\,b] \subset \mathbb{R}\;{\text{or}}\;{{\mathbb{R}}^{ + }})\) . Formulae relating the exponent \(n\) in the desired order \((O{{(10}^{{ - n}}}))\) of accuracy, the resolution \(J\) of the bandwidth of the approximation space, the dependences of the lower and upper limits in the finite sum in the approximation and a formula for a posteriori error in the approximate solution are provided. The efficiency and elegance of the scheme have been examined for various second-order, non-local, non-linear ordinary differential equations of physical interest and found efficient.