<p>This paper presents a uniformly convergent numerical method for solving second-order singularly perturbed Volterra integro-differential equations. The problem features a small perturbation parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varepsilon \in (0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>, which induces a boundary layer characterized by sharp gradients that classical numerical schemes fail to resolve without excessive mesh refinement. To address this, we proposed a Scharfetter–Gummel method for the differential operator, which embeds the asymptotic structure of the solution directly into the discrete flux, thereby capturing the boundary layer accurately even on a uniform mesh. The nonlocal Volterra integral term is approximated using the composite trapezoidal rule. A rigorous stability and convergence analysis establishes that the scheme is <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-uniformly convergent achieving first-order convergence with error constants independent of both <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation> and the mesh width <i>h</i>. Numerical experiments on several test problems confirm the theoretical analysis, demonstrating uniform convergence and superior performance over conventional methods that deteriorate as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varepsilon \rightarrow 0^{+}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>. The results shows the efficiency, accuracy, and robustness of the proposed method.</p>

Scharfetter–Gummel method for singularly perturbed volterra integro-differential equations

期刊论文
发表日期：

2026年8月4日

查看全文

Muath Awadalla,
Mesfin Mekuria Woldaregay

摘要

This paper presents a uniformly convergent numerical method for solving second-order singularly perturbed Volterra integro-differential equations. The problem features a small perturbation parameter $\varepsilon \in (0,1]$ $ε \in (0, 1]$ , which induces a boundary layer characterized by sharp gradients that classical numerical schemes fail to resolve without excessive mesh refinement. To address this, we proposed a Scharfetter–Gummel method for the differential operator, which embeds the asymptotic structure of the solution directly into the discrete flux, thereby capturing the boundary layer accurately even on a uniform mesh. The nonlocal Volterra integral term is approximated using the composite trapezoidal rule. A rigorous stability and convergence analysis establishes that the scheme is $\varepsilon $ $ε$ -uniformly convergent achieving first-order convergence with error constants independent of both $\varepsilon $ $ε$ and the mesh width h. Numerical experiments on several test problems confirm the theoretical analysis, demonstrating uniform convergence and superior performance over conventional methods that deteriorate as $\varepsilon \rightarrow 0^{+}$ $ε \to 0^{+}$ . The results shows the efficiency, accuracy, and robustness of the proposed method.