<p>In this paper, we propose and analyze a nonstandard finite difference (NSFD) scheme for a class of coupled nonlinear singular boundary value problems. The simultaneous presence of nonlinearity and singularity presents substantial analytical and computational challenges for classical discretization techniques. To overcome these difficulties, we construct an NSFD scheme incorporating nonlinear denominator functions such as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\sin (\Delta t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>sin</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\tanh {(\Delta t)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>tanh</mo> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, etc., in place of the conventional step size <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Delta t$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </math></EquationSource> </InlineEquation>. Two auxiliary parameters, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$k_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$k_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>, are introduced within the denominator functions to enhance stability and accuracy. A rigorous convergence analysis of the proposed method is carried out, establishing sufficient conditions under which the discrete solution converges to the continuous solution. The theoretical results demonstrate the consistency and stability properties of the scheme. The role of parameters <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$k_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$k_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> in improving numerical behavior is examined, and appropriate choices are identified. Numerical experiments on representative test problems confirm the theoretical findings and show that the proposed method attains improved accuracy and computational efficiency compared to existing numerical approaches.</p>

A parameterized NSFD framework for coupled nonlinear singular BVPs

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发表日期：

2026年7月8日

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Vamika Rathi,
Sheerin Kayenat,
Amit K. Verma

摘要

In this paper, we propose and analyze a nonstandard finite difference (NSFD) scheme for a class of coupled nonlinear singular boundary value problems. The simultaneous presence of nonlinearity and singularity presents substantial analytical and computational challenges for classical discretization techniques. To overcome these difficulties, we construct an NSFD scheme incorporating nonlinear denominator functions such as $\sin (\Delta t)$ $sin (Δ t)$ , $\tanh {(\Delta t)}$ $tanh (Δ t)$ , etc., in place of the conventional step size $\Delta t$ $Δ t$ . Two auxiliary parameters, $k_1$ $k_{1}$ and $k_2$ $k_{2}$ , are introduced within the denominator functions to enhance stability and accuracy. A rigorous convergence analysis of the proposed method is carried out, establishing sufficient conditions under which the discrete solution converges to the continuous solution. The theoretical results demonstrate the consistency and stability properties of the scheme. The role of parameters $k_1$ $k_{1}$ and $k_2$ $k_{2}$ in improving numerical behavior is examined, and appropriate choices are identified. Numerical experiments on representative test problems confirm the theoretical findings and show that the proposed method attains improved accuracy and computational efficiency compared to existing numerical approaches.