The main focus of this work is to develop a hybrid numerical method based on the L1 scheme and the multi-dimensional Hermite wavelets. We discuss the stability and convergence analysis on a newly designed time-graded mesh to address a class of time-fractional delay partial differential equations involving multi-singularities. In the context of multi-singularities, there are significant challenges for higher-dimensional problems, and the available analytical framework exhibits substantial limitations. Addressing these challenges requires innovative approaches that can effectively navigate the increased complexity of higher-dimensional problems while maintaining analytical rigor and computational efficiency. We use the L1 scheme to convert the proposed problem into a semi-discrete form. The stability and convergence of the temporal semi-discretization on the newly constructed graded mesh are analyzed based on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {L}_{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">L</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation>-norm that leads to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {O}(N^{-(2-\alpha )})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>N</mi> <mrow> <mo>-</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>-</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> temporal rate of accuracy for a suitably chosen grading parameter. The strength of the newly constructed mesh is that it provides a more robust and accurate approach to address multi-singularities and has less computational cost to achieve the desired accuracy compared to other meshes available in the literature. The multi-dimensional Hermite wavelet approximation is taken into account to solve the semi-discrete problem and we use uniformly distributed collocation points in the spatial direction to estimate the unknown wavelet coefficients. Further, the convergence analysis of the proposed hybrid numerical approximation leads to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {O}(N^{-(2-\alpha )})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>N</mi> <mrow> <mo>-</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>-</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> rate of accuracy over the space-time domain based on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {L}_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">L</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-norm for a suitable choice of the grading parameter. In particular, the performance of the hybrid numerical approach is verified through numerous complex problems involving multiple delay parameters.

An efficient hybrid numerical approach for time-fractional sub-diffusion equations with multi-singularities

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发表日期：

2025年6月18日

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Subham Patel,
Sudarshan Santra,
Ratikanta Behera

摘要

The main focus of this work is to develop a hybrid numerical method based on the L1 scheme and the multi-dimensional Hermite wavelets. We discuss the stability and convergence analysis on a newly designed time-graded mesh to address a class of time-fractional delay partial differential equations involving multi-singularities. In the context of multi-singularities, there are significant challenges for higher-dimensional problems, and the available analytical framework exhibits substantial limitations. Addressing these challenges requires innovative approaches that can effectively navigate the increased complexity of higher-dimensional problems while maintaining analytical rigor and computational efficiency. We use the L1 scheme to convert the proposed problem into a semi-discrete form. The stability and convergence of the temporal semi-discretization on the newly constructed graded mesh are analyzed based on $\mathcal {L}_{\infty }$ $L_{\infty}$ -norm that leads to $\mathcal {O}(N^{-(2-\alpha )})$ $O (N^{- (2 - α)})$ temporal rate of accuracy for a suitably chosen grading parameter. The strength of the newly constructed mesh is that it provides a more robust and accurate approach to address multi-singularities and has less computational cost to achieve the desired accuracy compared to other meshes available in the literature. The multi-dimensional Hermite wavelet approximation is taken into account to solve the semi-discrete problem and we use uniformly distributed collocation points in the spatial direction to estimate the unknown wavelet coefficients. Further, the convergence analysis of the proposed hybrid numerical approximation leads to $\mathcal {O}(N^{-(2-\alpha )})$ $O (N^{- (2 - α)})$ rate of accuracy over the space-time domain based on $\mathcal {L}_{2}$ $L_{2}$ -norm for a suitable choice of the grading parameter. In particular, the performance of the hybrid numerical approach is verified through numerous complex problems involving multiple delay parameters.