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<p>This paper demonstrates a numerical stratagem for the solution of two dimensional single and coupled partial differential equations, using the new version of the Haar wavelets namely: the scale-3 Haar wavelets (S3HW), combined with the finite difference formulation. The proposed method consists of two phases. The first phase deals with the numerical estimation of the temporal derivative via finite difference which converts the problem to time discrete form. The second phase describes, the approximation of the spatial derivatives along with solution, adopting S3HW. Then, the collocation technique is implemented to transform the resultant system to the set of linear algebraic equations. Solution of the linear system gives the unknown wavelet coefficients which utilized to determine the numerical solutions. Afterwards, the error, convergence, and stability analysis are conducted and deduced a new error estimate. Besides, the numerical simulations are done to verify the scheme and the obtained theoretical findings (convergence and stability). To validate, the performance of the present scheme different error measures <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44198_2025_277_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="247" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathcal {L}}_{\infty },~{\mathcal {L}}_{2},~\text {root mean square (RMS)}\)</EquationSource> </InlineEquation>, and relative error <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44198_2025_277_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">\((\text {RE})\)</EquationSource> </InlineEquation> are determined numerically. The scheme is also compared in terms of error with the scale-2 Haar wavelets and radial basis functions based algorithms. Overall judgement shows, that the numerical results of the developed scheme are in good agreement with the exact solution and the aforementioned methods in the literature.</p>

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Numerical Scheme for the Computational Study of Two Dimensional Diffusion and Burgers’ Systems with Stability and Error Estimate

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年4月22日

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  • Muhammad Bilal,
  • Abdul Ghafoor,
  • Manzoor Hussain,
  • Kamal Shah,
  • Thabet Abdeljawad

摘要

This paper demonstrates a numerical stratagem for the solution of two dimensional single and coupled partial differential equations, using the new version of the Haar wavelets namely: the scale-3 Haar wavelets (S3HW), combined with the finite difference formulation. The proposed method consists of two phases. The first phase deals with the numerical estimation of the temporal derivative via finite difference which converts the problem to time discrete form. The second phase describes, the approximation of the spatial derivatives along with solution, adopting S3HW. Then, the collocation technique is implemented to transform the resultant system to the set of linear algebraic equations. Solution of the linear system gives the unknown wavelet coefficients which utilized to determine the numerical solutions. Afterwards, the error, convergence, and stability analysis are conducted and deduced a new error estimate. Besides, the numerical simulations are done to verify the scheme and the obtained theoretical findings (convergence and stability). To validate, the performance of the present scheme different error measures \({\mathcal {L}}_{\infty },~{\mathcal {L}}_{2},~\text {root mean square (RMS)}\) , and relative error \((\text {RE})\) are determined numerically. The scheme is also compared in terms of error with the scale-2 Haar wavelets and radial basis functions based algorithms. Overall judgement shows, that the numerical results of the developed scheme are in good agreement with the exact solution and the aforementioned methods in the literature.