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<p>In this work, we propose a computational approach for solving a multi term time-fractional convection diffusion reaction (MTTF-CDR) equation with variable coefficients. The Caputo sense is used to characterise the temporal fractional derivatives. In general, the solutions to these problems often exhibits a weak singularity at the initial time. To deal with the initial weak singularity, we consider a graded mesh on the time domain. The proposed method involves applying the non-uniform <i>L</i>1 formula on the graded mesh for the discretization of time-fractional derivative and a fourth-order compact difference approximation on the uniform mesh for the spatial discretization. The stability and error estimates of the proposed numerical scheme are thoroughly analyzed in the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {L}^2\)</EquationSource> </InlineEquation>-norm. The theory shows that the method is unconditionally stable for all <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\beta _l \in (0, 1)\)</EquationSource> </InlineEquation> with a convergence order of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal {O}( N^{-\min \{2-\beta _l,\gamma \beta _l\}}+h^4)\)</EquationSource> </InlineEquation>. To demonstrate the theoretical results, numerical experiments are carried out.</p>

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Numerical solution of a multi term time-fractional convection diffusion reaction equation with variable coefficients subjected to weakly singular solution

  • 期刊论文

  • 发表日期：

    2025年11月8日

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  • Jyoti Yadav,
  • Pradip Roul

摘要

In this work, we propose a computational approach for solving a multi term time-fractional convection diffusion reaction (MTTF-CDR) equation with variable coefficients. The Caputo sense is used to characterise the temporal fractional derivatives. In general, the solutions to these problems often exhibits a weak singularity at the initial time. To deal with the initial weak singularity, we consider a graded mesh on the time domain. The proposed method involves applying the non-uniform L1 formula on the graded mesh for the discretization of time-fractional derivative and a fourth-order compact difference approximation on the uniform mesh for the spatial discretization. The stability and error estimates of the proposed numerical scheme are thoroughly analyzed in the \(\mathcal {L}^2\) -norm. The theory shows that the method is unconditionally stable for all \(\beta _l \in (0, 1)\) with a convergence order of \(\mathcal {O}( N^{-\min \{2-\beta _l,\gamma \beta _l\}}+h^4)\) . To demonstrate the theoretical results, numerical experiments are carried out.