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<p>We&#xa0;construct a&#xa0;monotone finite-difference scheme for an&#xa0;initial-boundary value problem for a&#xa0;time-fractional quasilinear parabolic equation.We&#xa0;establish a&#xa0;maximum principle for the corresponding differential problem and derive two-sided estimates for the numerical solution directly from the input data without imposing sign conditions on them.We&#xa0;also obtain a&#xa0;priori estimates in the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ C $</EquationSource> </InlineEquation>-norm for both the exact and approximate solutions.These estimates coincide with the corresponding bounds for the exact solution to the original differential problem, showing that the proposed difference scheme preserves the qualitative properties of the continuous problem.Convergence of the scheme in the grid <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ L_{2} $</EquationSource> </InlineEquation>-norm is proved by combining the energy inequality method with a&#xa0;Gronwall-type inequality.In&#xa0;addition, a&#xa0;convergence criterion is used to establish the stability of the proposed scheme.</p>

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Second-Order Monotone Finite-Difference Schemes for Time-Fractional Quasilinear Parabolic Equations

  • 期刊论文

  • 发表日期：

    2026年7月30日

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  • L. M. Hieu

摘要

We construct a monotone finite-difference scheme for an initial-boundary value problem for a time-fractional quasilinear parabolic equation.We establish a maximum principle for the corresponding differential problem and derive two-sided estimates for the numerical solution directly from the input data without imposing sign conditions on them.We also obtain a priori estimates in the $ C $ -norm for both the exact and approximate solutions.These estimates coincide with the corresponding bounds for the exact solution to the original differential problem, showing that the proposed difference scheme preserves the qualitative properties of the continuous problem.Convergence of the scheme in the grid $ L_{2} $ -norm is proved by combining the energy inequality method with a Gronwall-type inequality.In addition, a convergence criterion is used to establish the stability of the proposed scheme.