<p>This paper introduces a high-order efficient algorithm for approximating a nonlinear time-fractional biharmonic equation with an initial singularity. The Caputo fractional derivative is employed, and a second-order scheme is developed to discretize the time derivative on nonuniform time steps, effectively addressing the initial singularity. For the spatial derivative, a high-order non-polynomial parametric quintic spline method is considered. The proposed approach efficiently handles the initial singularity and reduces computational cost through a fast nonuniform time discretization scheme. The resulting method is computationally efficient, with a complexity of approximately <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal{O}(MN \log^2 N)\)</EquationSource> </InlineEquation> and storage requirements of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal{O}(M \log^2 N)\)</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(N\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(M\)</EquationSource> </InlineEquation> denote the total number of grid points in the time and spatial directions, respectively. Furthermore, the method is proven to be unconditionally stable and convergent, with an error of order <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathcal{O}\left(h^{4.5} + N^{-\min\{r\mu, 2\}}+\epsilon\right)\)</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mu\)</EquationSource> </InlineEquation> is the fractional derivative order, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(r\)</EquationSource> </InlineEquation> is the mesh grading parameter, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(h\)</EquationSource> </InlineEquation> is the spatial mesh size and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\epsilon\)</EquationSource> </InlineEquation> represents the approximation tolerance introduced by the fast sum-of-exponentials technique. Numerical experiments are presented to validate the theoretical analysis and demonstrate the method’s effectiveness in achieving high accuracy and computational efficiency.</p>

Efficient higher-order approximations for a nonlinear time-fractional biharmonic equation with initial singularity

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2026年4月24日

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Richa Singh,
Sunil Kumar,
Higinio Ramos

摘要

This paper introduces a high-order efficient algorithm for approximating a nonlinear time-fractional biharmonic equation with an initial singularity. The Caputo fractional derivative is employed, and a second-order scheme is developed to discretize the time derivative on nonuniform time steps, effectively addressing the initial singularity. For the spatial derivative, a high-order non-polynomial parametric quintic spline method is considered. The proposed approach efficiently handles the initial singularity and reduces computational cost through a fast nonuniform time discretization scheme. The resulting method is computationally efficient, with a complexity of approximately \(\mathcal{O}(MN \log^2 N)\) and storage requirements of \(\mathcal{O}(M \log^2 N)\) , where \(N\) and \(M\) denote the total number of grid points in the time and spatial directions, respectively. Furthermore, the method is proven to be unconditionally stable and convergent, with an error of order \(\mathcal{O}\left(h^{4.5} + N^{-\min\{r\mu, 2\}}+\epsilon\right)\) , where \(\mu\) is the fractional derivative order, \(r\) is the mesh grading parameter, \(h\) is the spatial mesh size and \(\epsilon\) represents the approximation tolerance introduced by the fast sum-of-exponentials technique. Numerical experiments are presented to validate the theoretical analysis and demonstrate the method’s effectiveness in achieving high accuracy and computational efficiency.