A fast ADI orthogonal spline collocation method with graded meshes for the two-dimensional fractional integro-differential equation

• Published in: Advances in computational mathematics Vol. 47; no. 5
• Main Authors: Qiao, Leijie, Xu, Da
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2021; Springer Nature B.V
• Subjects: Collocation methods; Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Differential equations; Error analysis; Kernels; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Singularities; Stability analysis; Visualization; Fractional integro-differential equation; 65M15; 65M70; Alternating direction implicit method; 45E10; Orthogonal spline collocation method; 35R11; Second-order backward differentiation formula; Convergence
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-021-09884-5

We propose and analyze a time-stepping Crank-Nicolson(CN) alternating direction implicit(ADI) scheme combined with an arbitrary-order orthogonal spline collocation (OSC) methods in space for the numerical solution of the fractional integro-differential equation with a weakly singular kernel. We prove the stability of the numerical scheme and derive error estimates. The analysis presented allows variable time steps which, as will be shown, can efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term. Finally, some numerical tests are given.