An integral discretization scheme on a graded mesh for a fractional differential equation with integral boundary conditions

• Published in: Journal of mathematical chemistry Vol. 62; no. 6; pp. 1384 - 1398
• Main Authors: Cen, Zhongdi, Huang, Jian, Xu, Aimin
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2024; Springer Nature B.V
• Subjects: Boundary conditions; Chemistry; Chemistry and Materials Science; Differential equations; Discretization; Exact solutions; Fractional calculus; Integral equations; Math. Applications in Chemistry; Mathematical analysis; Original Paper; Physical Chemistry; Quasi Newton methods; Theoretical and Computational Chemistry; Truncation errors; Fractional differential equation; 65L05; Graded mesh; Grönwall inequality; 65L20; Integral boundary conditions; 65L12; Caputo fractional derivative
• ISSN: 0259-9791; 1572-8897
• DOI: 10.1007/s10910-024-01596-7

In this paper, a fractional differential equation with integral conditions is studied. The fractional differential equation is transformed into an integral equation with two initial values, where the initial values needs to ensure that the exact solution satisfies the integral boundary conditions. A graded mesh based on a priori information of the exact solution is constructed and the linear interpolation is used to approximate the functions in the fractional integral. The rigorous analysis about the convergence of the discretization scheme is derived by using the truncation error estimate techniques and the generalized Grönwall inequality. A quasi-Newton method is used to determine the initial values so that the numerical solution satisfies two integral boundary conditions within a prescribed precision. It is shown that the scheme is second-order convergent, which improves the results on the uniform mesh.