Analysis and numerical solution of a Riemann-Liouville fractional derivative two-point boundary value problem

作者：Kopteva Natalia; Stynes Martin^*

来源：Advances in Computational Mathematics, 2017, 43(1): 77-99.

DOI：10.1007/s10444-016-9476-x

• Fractional differential equation
• Riemann-Liouville fractional derivative
• Boundary value problem
• Weakly singular Volterra integral equation
• Collocation method

A two-point boundary value problem is considered on the interval [0, 1], where the leading term in the differential operator is a Riemann-Liouville fractional derivative of order 2 - delta with 0 < delta < 1. It is shown that any solution of such a problem can be expressed in terms of solutions to two associated weakly singular Volterra integral equations of the second kind. As a consequence, existence and uniqueness of a solution to the boundary value problem are proved, the structure of this solution is elucidated, and sharp bounds on its derivatives (in terms of the parameter delta) are derived. These results show that in general the first-order derivative of the solution will blow up at x = 0, so accurate numerical solution of this class of problems is not straightforward. The reformulation of the boundary problem in terms of Volterra integral equations enables the construction of two distinct numerical methods for its solution, both based on piecewise-polynomial collocation. Convergence rates for these methods are proved and numerical results are presented to demonstrate their performance.

• 出版日期2017-2
• 单位北京计算科学研究中心