Numerical schemes for integro-differential equations with Erdélyi-Kober fractional operator
Łukasz PłociniczakFaculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wyb. Wyspiańskiego 27, 50370, Wrocław, PolandSzymon SobieszekFaculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wyb. Wyspiańskiego 27, 50370, Wrocław, Poland
2016en
ABI
Abstract
This work investigates several discretizations of the Erdélyi-Kober fractional operator and their use in integro-differential equations. We propose two methods of discretizing E-K operator and prove their errors asymptotic behaviour for several different variants of each discretization. We also determine the exact form of error constants. Next, we construct a finite-difference scheme based on a trapezoidal rule to solve a general first order integro-differential equation. As is known from the theory of Abel integral equations, the rate of convergence of any finite-different method depends on the severity of kernel’s singularity. We confirm these results in the E-K case and illustrate our considerations with numerical examples.
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