Numerical schemes for integro-differential equations with Erdélyi-Kober fractional operator

This work investigates several discretizations of the Erdelyi-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.