Scientific African (Jun 2025)
Construction and analysis of numerical algorithms for Erdelyi–Kober fractional integral equations
Abstract
Erdelyi–Kober fractional integral (EKI) operator is a generalization of Riemann–Liouville integral (RLI) and has applications in diverse areas. However, unlike the RLI whose singular kernel is linear in the integration variable, the singular kernel of the EKI is nonlinear in the integration variable. This makes approximating EKI equations (EKIEs) quite challenging and studies on them are scanty. The few existing methods are mostly based on change of variables which increases computational complexity. In this paper, we propose two methods, by using linear interpolations to reduce the problem to computing beta functions which are readily available in almost all scientific computing platforms. This culminated into one explicit and one predictor–corrector method. Our approach facilitates software re-usability and eliminates the computational inefficiency arising from increased degrees of freedom and use of non-uniform grids. Stability and convergence results are established using interpolation theory and discrete Gronwall inequality. Our theoretical results suggest that any first order method can be used as a predictor in the Adams method; three predictors are investigated. Numerical experiments are provided to support the theoretical results — that the methods converge. The results further revealed that although all the three investigated predictors can be used in the Adams method and still achieve second order of convergence, only the Euler’s predictor can serve as a stand-alone first order numerical method. The other two predictors are not recommended as stand-alone numerical methods for fractional EKIEs. The implication of this result is that not all successful predictors in fractional calculus are necessarily successful stand-alone methods. This will help practitioners to be circumspect in choosing methods for fractional calculus problems.
