ON FRACTIONAL INTEGRALS AND DERIVATIVES OF A FUNCTION WITH RESPECT TO ANOTHER FUNCTION
Juan J. Nieto, Madeaha Alghanmi, Bashir Ahmad, Ahmed Alsaedi, Boshra Alharbi
Abstract
In this paper, we present new definitions of generalized fractional integrals and derivatives with respect to another function and derive some of their properties, such as their inter-relationship and semigroup law. Caputo-type generalized fractional derivative with respect to another function is also defined and its properties are derived. A Cauchy problem involving the new Caputo-type generalized fractional derivative is also studied. We also provide an expansion formula for Caputo-type derivative and apply it to solve a fractional-order problem.
Topics & Concepts
Fractional calculusMathematicsMittag-Leffler functionSemigroupType (biology)Cauchy distributionFunction (biology)Applied mathematicsOrder (exchange)Derivative (finance)Pure mathematicsMathematical analysisEvolutionary biologyEcologyBiologyEconomicsFinanceFinancial economicsFractional Differential Equations SolutionsNonlinear Differential Equations AnalysisIterative Methods for Nonlinear Equations