Affiliation:
1. College of Mathematics and Information Science, Hebei University, Baoding 071002, China
Abstract
This study investigates the initial value problem of high-order variable-order φ-Hilfer fractional implicit integro-differential equations. Due to the lack of the semigroup property in variable-order fractional integrals, solving these equations presents significant challenges. We introduce a novel approach that approximates variable-order fractional derivatives using a piecewise constant approximation method. This method facilitates an equivalent integral representation of the equations and establishes the Ulam stability criterion. In addition, we explore higher-order forms of fractional-order equations, thereby enriching the qualitative and stability results of their solutions.
Funder
National Natural Science Foundation of China
Reference14 articles.
1. Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier.
2. Basic theory of fractional differential equations;Lakshmikantham;Nonlinear Anal. Theory Methods Appl.,2008
3. Analysis of fractional differential equations;Diethelm;J. Math. Anal. Appl.,2002
4. Analytical and numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations;Phys. A Stat. Mech. Appl.,2018
5. A review on variable-order fractional differential equations: Mathematical foundations, physical models, numerical methods and applications;Sun;Fract. Calc. Appl. Anal.,2019