Analytical and numerical analysis for a nonlinear fractional differential equations involving the new Caputo–Fabrizio integral

S. Lemita^a, N. Chihi^b, S. Lemouchi<sup>c

a</sup> Department of Mathematics, Echahid Cheikh Larbi Tebessi University, Tebessa, 12000, Algeria
^b Department of Mathematics, Higher Normal School of Technological Education of Skikda, Skikda, 21000, Algeria
^c Department of Mathematics, Ecole Normale Supérieure de Ouargla, Ouargla, 30000, Algeria

Abstract: In this paper we consider nonlinear fractional differential equations involving the new Caputo–Fabrizio derivative of order $\gamma\in ]1, 2[$. We convert the fractional problem to an equivalent nonlinear Volterra integro-differential equation of the second kind, then we investigate the existence and uniqueness of its solution under certain given conditions by using the Schauder fixed point theorem. Finally, we numerically solve the proposed fractional problem by applying the Nyström method, and we provide some suitable examples to support our study.

Key words: fractional differential equation, Caputo–Fabrizio derivative, integro-differential Volterra equation, Schauder fixed point theorem, Nyström method.

MSC: 34K37, 45D05, 47G20, 47H10, 65R20

Received: 23.01.2025
Revised: 31.01.2025
Accepted: 16.06.2025

DOI: 10.15372/SJNM20250401