A mean-value Approach to solve fractional differential and integral equations

• Published in: Chaos, solitons and fractals solitons and fractals, 2020-09, Vol.138, p.109895, Article 109895
• Main Authors: De Angelis, Paolo, De Marchis, Roberto, Martire, Antonio Luciano, Oliva, Immacolata
• Format: Article
• Language: English
• Subjects: Fractional differential equation; Fractional Vvolterra integral equations; Mean-value theorem
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2020.109895

•We propose a new numerical method to solve integral partial and/or differential equations.•The rate of converge of the approximated solution depends on the non-integer order of the integral equation.•The algorithm is based on an application of the Mean-Value Theorem and turns out to be effective and flexible. In this paper we provide a new numerical method to solve nonlinear fractional differential and integral equations. The algorithm proposed is based on an application of the fractional Mean-Value Theorem, which allows to transform the initial problem into a suitable system of nonlinear equations. The latter is easily solved through standard methods. We prove that the approximated solution converges to the exact (unknown) one, with a rate of convergence depending on the non-integer order characterizing the fractional equation. To test the effectiveness of our proposal, we produce several examples and compare our results with already existent procedures.