A class of fractional differential equations via power non-local and non-singular kernels: Existence, uniqueness and numerical approximations

• Autor(a) principal: Zitane, Hanaa
• Data de Publicação: 2023
• Outros Autores: Torres, Delfim F. M.
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10773/39889
• topic: Fractional initial value problems; Gronwall’s inequality; Non-singular kernels; Numerical methods; Power fractional calculus

We prove a useful formula and new properties for the recently introduced power fractional calculus with non-local and non-singular kernels. In particular, we prove a new version of Gronwall’s inequality involving the power fractional integral; and we establish existence and uniqueness results for nonlinear power fractional differential equations using fixed point techniques. Moreover, based on Lagrange polynomial interpolation, we develop a new explicit numerical method in order to approximate the solutions of a rich class of fractional differential equations. The approximation error of the proposed numerical scheme is analyzed. For illustrative purposes, we apply our method to a fractional differential equation for which the exact solution is computed, as well as to a nonlinear problem for which no exact solution is known. The numerical simulations show that the proposed method is very efficient, highly accurate and converges quickly.