Properties of fractional calculus with respect to a function and Bernstein type polynomials

• Published in: Mathematical methods in the applied sciences 2023-01, Vol.46 (1), p.930-960
• Main Authors: Sousa, José Vanterler da C., Frederico, Gastão S. F., Oliveira, Daniela S., Capelas de Oliveira, Edmundo
• Format: Article
• Language: English
• Subjects: Bernstein type polynomials; Derivatives; Differential calculus; Differential equations; Fractional calculus; fractional mean value theorem; Hypergeometric functions; Integrals; Mathematical analysis; Polynomials; ψ$$ \psi $$‐Hilfer fractional derivative; ψ$$ \psi $$‐Riemann–Liouville fractional integral
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.8557

This paper is divided into two stages. In the first stage, we investigated a new approach for the ψ$$ \psi $$‐Riemann–Liouville fractional integral and the Faa di Bruno formula for the ψ$$ \psi $$‐Hilfer fractional derivative. In addition, we discussed other properties involving the ψ$$ \psi $$‐Hilfer fractional derivative and the ψ$$ \psi $$‐Riemann–Liouville fractional integral. In the second stage, Bernstein polynomials involving the ψ(·)$$ \psi \left(\cdotp \right) $$ function are investigated and the ψ$$ \psi $$‐Riemann–Liouville fractional integral and ψ$$ \psi $$‐Hilfer fractional derivative from the Bernstein polynomials are evaluated. We also discussed the relationship between the ψ$$ \psi $$‐Hilfer fractional derivative with Laguerre polynomials and hypergeometric functions, and a version of the fractional mean value theorem with respect to a function. Motivated by the Bernstein polynomials, the second stage uses the Bernstein polynomials to approximate the solution of a fractional integro‐differential equation with Hilfer fractional derivative and concluding with a numerical approach with its respective graph.