Properties of fractional calculus with respect to a function and Bernstein type polynomials |
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| Authors: | José Vanterler da C Sousa Gastão S F Frederico Daniela S Oliveira Edmundo Capelas de Oliveira |
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| Institution: | 1. Center for Mathematics, Computing and Cognition, Federal University of ABC, Avenida dos Estados, Santo Andre, Brazil;2. Federal University of Ceará, Campus de Russas, Russas, Brazil;3. Coordination of Civil Engineering, UTFPR, Guarapuava, Brazil;4. Department of Applied Mathematics, Imecc-State University of Campinas, Campinas, Brazil |
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| Abstract: | This paper is divided into two stages. In the first stage, we investigated a new approach for the -Riemann–Liouville fractional integral and the Faa di Bruno formula for the -Hilfer fractional derivative. In addition, we discussed other properties involving the -Hilfer fractional derivative and the -Riemann–Liouville fractional integral. In the second stage, Bernstein polynomials involving the function are investigated and the -Riemann–Liouville fractional integral and -Hilfer fractional derivative from the Bernstein polynomials are evaluated. We also discussed the relationship between the -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. |
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| Keywords: | ψ$$ \psi $$-Hilfer fractional derivative ψ$$ \psi $$-Riemann–Liouville fractional integral Bernstein type polynomials fractional mean value theorem |
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