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On the nonlinear ( k , Ψ ) -Hilfer fractional differential equations

2021; Elsevier BV; Volume: 152; Linguagem: Inglês

10.1016/j.chaos.2021.111335

1873-2887

Kishor D. Kucche, Audumbar Mali,

Iterative Methods for Nonlinear Equations

In the current paper, we present the most generalized variant of the Hilfer derivative so-called (k,Ψ)-Hilfer fractional derivative operator. The (k,Ψ)-Riemann-Liouville and (k,Ψ)-Caputo fractional derivatives are obtained as a special case of (k,Ψ)-Hilfer fractional derivative. We demonstrate a few properties of (k,Ψ)-Riemann-Liouville fractional integral and derivative that expected to build up the calculus of (k,Ψ)-Hilfer fractional derivative operator. We present some significant outcomes about (k,Ψ)-Hilfer fractional derivative operator that require to derive the equivalent fractional integral equation to nonlinear (k,Ψ)-Hilfer fractional differential equation. We prove the existence and uniqueness for the solution of nonlinear (k,Ψ)-Hilfer fractional differential equation. In the conclusion section, we list the various k-fractional derivatives that are specific cases of (k,Ψ)-Hilfer fractional derivative.