Sentence examples for derivative by using from inspiring English sources

Recently, Klimek investigated existence and uniqueness of the solution of sequential fractional differential equations with Hadamard derivative by using the contraction principle in [16].

Romero et al. [20] introduced a new fractional operator called the k-Riemann-Liouville fractional derivative by using gamma k-function.

Ahmadi et al. investigated the Laplace transform formula on the fuzzy nth-order derivative by using the strongly generalized differentiability concept in [17].

Yang and Wang [20] established the existence and uniqueness of mild solutions of fractional evolution equations involving the Hilfer derivative by using the noncompact measure method.

Secondly, we propose a temporal second order numerical method for a fractional reaction-dispersion equation, where we discretize the Riesz fractional derivative by using two numerical schemes.

In this paper, we present some results for the attractivity of solutions for a k-dimensional system of fractional functional differential equations involving the Caputo fractional derivative by using the classical Schauder's fixed-point theorem.

A new fourth-order difference approximation is derived for the space fractional derivatives by using the weighted average of the shifted Grünwald formulae combining the compact technique.

The shape derivatives are derived by using a Lagrangian function and the adjoint method.

Recently, Allahviranloo et al. [14] and Hoa [15] independently introduced the concept of Caputo gH-derivative by using the generalized Hukuhara difference (or gH-difference).

Subsequently, Salahshour et al. [10] considered the solutions of fuzzy fractional differential equations under Riemann-Liouville H-derivative by using the fuzzy Laplace transform method.

The method is based on approximating functions and their derivatives by using the Whittaker cardinal function.