The modification of the Clarke generalized subdiNerentia1 due to Michel and Penot is a useful tool in determining differentiability properties for certain classes of real functions on a normed linear space. The Glteaux differentiability of any real function can be deduced from the GBteaux differentiability of the norm if the function has a directional derivative which attains a constant related...

and Applied Analysis 3 Let Γ(⋅) denote the gamma function. For any positive integer n and real number θ (n − 1 < θ < n), there are different definitions of fractional derivatives with order θ in [8]. During this paper, we consider the left, (right) Caputo derivative and left (right) Riemann-Liouville derivative defined as follows: (i) the left Caputo derivative: C 0 D θ t V (t) = 1 Γ (n − θ) ∫ ...

2. Preliminaries Fractional differintegration’s set foot in control compass, also consistency of Caputo definition among the other fractional derivative’s definitions with engineering applications, is caused to devote this section to the basic definition of fractional calculus and its properties (Table 1). The Riemann–Liouville (RL) and the Caputo are the most popular fractional derivatives def...

In this paper, we defined the interval form of \(\alpha, \beta\) level set for triangular q-rung orthopair fuzzy (qROPF) number. To obtain a differentiability notion qROPF valued functions, Hukuhara (H-differentiability) and level-wise H-differentiability is defined. Using KKT optimality condition optimization problem with objective function are formulated.

Here we state the main properties of the Caputo, Riemann-Liouville and the Caputo via Riemann-Liouville fractional derivatives and give some general notes on these properties. Some properties given in some recent literatures and used to solve fractional nonlinear partial differential equations will be proved that they are incorrect by giving some counter examples.

Zaj́ıček has recently shown that for a lower semi-continuous real-valued function on an Asplund space, the set of points where the function is Fréchet subdifferentiable but not Fréchet differentiable is first category. We introduce another variant of Fréchet differentiability, called Fréchet directional differentiability, and show that for any realvalued function on a normed linear space, the se...

In this paper, we study the numerical method for solving second-order fuzzy differential equations using Adomian method under strongly generalized differentiability. And, we present an example with initial condition having four different solutions to illustrate the efficiency of the proposed method under strongly generalized differentiability. Keywords—Fuzzy-valued function, Fuzzy initial value...

There are multiple reasons why anisotropic basis functions may be needed or be more appropriate. The most obvious is that if the basis function is to be defined on Rn × T then there is no natural norm on this space that would reflect the unique properties of time. A second reason is that function being interpolated or approximated may incorporate a directional dependence. Thirdly, differentiabi...

This manuscript assesses a semi-analytical method in connection with new hybrid fuzzy integral transform and the Adomian decomposition via notion of fuzziness known as Elzaki (briefly, EADM). Moreover, we use aforesaid strategy to address time-fractional Fornberg–Whitham equation (FWE) under gH-differentiability by employing different initial conditions (IC). Several algebraic aspects Caputo fr...

Most of the existing approaches for fuzzy reliability analysis are based on fuzzy probability.  The aim of this paper is to describe fuzzy reliability using fuzzy differential equation. The reliability of a system in real world applications is affected by some uncertain parameters. Fuzzy reliability is a way to present the reliability function uncertainly using fuzzy parameters. In the proposed...