Mixed Fractional Derivatives Research Articles

Mixed Fractional Differential Equations and Generalized Operator-Valued Mittag-Leffler Functions

We introduce the most general mixed fractional derivatives and integrals from three points of views: probability, the theory of operator semigroups, and the theory of generalized functions. The solutions to the resulting mixed fractional PDEs turned out to be representable in terms of of completely monotone functions in a certain class generalizing the usual Mittag-Leffler functions.

Numerical Double Sumudu Transform for Nonlinear Mixed Fractional Partial Differential Equations

In this paper, we are going to study a nonlinear fractional partial differential equations by double Sumudu transformation coupled with Adomian decomposition method or with variational iteration method. For this case, we choose an important fractional partial differential equations, such as parabolic-hyperbolic with mixed fractional derivatives (homogeneous and nonhomogeneous) types. Some examples are given here to illustrate efficiency of this method.

On generalized fractional integration by parts formulas and their applications to boundary value problems

Abstract In the paper, we derive a formula of integration by parts for the left-sided Hilfer derivative. Moreover, we present a formula of such a type for the right-sided Hilfer derivative. These results generalize the well-known rules of integration by parts for the Riemann–Liouville and Caputo derivatives. Finally, we use these rules to investigate some boundary value problems involving mixed fractional derivatives.

Existence and Stability Analysis for Fractional Differential Equations with Mixed Nonlocal Conditions

In this paper, we study the existence and uniqueness of solution for fractional differential equations with mixed fractional derivatives, integrals and multi-point conditions. After that, we also establish different kinds of Ulam stability for the problem at hand. Examples illustrating our results are also presented.

MIXED FRACTIONAL DIFFERENTIATION OPERATORS IN HÖLDER SPACES DEFINED BY USUAL HÖLDER CONDITION

We study mixed fractional derivative in Marchaud form of function of two variables in Hölder spaces of different orders in each variables. The main interest being in the evaluation of the latter for the mixed fractional derivative in the cases Hölder class defined by usual Hölder condition

An analytical framework of 2D diffusion, wave-like, telegraph, and Burgers’ models with twofold Caputo derivatives ordering

The purpose of the current work is to provide an analytical solution framework based on extended fractional power series expansion to solve 2D temporal–spatial fractional differential equations. For this purpose, we first present a new trivariate expansion endowed with twofold Caputo-fractional derivatives ordering, namely $$\alpha ,\,\beta \in (0,1]$$ , to study the combined effect of fractional derivatives on both temporal and spatial coordinates. Then, by virtue of this expansion, a parallel scheme of the Taylor power series solution method is utilized to extract both closed-form and supportive approximate series solutions of 2D temporal–spatial fractional diffusion, wave-like, telegraph, and Burgers’ models. The obtained closed-form solutions are found to be in harmony with the exact solutions exist in the literature when $$\alpha =\beta =1$$ , which exhibits the legitimacy and the validity of the proposed method. Moreover, the accuracy of the approximate series solutions is validated using graphical and tabular tools. Finally, a version of Taylor’s Theorem that associated with our proposed expansion is derived in terms of mixed fractional derivatives.

An analytical study of physical models with inherited temporal and spatial memory

Du et al. (Sci. Reb. 3, 3431 (2013)) demonstrated that the fractional derivative order can be physically interpreted as a memory index by fitting the test data of memory phenomena. The aim of this work is to study analytically the joint effect of the memory index on time and space coordinates simultaneously. For this purpose, we introduce a novel bivariate fractional power series expansion that is accompanied by twofold fractional derivatives ordering $\alpha$ , $\beta\in(0,1]$ . Further, some convergence criteria concerning our expansion are presented and an analog of the well-known bivariate Taylor’s formula in the sense of mixed fractional derivatives is obtained. Finally, in order to show the functionality and efficiency of this expansion, we employ the corresponding Taylor’s series method to obtain closed-form solutions of various physical models with inherited time and space memory.

Mixed Order Fractional Differential Equations

This paper studies fractional differential equations (FDEs) with mixed fractional derivatives. Existence, uniqueness, stability, and asymptotic results are derived.

Two generalized Lyapunov-type inequalities for a fractional p-Laplacian equation with fractional boundary conditions

In this paper, we investigate the existence of positive solutions for the boundary value problem of nonlinear fractional differential equation with mixed fractional derivatives and p-Laplacian operator. Then we establish two smart generalizations of Lyapunov-type inequalities. Some applications are given to demonstrate the effectiveness of the new results.

The method of lower and upper solutions for mixed fractional four-point boundary value problem with p-Laplacian operator

Based on the monotone iterative technique, a new method of lower and upper solutions which is used to study the multi-point boundary value problem of nonlinear fractional differential equations with mixed fractional derivatives and p -Laplacian operator is proposed. Some new results on the existence of positive solutions for the four-point boundary value problem are established. Finally, an example is presented to illustrate the wide range of potential applications of our main results.

A Second-Order Finite Difference Method for Two-Dimensional Fractional Percolation Equations

AbstractA finite difference method which is second-order accurate in time and in space is proposed for two-dimensional fractional percolation equations. Using the Fourier transform, a general approximation for the mixed fractional derivatives is analyzed. An approach based on the classical Crank-Nicolson scheme combined with the Richardson extrapolation is used to obtain temporally and spatially second-order accurate numerical estimates. Consistency, stability and convergence of the method are established. Numerical experiments illustrating the effectiveness of the theoretical analysis are provided.

Existence of solutions for impulsive fractional Langevin functional differential equations with variable parameter

In this paper, by using some fixed point theorems, we discuss a class of impulsive fractional Langevin functional differential equations with mixed fractional derivatives and variable parameter. The existence of solutions are obtained. Examples are also given to illustrate our theoretical results.

KOLMOGOROV WIDTHS IN THE SPACE ${\tilde L}_q$ OF THE CLASSES ${\tilde W}_p^{\overline \alpha}$ AND ${\tilde H}_p^{\overline \alpha}$ OF PERIODIC FUNCTIONS OF SEVERAL VARIABLES

The author finds the order of the Kolmogorov widths for all , where is the class of periodic functions of several variables determined by a Weyl mixed fractional derivative, and for or , where is the class determined by a mixed difference. Bibliography: 28 titles.