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This paper focuses on revisiting and improving the results regarding the existence of a solution to the implicit fractional differential equations (FDEs) given in the following form: DCρω(ϰ)=fϰ,ω(ϰ),DCρω(ϰ)$$ {\mathcal{D}}_C&#x0005E;{\rho}\omega \left(\mathrm{\varkappa}\right)&#x0003D;f\left(\mathrm{\varkappa},\omega \left(\mathrm{\varkappa}\right),{\mathcal{D}}_C&#x0005E;{\rho}\omega \left(\mathrm{\varkappa}\right)\right) $$for all ϰ∈[0,T]$$ \mathrm{\varkappa}\in \left[0,T\right] $$ with the initial condition ω(0)=ω0$$ \omega (0)&#x0003D;{\omega}_0 $$, where 0<ρ<1,DCρ$$ 0<\rho <1,{\mathcal{D}}_C&#x0005E;{\rho } $$ is the Caputo fractional derivative (CFD). Mainly, we aim to weaken the necessary conditions often found in the literature for guaranteeing the existence of solution. [ABSTRACT FROM AUTHOR]

In this paper, we establish global fractional Halanay‐type inequalities with distributed delays in differential and integral forms and consequently give a simple sufficient condition for Mittag–Leffler stability of solutions to fractional differential equations with this type of delays. Our essential tool is the submultiplicative property of the Mittag–Leffler functions which imitates the multiplicative property of the exponential functions in the classical theory of differential equations. The estimation with explicit decaying rate and amplitude constant is a faithful extension of the classical Halanay inequality. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the multiterm time‐fractional partial differential equation involving the Caputo operator associated with the Laplace operator, which includes the momentum equations of the fractional Oldroyd‐B fluid and the fractional Burgers fluid under some suitable conditions. By using several properties of multivariate Mittag–Leffler functions, the well‐posedness and the long‐time behavior for the Dirichlet problem are obtained. In addition, the uniqueness in inverse problem of determining orders of time‐fractional derivatives of the equation is proved. [ABSTRACT FROM AUTHOR]

This paper is in concern with the study of differential problems involving the ψ$$ \psi $$‐shifted fractional derivatives, where ψ$$ \psi $$ is a scaling function. Such operators can be thought of as a generalization of several fractional derivatives such as the classical Riemann–Liouville and Caputo operators, the Hadamard operators, the generalized fractional operators, and the Erdélyi–Kober operators, among others. Two types of boundary conditions are considered: the Cauchy problems and the integral boundary conditions. [ABSTRACT FROM AUTHOR]

Neural network has good self‐learning and adaptive capabilities. In this paper, a wavelet neural network is proposed to be used to solve the value problem of fractional differential equations (FDE). We construct a wavelet neural network (WNN) with the structure 1 ×N× 1 based on the wavelet function and give the conditions for the convergence of the given algorithm. This method uses the truncated power series of the solution function to transform the original differential equation into an approximate solution, then, using WNN, update the parameters, and finally get the FDE solution. Simulation results prove the validity of WNN. [ABSTRACT FROM AUTHOR]

This paper is devoted to the implementation of fractional differential transform method (FDTM) for the evaluation of new approximations for the fractional models of the system of multi‐pantograph differential equations. Some new properties of FDTM have also been proposed for the fractional integro‐differential equations. It is noteworthy that the integro‐differential equations are studied for integers in the literature, and we have proposed the solutions of proportional delayed integro‐differential equations for fractional order. The evaluated results are verified numerically by considering some test examples of fractional integro‐differential equations and fractional models of the system of multi‐pantograph differential equations. The outcomes conclude that evaluated series solutions via FDTM promise more appropriate approximations possess very negligible error and represent the convergent approximations. Thus, evaluated series solutions are comparatively more accurate and require less computational time. [ABSTRACT FROM AUTHOR]

In the present paper, a nonlinear fractional Schrödinger integro‐differential equation is considered in a Hilbert space. Operator approach is applied on multidimensional problems with nonlinearity that deserve a studious treatment. In this paper, theorems on existence and uniqueness of a bounded solution for the abstract problem are achieved. Additionally, existence theorems are obtained for first and second orders of accuracy difference schemes of the abstract problem. Furthermore, theorems are applied on a one‐dimensional problem with nonlocal condition and a multidimensional problem with Dirichlet boundary condition. Numerical results and illustrations are presented to show the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is a study about a fractional‐order hybrid system of two fractional‐order differential equations. We have given the existence of solution and uniqueness of solutions, and finally, the numerical solutions for an application is presented. The existence results are based on the classical fixed‐point approach while the numerical results are obtained with the help of fractional‐order Euler's technique. In the numerical solutions, we can see that the orders of the differential equations are playing important role. We have given related literature for the readers for their detail explanation. [ABSTRACT FROM AUTHOR]

In this paper, the Hydon method to determine discrete symmetries for a differential equation is employed to construct discrete symmetries for a family of ordinary, partial, and fractional differential equations. An application of those discrete symmetries to construct other solutions from known ones is illustrated. [ABSTRACT FROM AUTHOR]

In this paper, we investigate in more detail some useful theorems related to conformable fractional derivative (CFD) and integral and introduce two classes of conformable fractional Sturm‐Liouville problems (CFSLPs): namely, regular and singular CFSLPs. For both classes, we study some of the basic properties of the Sturm‐Liouville theory. In the class of r‐CFSLPs, we discuss two types of CFSLPs which include left‐ and right‐sided CFDs, each of order α∈(n,n+1], and prove properties of the eigenvalues and the eigenfunctions in a certain Hilbert space. Also, we apply a fixed‐point theorem for proving the existence and uniqueness of the eigenfunctions. As an operator for the class of s‐CFSLPs, we first derive two fractional types of the hypergeometric differential equations of order α∈(0,1] and obtain their analytical eigensolutions as Gauss hypergeometric functions. Afterwards, we define the conformable fractional Legendre polynomial/functions (CFLP/Fs) as Jacobi polynomial and investigate their basic properties. Moreover, the conformable fractional integral Legendre transforms (CFILTs) based on CFLP/Fs‐I and ‐II are introduced, and using these new transforms, an effective procedure for solving explicitly certain ordinary and partial conformable fractional differential equations (CFDEs) are given. Finally, as a theoretical application, some fractional diffusion equations are solved. [ABSTRACT FROM AUTHOR]

In this paper, a solution to initial value problems for fractional‐order linear commensurate multi‐term differential equations with Caputo derivatives is presented. The solution is obtained in the form of a finite sum of the Mittag‐Leffler–type functions and the meta‐trigonometric cosine function by using a numerical‐analytical method. The results of presented numerical experiments show that for high accuracy calculations of these functions, the multi‐precision arithmetic must be applied. The approach for solving of the initial value problems for generalized Basset equation, generalized Bagley‐Torvik equation, and multi‐term fractional equation is demonstrated. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some fixed‐point theorems for a generalized almost Hardy‐Rogers‐type F contraction in a metric‐like space and give an example to illustrate these main results. Moreover, we show the applications of electric circuit equations, second‐order differential equations, and fractional differential equations. Our results improve, generalize, and extend the corresponding results in literature. [ABSTRACT FROM AUTHOR]

In this paper, we study relative controllability of fractional differential equations with pure delay. Delayed Gram‐type matrix criterion and rank criterion for relative controllability are established with the help of the explicit solution formula. An example is given to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the study of a family of nonlinear Volterra equations coming from the theory of viscoelasticity. We analyze the existence of local mild solutions to the problem and their possible continuation to a maximal interval of existence. We also consider the problem of continuous dependence with respect to initial data. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of ground state solution for the following fractional differential equations with tempered fractional derivative: [ABSTRACT FROM AUTHOR]

The paper is devoted to study the stability of nonlinear fractional order difference systems by their linear approximation. Additionally, we show the relation between the stability of linear fractional order differential systems and their discretizations. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, the Bäcklund transformation of fractional Riccati equation is presented to establish travelingwave solutions for two nonlinear space-time fractional differential equations in the sense of modified Riemann-Liouville derivatives, namely, the space-time fractional generalized reaction duffing equation and the space-time fractional diffusion reaction equation with cubic nonlinearity. The proposed method is effective and convenient for solving nonlinear evolution equations with fractional order. [ABSTRACT FROM AUTHOR]

In the paper, fractional systems with Riemann-Liouville derivatives are studied. A theorem on the existence and uniqueness of a solution of a fractional ordinary Cauchy problem is given. Next, the Pontryagin maximum principle for nonlinear fractional control systems with a nonlinear integral performance index is proved. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]