Related papers: Why fractional derivatives with nonsingular kernel…

In fractional calculus there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and…

In this paper, a new fractional derivative involving the normalized sinc function without singular kernel is proposed. The Laplace transform is used to find the analytical solution of the anomalous heat-diffusion problems. The comparative…

We demonstrate that a violation of the Leibniz rule is a characteristic property of derivatives of non-integer orders. We prove that all fractional derivatives D^a, which satisfy the Leibniz rule D^(fg)=(D^a f) g + f (D^a g), should have…

The Riemann-Liouville formula for fractional derivatives and integrals (differintegration) is used to derive formulae for matrix order derivatives and integrals. That is, the parameter for integration and differentiation is allowed to…

This paper presents a self-contained new theory of weak fractional differential calculus in one-dimension. The crux of this new theory is the introduction of a weak fractional derivative notion which is a natural generalization of integer…

We consider here convolution operators, in the Caputo sense, with non-singular kernels. We prove that the solutions to some integro-differential equations with such operators (acting on the space variable) coincide with the transition…

The following material was created with the idea of being used for an introductory fractional calculus course. A recapitulation of the history of fractional calculus is presented, as well as the different attempts at fractional derivatives…

In this paper we show different inequalities for fractional order differential operators. In particular, the Sobolev, Hardy, Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg type inequalities for the Caputo, Riemann-Liouville and Hadamard…

We study two generalizations of fractional variational problems by considering higher-order derivatives and a state time delay. We prove a higher-order integration by parts formula involving a Caputo fractional derivative of variable order…

This paper is devoted to study the nonlinear sequential fractional boundary value problems involving generalized $\psi$-Caputo fractional derivatives with nonlocal boundary conditions. We investigate the Green function and some of its…

Many different types of fractional calculus have been proposed, which can be organised into some general classes of operators. For a unified mathematical theory, results should be proved in the most general possible setting. Two important…

The concept of nonlinear self-adjointness is employed to construct the conservation laws for fractional evolution equations using its Lie point symmetries. The approach is demonstrated on subdiffusion and diffusion-wave equations with the…

In this paper we present a new type of fractional operator, which is a generalization of the Caputo and Caputo--Hadamard fractional derivative operators. We study some properties of the operator, namely we prove that it is the inverse…

We Investigate two types of dual identities for Caputo fractional differences. The first type relates nabla and delta type fractional sums and differences. The second type represented by the Q-operator relates left and right fractional sums…

In this paper we study isoperimetric problems of the calculus of variations with left and right Riemann-Liouville fractional derivatives. Both situations when the lower bound of the variational integrals coincide and do not coincide with…

This paper focuses on the fractional difference of Lyapunov functions related to Riemann-Liouville, Caputo and Grunwald-Letnikov definitions. A new way of building Lyapunov functions is introduced and then five inequalities are derived for…

We prove necessary optimality conditions, in the class of continuous functions, for variational problems defined with Jumarie's modified Riemann-Liouville derivative. The fractional basic problem of the calculus of variations with free…

We present sufficient conditions for the existence of positive solutions for a class of fractional singular boundary value problems in presence of Caputo fractional derivative. Further, the nonlinearity involved has singularity with respect…

The article investigates an inverse problem of determining the right-hand side of a subdiffusion equation with Caputo fractional derivative whose elliptic part has the most general form and is defined on an N-dimensional torus T N . The…

An economic interpretation of the Caputo derivatives of non-integer orders is proposed. The suggested economic interpretation of the fractional derivatives is based on a generalization of average and marginal values of economic indicators.…