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We study three types of generalized partial fractional operators. An extension of Green's theorem, by considering partial fractional derivatives with more general kernels, is proved. New results are obtained, even in the particular case…

We consider a class of numerical approximations to the Caputo fractional derivative. Our assumptions permit the use of nonuniform time steps, such as is appropriate for accurately resolving the behavior of a solution whose derivatives are…

This article is devoted to developing an abstract theory of time-fractional gradient flow equations for time-dependent convex functionals in real Hilbert spaces. The main results concern the existence of strong solutions to time-fractional…

We consider some possible approaches to the fractional-order generalization of definition of variation (functional) derivative. Some problems of formulation of a fractional-order variational derivative are discussed. To give a consistent…

In this paper, we consider the finite difference method for the generalized two-dimensional (2D) multi-term time-fractional Oldroyd-B fluid model, which is a subclass of non-Newtonian fluids. Different from the general multi-term time…

New integrability properties of a family of sequences of ordinary differential equations, which contains the Riccati and Abel chains as the most simple sequences, are studied. The determination of n generalized symmetries of the nth-order…

In this paper, we first propose an unconditionally stable implicit difference scheme for solving generalized time-space fractional diffusion equations (GTSFDEs) with variable coefficients. The numerical scheme utilizes the $L1$-type formula…

We prove a necessary optimality condition of Euler-Lagrange type for variational problems on time scales involving nabla derivatives of higher-order. The proof is done using a new and more general fundamental lemma of the calculus of…

This paper proposes a new generalized Seikkala derivative (gS-derivative) of a fuzzy-valued function. We see that, there are many elementary fuzzy-valued functions which occur frequently as solution of fuzzy differential equation, are not…

Fractional mechanics describes both conservative and non-conservative systems. The fractional variational principles gained importance in studying the fractional mechanics and several versions are proposed. In classical mechanics the…

We present a novel generalized convolution quadrature method that accurately approximates convolution integrals. During the late 1980s, Lubich introduced convolution quadrature techniques, which have now emerged as a prevalent methodology…

In this PhD thesis we introduce a generalized fractional calculus of variations. We consider variational problems containing generalized fractional integrals and derivatives, and study them using standard (indirect) and direct methods. In…

We discuss the derivation of a ``cluster sum rule'' from the Gellmann-Goldberger-Thirring (GGT) sum rule as an alternative to the Thomas-Reiche-Kuhn (TRK) sum rule, which was used as the basis up to now. We compare differences in the…

This article provides an accessible introduction to fractional derivatives, a concept that extends classical calculus by allowing derivatives of non-integer order. It explores both the fundamental definitions and some of the most relevant…

In this work, a generalization of the well known Bernoulli inequality is obtained by using the theory of discrete fractional calculus. As far as we know our approach is novel.

The mean field methods, which entail approximating intractable probability distributions variationally with distributions from a tractable family, enjoy high efficiency, guaranteed convergence, and provide lower bounds on the true…

We construct the existence theory for generalized fractional Bessel differential equations and find the solutions in the form of fractional or logarithmic fractional power series. We figure out the cases when the series solution is unique,…

An algebraic technique is presented that does not use results of model theory and makes it possible to construct a general Galois theory of arbitrary nonlinear systems of partial differential equations. The algebraic technique is based on…

We study incommensurate fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives and generalized fractional integrals and derivatives. We obtain necessary optimality…

Normal and anomalous diffusion are ubiquitous in many complex systems [1] . Here, we define a time and space generalized diffusion equation (GDE), which uses fractional-time derivatives and transformed d-path Laplacian operators on…