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We have recently presented an extension of the standard variational calculus to include the presence of deformed derivatives in the Lagrangian of a system of particles and in the Lagrangian density of field-theoretic models. Classical…
An adaptive finite difference scheme for variable-order fractional-time subdiffusion equations in the Caputo form is studied. The fractional time derivative is discretized by the L1 procedure but using nonhomogeneous timesteps. The size of…
We give a proper fractional extension of the classical calculus of variations by considering variational functionals with a Lagrangian depending on a combined Caputo fractional derivative and the classical derivative. Euler-Lagrange…
This paper investigates the global stability and the global asymptotic stability independent of the sizes of the delays of linear time-varying Caputo fractional dynamic systems of real fractional order possessing internal point delays. The…
We review some recent results of the fractional variational calculus. Necessary optimality conditions of Euler-Lagrange type for functionals with a Lagrangian containing left and right Caputo derivatives are given. Several problems are…
We introduce the delayed Mittag-Leffler type matrix functions, delayed fractional cosine, delayed fractional sine and use the Laplace transform to obtain an analytical solution to the IVP for a Hilfer type fractional linear time-delay…
The application of the approximation-operational approach to solving linear differential equations of fractional order with variable coefficients is considered. It is shown that the method can also be applied to solving differential…
This article deals with higher order Caputo fractional variational problems with the presence of delay in the state variables and their integer higher order derivatives.
Delay differential equations are of great importance in science, engineering, medicine and biological models. These type of models include time delay phenomena which is helpful for characterising the real-world applications in machine…
Caputo-Fabrizio fractional delta derivatives on an arbitrary time scale are presented. When the time scale is chosen to be the set of real numbers, then the Caputo-Fabrizio fractional derivative is recovered. For isolated or partly…
Main results and techniques of the fractional calculus of variations are surveyed. We consider variational problems containing Caputo derivatives and study them using both indirect and direct methods. In particular, we provide necessary…
In this paper, we are primarily concerned with the study of entire and analytical solutions of abstract degenerate (multi-term) fractional differential equations with Caputo time-fractional derivatives. We also analyze systems of such…
Fractional derivatives can be used to model time delays in a diffusion process. When the order of the fractional derivative is distributed over the unit interval, it is useful for modeling a mixture of delay sources. In some special cases…
We present a new numerical tool to solve partial differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one of them an approximation formula is…
We introduce a fractional calculus on time scales using the theory of delta (or nabla) dynamic equations. The basic notions of fractional order integral and fractional order derivative on an arbitrary time scale are proposed, using the…
Derivatives of fractional order are introduced in different ways: as left-inverse of the fractional integral or by generalizing the limit of the difference quotient defining integer-order derivatives. Although the two approaches lead (under…
We examine the fractional derivative of composite functions and present a generalization of the product and chain rules for the Caputo fractional derivative. These results are especially important for physical and biological systems that…
In this work, we rigorously investigate the robustness of graph neural fractional-order differential equation (FDE) models. This framework extends beyond traditional graph neural (integer-order) ordinary differential equation (ODE) models…
Based on the popular Caputo fractional derivative of order $\beta$ in $(0,1)$, we define the censored fractional derivative on the positive half-line $\mathbb R_+$. This derivative proves to be the Feller generator of the censored (or…
We generalize the fractional Caputo derivative to the fractional derivative ${^CD^{\alpha,\beta}_{\gamma}}$, which is a convex combination of the left Caputo fractional derivative of order $\alpha$ and the right Caputo fractional derivative…