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The inversion of nabla Laplace transform, corresponding to a causal sequence, is considered. Two classical methods, i.e., residual calculation method and partial fraction method are developed to perform the inverse nabla Laplace transform.…
Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately…
The relaxation functions introduced empirically by Debye, Cole-Cole, Cole-Davidson and Havriliak-Negami are, each of them, solutions to their respective kinetic equations. In this work, we propose a generalization of such equations by…
There are many resources useful for processing images, most of them freely available and quite friendly to use. In spite of this abundance of tools, a study of the processing methods is still worthy of efforts. Here, we want to discuss the…
The fractional calculus of variations is now a subject under strong research. Different definitions for fractional derivatives and integrals are used, depending on the purpose under study. In this paper the fractional operators are defined…
In this paper, we give a review of fractal calculus which is an expansion of standard calculus. Fractal calculus is applied for functions which are not differentiable or integrable on totally disconnected fractal sets such as middle-$\mu$…
Discrete maps with long-term memory are obtained from nonlinear differential equations with Riemann-Liouville and Caputo fractional derivatives. These maps are generalizations of the well-known universal map. The memory means that their…
In a recent paper, Cohl and Costas-Santos derived a number of interesting multi-derivative and multi-integral relations for associated Legendre and Ferrers functions in which the orders of those functions are changed in integral steps.…
Computational methods for fractional differential equations exhibit essential instability. Even a minor modification of the coefficients or other entry data may switch good results to the divergent. The goal of this paper is to suggest the…
Fractional calculus is a powerful and effective tool for modelling nonlinear systems. The M derivative is the generalization of alternative fractional derivative. This M derivative obey the properties of integer calculus. In this paper, we…
This study reexamines diffusive representations for fractional integrals with the goal of pioneering new variants of such representations. These variants aim to offer highly efficient numerical algorithms for the approximate computation of…
In this paper, the fractional differential matrices based on the Jacobi-Gauss points are derived with respect to the Caputo and Riemann-Liouville fractional derivative operators. The spectral radii of the fractional differential matrices…
Fractional derivatives are nonlocal differential operators of real order that often appear in models of anomalous diffusion and a variety of nonlocal phenomena. Recently, a version of the Schr\"odinger Equation containing a fractional…
The fundamental importance of functional differential equations has been recognized in many areas of mathematical physics, such as fluid dynamics (Hopf characteristic functional equation), quantum field theory (Schwinger-Dyson equations)…
We extend the classical Bernstein technique to the setting of integro-differential operators. As a consequence, we provide first and one-sided second derivative estimates for solutions to fractional equations, including some convex fully…
A general method for solving linear differential equations of arbitrary order, is used to arrive at new representations for the solutions of the known differential equations, both without and with a source term. A new quasi-solvable…
Fractional calculus of variation plays an important role to formulate the non-conservative physical problems. In this paper we use semi-inverse method and fractional variational principle to formulate the fractional order generalized…
In this article, the existence and uniqueness about the solution for a class of stochastic fractional-order differential equation systems are investigated, where the fractional derivative is described in Caputo sense. The fractional…
We obtain a new decomposition of the Riemann-Liouville operators of fractional integration as a series involving derivatives (of integer order). The new formulas are valid for functions of class $C^n$, $n \in \mathbb{N}$, and allow us to…
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…