Related papers: Diffusive Representations for the Numerical Evalua…
This paper is devoted to the general theory of systems of time-fractional differential-operator equations. The representation formulas for solutions of systems of ordinary differential equations with single (commensurate) fractional order…
In this paper we present in one-dimensional space a numerical solution of a partial differential equation of fractional order. This equation describes a process of anomalous diffusion. The process arises from the interactions within the…
In this paper, an easy-to-implement and computationally effective numerical method based on the new orthogonal hybrid functions is developed to solve system of fractional order differential equations numerically. The new orthogonal hybrid…
In this study the general formula for differential and integral operations of fractional calculus via fractal operators by the method of cumulative diminution and cumulative growth is obtained. The under lying mechanism in the success of…
This chapter presents some numerical methods to solve problems in the fractional calculus of variations and fractional optimal control. Although there are plenty of methods available in the literature, we concentrate mainly on approximating…
This discussion paper presents some parts of the work in progress. It is shown that G.W. Leibniz was the first who raised the question about geometric interpretation of fractional-order operators. Geometric interpretations of the…
Determining the unknown order of the fractional derivative in differential equations simulating various processes is an important task of modern applied mathematics. In the last decade, this problem has been actively studied by specialists.…
In this work, we develop a highly efficient representation of functions and differential operators based on Fourier analysis. Using this representation, we create a variational hybrid quantum algorithm to solve static, Schr\"odinger-type,…
In this paper we aim to construct an abstract model of a differential operator with a fractional integro-differential operator composition in final terms, where modeling is understood as an interpretation of concrete differential operators…
We consider the integral and derivative operators of tempered fractional calculus, and examine their analytic properties. We discover connections with the classical Riemann-Liouville fractional calculus and demonstrate how the operators may…
In this paper, we consider the strong convergence of the time-space fractional diffusion equation driven by fractional Gaussion noise with Hurst index $H\in(\frac{1}{2},1)$. A sharp regularity estimate of the mild solution and the numerical…
Some linear integro-differential operators have old and classical representations as the Dirichlet-to-Neumann operators for linear elliptic equations, such as the 1/2-Laplacian or the generator of the boundary process of a reflected…
We deal with the higher-order fractional Laplacians by two methods: the integral method and the system method. The former depends on the integral equation equivalent to the differential equation. The latter works directly on the…
This overview is devoted to splitting methods, a class of numerical integrators intended for differential equations that can be subdivided into different problems easier to solve than the original system. Closely connected with this class…
The differentiation by integration method with Jacobi polynomials was originally introduced by Mboup, Join and Fliess. This paper generalizes this method from the integer order to the fractional order for estimating the fractional order…
We introduce the concept of fractional derivative of Riemann-Liouville on time scales. Fundamental properties of the new operator are proved, as well as an existence and uniqueness result for a fractional initial value problem on an…
Fractional differential equations are powerful mathematical descriptors for intricate physical phenomena in a compact form. However, compared to integer ordinary or partial differential equations, solving fractional differential equations…
We develop an operator-theoretical method for the analysis on well posedness of partial differential equations that can be modeled in the form \begin{equation*} \left\{ \begin{array}{rll} \Delta^{\alpha} u(n) &= Au(n+2) + f(n,u(n)), \quad n…
In this article, we investigate partial integrals and partial derivatives of bivariate fractal interpolation functions. We prove also that the mixed Riemann-Liouville fractional integral and derivative of order $\gamma = (p, q); p > 0,q >…
We study a numerical approximation for a nonlinear variable-order fractional differential equation via an integral equation method. Due to the lack of the monotonicity of the discretization coefficients of the variable-order fractional…