Related papers: Modelling some real phenomena by fractional differ…
The Gr\"unwald and shifted Gr\"unwald formulas for the function $y(x)-y(b)$ are first order approximations for the Caputo fractional derivative of the function $y(x)$ with lower limit at the point $b$. We obtain second and third order…
In this paper we obtain new estimates of the sequential Caputo fractional derivatives of a function at its extremum points. We derive comparison principles for the linear fractional differential equations, and apply these principles to…
There has been considerable recent study in "sub-diffusion" models that replace the standard parabolic equation model by a one with a fractional derivative in the time variable. There are many ways to look at this newer approach and one…
Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…
For a class of tempered fractional terminal value problems of the Caputo type, we study the existence and uniqueness of the solution, analyse the continuous dependence on the given data and using a shooting method, we present and discuss…
The behaviour of the solutions of the time-fractional diffusion equation, based on the Caputo derivative, is studied and its dependence on the fractional exponent is analysed. The time-fractional convection-diffusion equation is also solved…
We introduce a new numerical method, based on Bernoulli polynomials, for solving multiterm variable-order fractional differential equations. The variable-order fractional derivative was considered in the Caputo sense, while the…
For terminal value problems of fractional differential equations of order $\alpha \in (0,1)$ that use Caputo derivatives, shooting methods are a well developed and investigated approach. Based on recently established analytic properties of…
This paper presents a better approach to model an engineering problem in fractal-time space based on local fractional calculus. Some examples are given to elucidate to establish governing equations with local fractional derivative.
This work presents an analytical and computational study of fractional-order delay differential equations formulated using both the conformable and Caputo derivatives. For the conformable case, we develop the associated integral,…
Efficient and fast predictor-corrector methods are proposed to deal with nonlinear Caputo-Fabrizio fractional differential equations, where Caputo-Fabrizio operator is a new proposed fractional derivative with a smooth kernel. The proposed…
Fractional order models have proven to be a very useful tool for the modeling of the mechanical behaviour of viscoelastic materials. Traditional numerical solution methods exhibit various undesired properties due to the non-locality of the…
In this paper we present a method to solve initial value problems for fractional growth models, such as generalizations of the exponential and logistic with periodic harvesting models. Using a discretization of the Caputo derivative we…
Fractional calculus generalizes the derivative and antiderivative operations of differential and integral calculus from integer orders to the entire complex plane. Methods are presented for using this generalized calculus with Laplace…
A fractional derivative is a temporally nonlocal operation which is computationally intensive due to inclusion of the accumulated contribution of function values at past times. In order to lessen the computational load while maintaining the…
The main purpose of this paper is to study the special fractional-order Chen-Lee system, using the Caputo fractional derivatives. For this fractional model we investigate the existence and uniqueness of solution of initial value problem,…
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…
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.…
We consider fractional differential equations of order $\alpha \in (0,1)$ for functions of one independent variable $t\in (0,\infty)$ with the Riemann-Liouville and Caputo-Dzhrbashyan fractional derivatives. A precise estimate for the order…
This paper is devoted to studying three-dimensional non-commensurate fractional order differential equation systems with Caputo derivatives. Necessary and sufficient conditions are for the asymptotic stability of such systems are obtained.