Fractional Pantograph Research Articles

Existence and stability results for random impulsive fractional pantograph equations

In this paper, we study the existence, uniqueness, stability through continuous dependence on initial conditions and Hyers-Ulam-Rassias stability results for random impulsive fractional pantograph differential systems by relaxing the linear growth conditions. Finally examples are given to illustrate the applications of the abstract results.

Numerical Approach for Solving Fractional Pantograph Equation

this article, we have investigate a Taylor collocation method, which is based on collocation method for solving fractional pantograph equation. This method is based on first taking the truncated fractional Taylor expansions of the solution function in the mathematical model and then substituting their matrix forms into the equation. Using the collocation points, we have the system of nonlinear algebraic equation. Then, we solve the system of linear algebraic equation using Maple 14 and we obtain the coefficients of Taylor expansion. In addition illustrative example is presented to demonstrate the effectiveness of the proposed method. Keywordspantograph equation; pantograph equation; fractional differential equation; mathematical model; collocation method; approximate solution i i t t f t u t t au t u 1 ' 0 ), ( ) ( ) ( ) ( ) (  

Existence of solutions for hybrid fractional pantograph equations

In this paper, we study the existence of the hybrid fractional pantograph equation {D?0+[x(t)/f(t,x(t),x(?t))= g(t,x(t), x(?t)), 0 < t < 1, x(0) = 0, where ?,?,? ?((0,1) and D?0+ denotes the Riemann-Liouville fractional derivative. The results are obtained using the technique of measures of noncompactness in the Banach algebras and a fixed point theorem for the product of two operators verifying a Darbo type condition. Some examples are provided to illustrate our results.

Existence of solutions of nonlinear fractional pantograph equations

This article deals with the existence of solutions of nonlinear fractional pantograph equations. Such model can be considered suitable to be applied when the corresponding process occurs through strongly anomalous media. The results are obtained using fractional calculus and fixed point theorems. An example is provided to illustrate the main result obtained in this article.

Spectral-Collocation Methods for Fractional Pantograph Delay-Integrodifferential Equations

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integrodifferential equations of Volterra type with pantograph delay. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collocation method, which shows that the error of approximate solution decays exponentially inL∞norm and weightedL2-norm. The numerical examples are given to illustrate the theoretical results.