© Hindawi Publishing Corp. LYAPUNOV STABILITY SOLUTIONS OF FRACTIONAL INTEGRODIFFERENTIAL EQUATIONS

Lyapunov stability and asymptotic stability conditions for the solutions of the fractional integrodiffrential equations x (α)(t) = f(t,x(t)) + ∫ t t0 K(t,s,x(s))ds, 0<α ≤ 1, with the initial condition x (α−1)(t0) = x0, have been investigated. Our methods are applications of Gronwall’s lemma and Schwartz inequality. 2000 Mathematics Subject Classification: 26A33, 34D20. 1. Introduction. Conside

On the existence and uniqueness of solutions of a class of fractional differential equations

AbstractIn this paper, we investigate the existence and uniqueness of solutions for the following class of multi-order fractional differential equationsDβ1γ1,δ1⋯Dβnγn,δnu(t):=∏i=1nDβiγi,δiu(t):=Dβi,nγi,δiu(t)=f(t,u(t)),t∈[0,1],u(0)=0,∑i=1nδi⩽1,γi>0,βi>0,1⩽i⩽n, where Dβi,nγi,δi denotes the generalized Erdélyi–Kober operator of fractional derivative of order δi. Moreover, some properties concerning the positive, maximal, minimal, and continuation of solutions are obtained

Solutions of non-linear oscillators by the modified differential transform method

AbstractA numerical method for solving nonlinear oscillators is proposed. The proposed scheme is based on the differential transform method (DTM), Laplace transform and Padé approximants. The modified differential transform method (MDTM) technique introduces an alternative framework designed to overcome the difficulty of capturing the periodic behavior of the solution, which is characteristic of oscillator equations, and give a good approximation to the true solution in a very large region. The numerical results demonstrate the validity and applicability of the new technique and a comparison is made with existing results

A generalized differential transform method for linear partial differential equations of fractional order

In this letter we develop a new generalization of the two-dimensional differential transform method that will extend the application of the method to linear partial differential equations with space- and time-fractional derivatives. The new generalization is based on the two-dimensional differential transform method, generalized Taylor’s formula and Caputo fractional derivative. Several illustrative examples are given to demonstrate the effectiveness of the present method. The results reveal that the technique introduced here is very effective and convenient for solving linear partial differential equations of fractional order

The modified homotopy perturbation method for solving strongly nonlinear oscillators

In this paper we propose a reliable algorithm for the solution of nonlinear oscillators. Our algorithm is based upon the homotopy perturbation method (HPM), Laplace transforms, and Padé approximants. This modified homotopy perturbation method (MHPM) utilizes an alternative framework to capture the periodic behavior of the solution, which is characteristic of oscillator equations, and to give a good approximation to the true solution in a very large region. The current results are compared with those derived from the established Runge–Kutta method in order to verify the accuracy of the MHPM. It is shown that there is excellent agreement between the two sets of results. Results also show that the numerical scheme is very effective and convenient for solving strongly nonlinear oscillators

Solution of an SEIR Epidemic Model in Fractional Order

In this paper, we consider the SEIR (Susceptible-Exposed-Infected-Recovered) epidemic model (with out of bilinear incidence rates) in fractional order. First the non-negative solution of the SEIR model in fractional order is discussed. Then calculate an approximate solution of the proposed model. The obtained results are compaired with those obtained by forth order Runge-Kutta method and nonstandard numerical method in the integer case. Finally, we present some numerical results

HOMOTOPY ANALYSIS METHOD FOR SYSTEMS OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

Abstract: In this article, based on the homotopy analysis method (HAM), a new analytic technique is proposed to solve systems of fractional integro-differential equations. Comparing with the exact solution, the HAM provides us with a simple way to adjust and control the convergence region of the series solution by introducing an auxiliary parameter h . Four examples are tested using the proposed technique. It is shown that the solutions obtained by the Adomian decomposition method (ADM) are only special cases of the HAM solutions. The present work shows the validity and great potential of the homotopy analysis method for solving linear and nonlinear systems of fractional integro-differential equations. The basic idea described in this article is expected to be further employed to solve other similar nonlinear problems in fractional calculus

Differential transform method for solving singularly perturbed Volterra integral equations

AbstractIn this work, the applications of differential transform method were extended to singularly perturbed Volterra integral equations. To show the efficiency of the method, some singularly perturbed Volterra integral equations are solved as numerical examples. Numerical results show that the differential transform method is very effective and convenient for solving a large number of singularly perturbed problems with high accuracy

Analytical Study of Fractional-Order Multiple Chaotic FitzHugh-Nagumo Neurons Model Using Multistep Generalized Differential Transform Method

The multistep generalized differential transform method is applied to solve the fractional-order multiple chaotic FitzHugh-Nagumo (FHN) neurons model. The algorithm is illustrated by studying the dynamics of three coupled chaotic FHN neurons equations with different gap junctions under external electrical stimulation. The fractional derivatives are described in the Caputo sense. Furthermore, we present figurative comparisons between the proposed scheme and the classical fourth-order Runge-Kutta method to demonstrate the accuracy and applicability of this method. The graphical results reveal that only few terms are required to deduce the approximate solutions which are found to be accurate and efficient