In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multiorder fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is ve...

In this paper, by introducing the fractional derivative in the sense of Caputo, the Adomian decomposition method is directly extended to study the coupled Burgers equations with timeand space-fractional derivatives. As a result, the realistic numerical solutions are obtained in a form of rapidly convergent series with easily computable components. The figures show the effectiveness and good acc...

We give a comparison of the efficiency of three alternative decomposition schemes for the approximate solution of multi-term fractional differential equations using the Caputo form of the fractional derivative. The schemes we compare are based on conversion of the original problem into a system of equations. We review alternative approaches and consider how the most appropriate numerical scheme...

In this paper we investigate optimality conditions for fractional variational problems, with a Lagrangian depending on the Riesz-Caputo derivative. First we prove a generalized Euler-Lagrange equation for the case when the interval of integration of the functional is different from the interval of the fractional derivative. Next we consider integral dynamic constraints on the problem, for sever...

Previously, based on the method of (radial powers) radial basis functions, we proposed a procedure for approximating derivative values from one-dimensional scattered noisy data. In this work, we show that the same approach also allows us to approximate the values of (Caputo) fractional derivatives (for orders between 0 and 1). With either an a priori or a posteriori strategy of choosing the reg...

There are many functions which are continuous everywhere but non-differentiable at someor all points such functions are termed as unreachable functions. Graphs representing suchunreachable functions are called unreachable graphs. For example ECG is such an unreachable graph. Classical calculus fails in their characterization as derivatives do not exist at the unreachable points. Such unreachabl...

*Correspondence: [email protected] School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, PR China Abstract In this paper, we study boundary-value problems for the following nonlinear fractional differential equations involving the Caputo fractional derivative: D0+x(t) = f (t, x(t), Dβ0+x(t)), t ∈ [0, 1], x(0) + x′(0) = y(x), ∫ 1 0 x(t)dt =m, x′′(0) = x′′′(0) = · · · = x(n–...

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 present state is determined by all past states with special forms of weights. To obtain discrete maps from fractional differential equations, we use the equival...

In this paper, we study the existence and uniqueness of solutions to the Cauchy problem for nonautonomous fractional differential equations involving Caputo derivative in Banach spaces. Definition for the solution in the Carathéodory sense and fundamental lemma are introduced. Some sufficient conditions for the existence and uniqueness of solutions are established by means of fractional calculu...

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of FredholmVolterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L∞ norm and weighted L2-norm. The numerical examp...