Solving systems of fractional differential equations by homotopy-perturbation method

In this Letter, approximate analytical solutions of systems of Fractional Differential Equations (FDEs) are derived by the Homotopy-Perturbation Method (HPM). The fractional derivatives are described in the Caputo sense. The solutions are obtained in the form of rapidly convergent infinite series with easily computable terms. Numerical results reveal that HPM is very effective and simple for obtaining approximate solutions of nonlinear systems of FDEs.     

Singular fractional evolution differential equations

We give an existence-uniqueness result for linear and nonlinear time fractional evolution equations with singularities in corresponding norm in extended Colombeau algebra of generalized functions using fractional analog for Duhamel principle. Paper deals with some nonlinear models with singularities appearing in viscoelasticity and in anomalous processes which have met great interest among researchers who consider them as a challenge in recent years.     

Numerical solution of fractional differential equations by using fractional B-spline

In this paper, we present fractional B-spline collocation method for the numerical solution of fractional differential equations. We consider this method for solving linear fractional differential equations which involve Caputo-type fractional derivatives. The numerical results demonstrate that the method is efficient and quite accurate and it requires relatively less computational work. For this reason one can conclude that this method has advantage on other methods and hence demonstrates the importance of this work.     

Lie and Noether symmetries of systems of complex ordinary differential equations and their split systems

The Lie and Noether point symmetry analyses of a kth-order system of m complex ordinary differential equations (ODEs) with m dependent variables are performed. The decomposition of complex symmetries of the given system of complex ODEs yields Lie- and Noether-like operators. The system of complex ODEs can be split into 2m coupled real partial differential equations (PDEs) and 2m Cauchy–Riemann (CR) equations. The classical approach is invoked to compute the symmetries of the 4m real PDEs and these are compared with the decomposed Lie- and Noether-like operators of the system of complex ODEs. It is shown that, in general, the Lie- and Noether-like operators of the system of complex ODEs and the symmetries of the decomposed system of real PDEs are not the same. A similar analysis is carried out for restricted systems of complex ODEs that split into 2m coupled real ODEs. We summarize our findings on restricted complex ODEs in two propositions.     

Dynamical modelling of superstatistical complex systems

We show how to construct the optimum superstatistical dynamical model for a given experimentally measured time series. For this purpose we generalise the superstatistics concept and study a Langevin equation with a memory kernel whose parameters fluctuate on a large time scale. We show how to construct a synthetic dynamical model with the same invariant density and correlation function as the experimental data. As a main example we apply our method to velocity time series measured in high-Reynolds-number turbulent Taylor-Couette flow, but the method can be applied to many other complex systems in a similar way.     

Exact solutions for nonlinear partial fractional differential equations

In this article,we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations.We use the improved(G’/G)-expansion function method to calculate the exact solutions to the time-and space-fractional derivative foam drainage equation and the time-and space-fractional derivative nonlinear KdV equation.This method is efficient and powerful for solving wide classes of nonlinear evolution fractional order equations.     

Initialized fractional differential equations with Riemann-Liouville fractional-order derivative

The initial value problem of fractional differential equations and its solving method are studied in this paper. Firstly, for easy understanding, a different version of the initialized operator theory is presented for Riemann-Liouville’s fractional-order derivative, addressing the initial history in a straightforward form. Then, the initial value problem of a single-term fractional differential equation is converted to an equivalent integral equation, a form that is easy for both theoretical and numerical analysis, and two illustrative examples are given for checking the correctness of the integral equation. Finally, the counter-example proposed in a recent paper, which claims that the initialized operator theory results in wrong solution of a fractional differential equation, is checked again carefully. It is found that solving the equivalent integral equation gives the exact solution, and the reason behind the result of the counter-example is that the calculation therein is based on the conventional Laplace transform for fractional-order derivative, not on the initialized operator theory. The counter-example can be served as a physical model of creep phenomena for some viscoelastic materials, and it is found that it fits experimental curves well.     

Matrix approach to discrete fractional calculus II: Partial fractional differential equations

A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of Podlubny’s matrix approach [I. Podlubny, Matrix approach to discrete fractional calculus, Fractional Calculus and Applied Analysis 3 (4) (2000) 359–386]. Four examples of numerical solution of fractional diffusion equation with various combinations of time-/space-fractional derivatives (integer/integer, fractional/integer, integer/fractional, and fractional/fractional) with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach. The fifth example illustrates that the method can be equally simply used for fractional differential equations with delays. A set of MATLAB routines for the implementation of the method as well as sample code used to solve the examples have been developed.     

A survey on the stability of fractional differential equations

Recently, fractional calculus has attracted much attention since it plays an important role in many fields of science and engineering. Especially, the study on stability of fractional differential equations appears to be very important. In this paper, a brief overview on the recent stability results of fractional differential equations and the analytical methods used are provided. These equations include linear fractional differential equations, nonlinear fractional differential equations, fractional differential equations with time-delay. Some conclusions for stability are similar to that of classical integer-order differential equations. However, not all of the stability conditions are parallel to the corresponding classical integer-order differential equations because of non-locality and weak singularities of fractional calculus. Some results and remarks are also included.     

Phase synchronization in fractional differential chaotic systems

We propose an analytical justification for phase synchronization of fractional differential equations. This justification is based upon a linear stability criterion for fractional differential equations. We then investigate the existence of phase synchronization in chaotic forced Duffing and Sprott-L fractional differential systems of equations. Our numerical results agree with those analytical justifications.     

Singular and non-topological soliton solutions for nonlinear fractional differential equations

In this article, the fractional derivatives are described in the modified Riemann–Liouville sense. We propose a new approach, namely an ansatz method, for solving fractional differential equations(FDEs) based on a fractional complex transform and apply it to solve nonlinear space–time fractional equations. As a result, the non-topological as well as the singular soliton solutions are obtained. This method can be suitable and more powerful for solving other kinds of nonlinear fractional FDEs arising in mathematical physics.     

Existence and approximation of solutions of fractional order iterative differential equations

In this paper, we investigate existence and approximation of solutions of fractional order iterative differential equations by virtue of nonexpansive mappings, fractional calculus and fixed point methods. Three existence theorems as well as convergence theorems for a fixed point iterative method designed to approximate these solutions are obtained in two different work spaces via Chebyshev’s norm, Bielecki’s norm and β norm. Finally, an example is given to illustrate the obtained results.     

Dynamical systems and complex systems theory to study unsteady combustion

Reacting flow fields are often subject to unsteadiness due to flow, reaction, diffusion, and acoustics. Further, flames can also exhibit inherent unsteadiness caused by various intrinsic instabilities. Interaction between various unsteady processes across multiple scales often makes combustion dynamics complex. Characterizing such complex dynamics is essential to ensure the safe and reliable operation of high efficiency combustion systems. Tools from nonlinear dynamics and complex systems theory provide new perspectives to analyze and interpret the data from real systems. They could also provide new ways of monitoring and controlling combustion systems. We discuss recent advances in studying unsteady combustion dynamics using the tools from dynamical systems theory and complex systems theory. We cover a range of problems involving unsteady combustion such as thermoacoustic instability, flame blowout, fire propagation, reaction chemistry and flow flame interaction.     

On multi-term fractional differential equations with multi-point boundary conditions

In this paper, we discuss the existence and uniqueness of solutions for a new class of multi-point boundary value problems of multi-term fractional differential equations by using standard fixed point theorems. We also demonstrate the application of the obtained results with the aid of examples. The paper concludes with the study of multi-term fractional integro-differential equations supplemented with multi-point boundary conditions. Our results are new and contribute significantly to the existing literature on the topic.     

On fractional Langevin differential equations with anti-periodic boundary conditions

In this paper, we provide existence criteria for the solutions of p-Laplacian fractional Langevin differential equations with anti-periodic boundary conditions. The Caputo fractional as well as Caputo q-fractional operators are used to address the derivatives. The main results are verified by the help of Leray–Schaefer’s fixed point theorem. We construct an example to illustrate the feasibility of the main theorems. Our results are new and provide extensions to some known theorems in the literature.     

On fractional order differential equations model for nonlocal epidemics

A fractional order model for nonlocal epidemics is given. Stability of fractional order equations is studied. The results are expected to be relevant to foot-and-mouth disease, SARS and avian flu.     

Lagrange equations of nonholonomic systems with fractional derivatives

This paper obtains Lagrange equations of nonholonomic systems with fractional derivatives. First, the exchanging relationships between the isochronous variation and the fractional derivatives are derived. Secondly, based on these exchanging relationships, the Hamilton's principle is presented for non-conservative systems with fractional derivatives. Thirdly, Lagrange equations of the systems are obtained. Furthermore, the d'Alembert–Lagrange principle with fractional derivatives is presented,and the Lagrange equations of nonholonomic systems with fractional derivatives are studied. An example is designed to illustrate these results.     

On systems of periodic differential equations

Restrictions on the characteristic exponents of periodic differential equations are derived from certain symmetry properties. In one case these restrictions amount to a stability theorem. In other cases they reduce the number of independent exponents.Supported by Fonds zur Förderung der wissenschaftlichen Forschung in österreich, Projekt Nr. 3225.     

Stability analysis of a class of fractional delay differential equations

In this paper we analyse stability of nonlinear fractional order delay differential equations of the form $D^{\alpha} y(t) = a f\left(y(t-\tau)\right) - b y(t)$ , where D ^α is a Caputo fractional derivative of order 0?<?α?≤?1. We describe stability regions using critical curves. To explain the proposed theory, we discuss fractional order logistic equation with delay.