We independently propose a new kind of the definition of fractional difference, fractional sum, and fractional difference equation, give some basic properties of fractional difference and fractional sum, and give some examples to demonstrate several methods of how to solve certain fractional difference equations.

An adaptive pseudospectral method is presented for solving a class of multiterm fractional boundary value problems (FBVP) which involve Caputo-type fractional derivatives. The multiterm FBVP is first converted into a singular Volterra integrodifferential equation (SVIDE). By dividing the interval of the problem to subintervals, the unknown function is approximated using a piecewise interpolation polynomial with unknown coefficients which is based on shifted Legendre-Gauss (ShLG) collocation points. Then the problem is reduced to a system of algebraic equations, thus greatly simplifying the problem. Further, some additional conditions are considered to maintain the continuity of the approximate solution and its derivatives at the interface of subintervals. In order to convert the singular integrals of SVIDE into nonsingular ones, integration by parts is utilized. In the method developed in this paper, the accuracy can be improved either by increasing the number of subintervals or by increasing the degree of the polynomial on each subinterval. Using several examples including Bagley-Torvik equation the proposed method is shown to be efficient and accurate.

In the paper, a class of perturbed Volterra equations of convolution type with three kernel functions is considered. The kernel functions $g_{\alpha }=t^{\alpha -1}/\mathrm{\Gamma }(\alpha )$, $t>0$, $\alpha \in [1,2]$, correspond to the class of equations interpolating heat and wave equations. The results obtained generalize our previous results from 2010.

A new robust fractional-order sliding mode controller (FOSMC) is proposed for the position control of a permanent magnet synchronous motor (PMSM). The sliding mode controller (SMC), which is insensitive to uncertainties and load disturbances, is studied widely in the application of PMSM drive. In the existing SMC method, the sliding surface is usually designed based on the integer-order integration or differentiation of the state variables, while in this proposed robust FOSMC algorithm, the sliding surface is designed based on the fractional-order calculus of the state variables. In fact, the conventional SMC method can be seen as a special case of the proposed FOSMC method. The performance and robustness of the proposed method are analyzed and tested for nonlinear load torque disturbances, and simulation results show that the proposed algorithm is more robust and effective than the conventional SMC method.

This paper is devoted to the study of abstract time-fractional equations of the following form: ${\mathbf{D}}_t^{{\alpha }_n}u(t)+{\sum }_{i=1}^{n-1}{A}_i{\mathbf{D}}_t^{{\alpha }_i}u(t)=A{\mathbf{D}}_t^{\alpha }u(t)+f(t)$, $t>0$, $u^{(k)}(0)=u_k$, $k=0,\mathrm{...},\lceil {\alpha }_n\rceil -1$, where $n\in \mathbb{N}\setminus \{1\}$, $A$ and $A_1,\mathrm{...},A_{n-1}$ are closed linear operators on a sequentially complete locally convex space , , $f(t)$ is an $E$-valued function, and ${\mathbf{\text{D}}}_t^{\alpha }$ denotes the Caputo fractional derivative of order $\alpha$ (Bazhlekova (2001)). We introduce and systematically analyze various classes of $k$-regularized ($C_1,C_2$)-existence and uniqueness (propagation) families, continuing in such a way the researches raised in (de Laubenfels (1999, 1991), Kostić (Preprint), and Xiao and Liang (2003, 2002). The obtained results are illustrated with several examples.

This paper deals with the existence of mild solutions for a class of fractional evolution equations with compact analytic semigroup. We prove the existence of mild solutions, assuming that the nonlinear part satisfies some local growth conditions in fractional power spaces. An example is also given to illustrate the applicability of abstract results.

We consider the numerical solution of a time-fractional heat equation, which is obtained from the standard diffusion equation by replacing the first-order time derivative with Riemann-Liouville fractional derivative of order α, where . The main purpose of this work is to extend the idea on Crank-Nicholson method to the time-fractional heat equations. We prove that the proposed method is unconditionally stable, and the numerical solution converges to the exact one with the order $O({\tau }^2+h^2)$. Numerical experiments are carried out to support the theoretical claims.

We study here different fractional versions of the compound Poisson process. The fractionality is introduced in the counting process representing the number of jumps as well as in the density of the jumps themselves. The corresponding distributions are obtained explicitly and proved to be solution of fractional equations of order less than one. Only in the final case treated in this paper, where the number of jumps is given by the fractional-difference Poisson process defined in Orsingher and Polito (2012), we have a fractional driving equation, with respect to the time argument, with order greater than one. Moreover, in this case, the compound Poisson process is Markovian and this is also true for the corresponding limiting process. All the processes considered here are proved to be compositions of continuous time random walks with stable processes (or inverse stable subordinators). These subordinating relationships hold, not only in the limit, but also in the finite domain. In some cases the densities satisfy master equations which are the fractional analogues of the well-known Kolmogorov one.

The reaction-diffusion equations have been widely used in physics, chemistry, and other areas. Forest fire can also be described by such equations. We here propose a fighting forest fire model. By using the normal form approach theory and center manifold theory, we analyze the stability of the trivial solution and Hopf bifurcation of this model. Finally, we give the numerical simulations to illustrate the effectiveness of our results.

We show that two recent definitions of discrete nabla fractional sum operators are related. Obtaining such a relation between two operators allows one to prove basic properties of the one operator by using the known properties of the other. We illustrate this idea with proving power rule and commutative property of discrete fractional sum operators. We also introduce and prove summation by parts formulas for the right and left fractional sum and difference operators, where we employ the Riemann-Liouville definition of the fractional difference. We formalize initial value problems for nonlinear fractional difference equations as an application of our findings. An alternative definition for the nabla right fractional difference operator is also introduced.

This paper presents the application of homotopy perturbation and variational iteration methods as numerical methods for Fredholm integrodifferential equation of fractional order with initial-boundary conditions. The fractional derivatives are described in Caputo sense. Some illustrative examples are presented.

This paper studies the information content of the chromosomes of twenty-three species. Several statistics considering different number of bases for alphabet character encoding are derived. Based on the resulting histograms, word delimiters and character relative frequencies are identified. The knowledge of this data allows moving along each chromosome while evaluating the flow of characters and words. The resulting flux of information is captured by means of Shannon entropy. The results are explored in the perspective of power law relationships allowing a quantitative evaluation of the DNA of the species.

This paper investigates the consensus problem for a class of fractional-order uncertain multiagent systems with general linear node dynamics. Firstly, an observer-type consensus protocol is proposed based on the relative observer states of neighboring agents. Secondly, based on property of the Kronecker product and stability theory of fractional-order system, some sufficient conditions are presented for robust asymptotical stability of the observer-based fractional-order control systems. Thirdly, robust stabilizing controllers are derived by using linear matrix inequality approach and matrix’s singular value decomposition. Our results are in the form of linear matrix inequalities which can easily be solved by LMI toolbox in MATLAB. Finally, a numerical simulation is performed to show the effectiveness of the theoretical results.

We consider the existence of positive solutions for a class of nonlinear integral boundary value problems for fractional differential equations. By using some fixed point theorems, the existence and multiplicity results of positive solutions are obtained. The results obtained in this paper improve and generalize some well-known results.

We introduce certain type of weighted variant of Riemann-Liouville fractional integral on ${ℝ}^n$ and obtain its sharp bounds on the central Morrey and $\lambda$-central BMO spaces. Moreover, we establish a sufficient and necessary condition of the weight functions so that commutators of weighted Hardy operators (with symbols in $\lambda$-central BMO space) are bounded on the central Morrey spaces. These results are further used to prove sharp estimates of some inequalities due to Weyl and Cesàro.

This paper is concerned with the existence of three solutions to a nonlinear fractional boundary value problem as follows: $(d/dt)((1/2){0}_{}{D}_{t}^{\alpha -1}{(}_{0}^{C}{D}_{t}^{\alpha }u(t))-(1/2){t}_{}{D}_{T}^{\alpha -1}{(}_{t}^{C}{D}_{T}^{\alpha }u(t)))+\lambda a(t)f(u(t))=0, \text{a}.\text{e}.\mathrm{ }t\in [0,T],$ $u(0)=u(T)=0,$ where $\alpha \in (1/2,1]$, and $\lambda $ is a positive real parameter. The approach is based on a critical-points theorem established by G. Bonanno.

We consider finite element Galerkin solutions for the space fractional diffusion equation with a nonlinear source term. Existence, stability, and order of convergence of approximate solutions for the backward Euler fully discrete scheme have been discussed as well as for the semidiscrete scheme. The analytical convergent orders are obtained as $O(k+h^{\tilde{\gamma }})$, where $\tilde{\gamma }$ is a constant depending on the order of fractional derivative. Numerical computations are presented, which confirm the theoretical results when the equation has a linear source term. When the equation has a nonlinear source term, numerical results show that the diffusivity depends on the order of fractional derivative as we expect.

We study the dynamic properties (equilibrium points, local and global stability, chaos and bifurcation) of the continuous dynamical system of the logistic equation of complex variables. The existence and uniqueness of uniformly Lyapunov stable solution will be proved.

This paper is concerned with the existence of integral solutions for nondensely defined fractional functional differential equations with impulse effects. Some errors in the existing paper concerned with nondensely defined fractional differential equations are pointed out, and correct formula of integral solutions is established by using integrated semigroup and some probability densities. Sufficient conditions for the existence are obtained by applying the Banach contraction mapping principle. An example is also given to illustrate our results.

We investigate the existence and uniqueness of positive solutions for the following singular fractional three-point boundary value problem , where , $D_{0^{+}}^{\alpha }$ is the standard Riemann-Liouville derivative and $f:(\mathrm{0,1}]\times [0,\infty )\to [0,\infty )$ with ${\lim \hspace{0.17em}}_{t\to 0^{+}}f(t,·)=\infty$ (i.e., $f$ is singular at $t=0$). Our analysis relies on a fixed point theorem in partially ordered metric spaces.

We study an antiperiodic boundary value problem of nonlinear fractional differential equations of order $q\in (4,5]$. Some existence results are obtained by applying some standard tools of fixed-point theory. We show that solutions for lower-order anti-periodic fractional boundary value problems follow from the solution of the problem at hand. Our results are new and generalize the existing results on anti-periodic fractional boundary value problems. The paper concludes with some illustrating examples.

We discuss the existence of positive solutions to the following fractional m-point boundary value problem with changing sign nonlinearity , where $\lambda$ is a positive parameter, , , with , $D_{0+}^{\alpha }$ is the standard Riemann-Liouville derivative, $f$ and may be singular at $t=0$ and/or $t=1$ and also may change sign. The work improves and generalizes some previous results.

We prove coupled coincidence point and coupled fixed point results of $F:X\times X\to X$ and $g:X\to X$ involving Meir-Keeler type contractions on the class of partially ordered metric spaces. Our results generalize some recent results in the literature. Also, we give some illustrative examples and application.

We investigate the existence of at least two positive solutions to eigenvalue problems of fractional differential equations with sign changing nonlinearities in more generalized boundary conditions. Our analysis relies on the Avery-Peterson fixed point theorem in a cone. Some examples are given for the illustration of main results.

We discuss the initial value problem for the nonlinear fractional differential equation $L(D)u=f(t,u),t\in (\mathrm{0,1}],u(0)=0$, where $L(D)=D^{s_n}-a_{n-1}{D}^{s_{n-1}}-\cdots -a_1{D}^{s_1}$, , and , $j=\mathrm{1,2},\dots ,n-1$, $D^{s_j}$ is the standard Riemann-Liouville fractional derivative and $f:[\mathrm{0,1}]\times ℝ\to ℝ$ is a given continuous function. We extend the basic theory of differential equation, the method of upper and lower solutions, and monotone iterative technique to the initial value problem. Some existence and uniqueness results are established.

We prove an integral inequality with singularity, which complements some known results. This inequality enables us to study the dependence of the solution on the initial condition to a fractional differential equation in the weighted space.

This paper studies musical opus from the point of view of three mathematical tools: entropy, pseudo phase plane (PPP), and multidimensional scaling (MDS). The experiments analyze ten sets of different musical styles. First, for each musical composition, the PPP is produced using the time series lags captured by the average mutual information. Second, to unravel hidden relationships between the musical styles the MDS technique is used. The MDS is calculated based on two alternative metrics obtained from the PPP, namely, the average mutual information and the fractal dimension. The results reveal significant differences in the musical styles, demonstrating the feasibility of the proposed strategy and motivating further developments towards a dynamical analysis of musical sounds.

We discuss the existence and uniqueness of solution to nonlinear fractional order ordinary differential equations $\left({\mathcal{D}}^{\alpha }-\rho t{\mathcal{D}}^{\beta }\right)x\left(t\right)=f\left(t,x\left(t\right),{\mathcal{D}}^{\gamma }x\left(t\right)\right)$, $t\in (0,1)$ with boundary conditions $x\left(0\right)=x_0,x\left(1\right)=x_1$ or satisfying the initial conditions $x\left(0\right)=0,x^{\prime }\left(0\right)=1$, where ${\mathcal{D}}^{\alpha }$ denotes Caputo fractional derivative, $\rho$ is constant, and . Schauder's fixed-point theorem was used to establish the existence of the solution. Banach contraction principle was used to show the uniqueness of the solution under certain conditions on $f$.

We investigate the existence and multiplicity of positive solutions for the nonlinear fractional differential equation initial value problem $D_{0+}^{\alpha }u\left(t\right)+D_{0+}^{\beta }u\left(t\right)=f(t,u\left(t\right)),\hspace{0.17em}$ , where is the standard Riemann-Liouville differentiation and $f:\left[\mathrm{0,1}\right]\times \left[0,\infty \right)\to \left[0,\infty \right)$ is continuous. By using some fixed-point results on cones, some existence and multiplicity results of positive solutions are obtained.

This paper investigates the existence of solutions for a coupled system of nonlinear fractional differential equations with m-point fractional boundary conditions on an unbounded domain. Some standard fixed point theorems are applied to obtain the main results. The paper concludes with two illustrative examples.

This paper studies a fractional differential inequality involving a new fractional derivative (Hilfer-Hadamard type) with a polynomial source term. We obtain an exponent for which there does not exist any global solution for the problem. We also provide an example to show the existence of solutions in a wider space for some exponents.

Variational iteration method and homotopy perturbation method are used to solve the fractional Fredholm integrodifferential equations with constant coefficients. The obtained results indicate that the method is efficient and also accurate.

Some Caputo q-fractional difference equations are solved. The solutions are expressed by means of a new introduced generalized type of q-Mittag-Leffler functions. The method of successive approximation is used to obtain the solutions. The obtained q-version of Mittag-Leffler function is thought as the q-analogue of the one introduced previously by Kilbas and Saigo (1995).