The existence of asymptotically almost automorphic mild solutions to an abstract stochastic fractional partial integrodifferential equation is considered. The main tools are some suitable composition results for asymptotically almost automorphic processes, the theory of sectorial linear operators, and classical fixed point theorems. An example is also given to illustrate the main theorems.

$\nabla$-Laplace transform, fractional $\nabla$-power function, $\nabla$-Mittag-Leffler function, fractional $\nabla$-integrals, and fractional $\nabla$-differential on time scales are defined. Some of their properties are discussed in detail. After then, by using Laplace transform method, the existence of the solution and the dependency of the solution upon the initial value for Cauchy-type problem with the Riemann-Liouville fractional $\nabla$-derivative are studied. Also the explicit solutions to homogeneous equations and nonhomogeneous equations are derived by using Laplace transform method.

Some sufficient conditions for the nonlinear integral operator of the Hammerstein type to be a diffeomorphism defined on a certain Sobolev space are formulated. The main result assures the invertibility of the Hammerstein operator and in consequence the global solvability of the nonlinear Hammerstein equations. The applications of the result to nonlinear Dirichlet BVP involving the fractional Laplacian and to some specific Hammerstein equation are presented.

This paper presents a general synchronization technique and an amplitude modulation of chaotic generators. Conventional synchronization and antisynchronization are considered a very narrow subset from the proposed technique where the scale between the output response and the input response can be controlled via control functions and this scale may be either constant (positive, negative) or time dependent. The concept of the proposed technique is based on the nonlinear control theory and Lyapunov stability theory. The nonlinear controller is designed to ensure the stability and convergence of the proposed synchronization scheme. This technique is applied on the synchronization of two identical fractional-order Chua's circuit systems with memristor. Different examples are studied numerically with different system parameters, different orders, and with five alternative cases where the scaling functions are chosen to be positive/negative and constant/dynamic which covers all possible cases from conventional synchronization to the amplitude modulation cases to validate the proposed concept.

We obtain the existence of a weak solution to a fractional nonlinear hyperbolic equation arising from relative theory by the Galerkin method. Its uniqueness is also discussed. Furthermore, we show the regularity of the obtained solution. In our proof, we use harmonic analysis techniques and compactness arguments.

This paper deals with robust synchronization of the fractional-order unified chaotic systems. Firstly, control design for synchronization of nominal systems is proposed via fractional sliding mode technique. Then, systematic uncertainties and external disturbances are considered in the fractional-order unified chaotic systems, and adaptive sliding mode control is designed for the synchronization issue. Finally, numerical simulations are carried out to verify the effectiveness of the two proposed control techniques.

A version of one-sided weighted Morrey space is introduced. The boundedness of some classical one-sided operators in harmonic analysis and PDE on these spaces are discussed, including the Riemann-Liouville fractional integral.

We discuss the existence of positive solutions for the coupled system of multiterm singular fractional integrodifferential boundary value problems $D_{0^{+}}^{\alpha }u(t)+f_1(t,u(t),v(t),({\varphi }_{1}u)(t),({\psi }_{1}v)(t),D_{0^{+}}^{p}u(t),D_{{\mathrm{0}}^{+}}^{{\mu }_{\mathrm{1}}}v(t),D_{{\mathrm{0}}^{+}}^{{\mu }_{\mathrm{2}}}v(t),\dots ,D_{{\mathrm{0}}^{+}}^{{\mu }_m}v(t))=\mathrm{0},D_{{\mathrm{0}}^{+}}^{\beta }v(t)+f_{\mathrm{2}}(t,u(t),v(t),({\varphi }_{\mathrm{2}}u)(t),({\psi }_{\mathrm{2}}v)(t),D_{{\mathrm{0}}^{+}}^{q}v(t),D_{{\mathrm{0}}^{+}}^{{\nu }_{\mathrm{1}}}u(t),D_{{\mathrm{0}}^{+}}^{{\nu }_{\mathrm{2}}}u(t),\dots ,D_{{\mathrm{0}}^{+}}^{{\nu }_m}u(t))=\mathrm{0}$, $u^{(i)}(\mathrm{0})=\mathrm{0}$ and $v^{(i)}(\mathrm{0})=\mathrm{0}$ for all $\mathrm{0}\le i\le n-\mathrm{2}$, $[D_{{\mathrm{0}}^{+}}^{{\delta }_{\mathrm{1}}}u(t){]}_{t=\mathrm{1}}=\mathrm{0}$ for and $\alpha -{\delta }_{\mathrm{1}}\ge \mathrm{1}$, $[D_{{\mathrm{0}}^{+}}^{{\delta }_{\mathrm{2}}}v(t){]}_{t=\mathrm{1}}=\mathrm{0}$ for and $\beta -{\delta }_{\mathrm{2}}\ge \mathrm{1}$, where $n\ge \mathrm{4}$, , , $(i=\mathrm{1,2},\dots ,m)$, ${\gamma }_j,{\lambda }_j:[\mathrm{0,1}]\times [\mathrm{0,1}]\to (\mathrm{0},\mathrm{\infty })$ are continuous functions $(j=\mathrm{1,2})$ and $({\varphi }_{j}u)(t)={\int }_{\mathrm{0}}^t\mathrm{‍}{\gamma }_j(t,s)u(s)ds,\mathrm{}\mathrm{}\mathrm{}({\psi }_{j}v)(t)={\int }_{\mathrm{0}}^t\mathrm{‍}{\lambda }_j(t,s)v(s)ds.$ Here $D$ is the standard Riemann-Liouville fractional derivative, $f_j$ $(j=\mathrm{1,2})$ is a Caratheodory function, and $f_j(t,x,y,z,w,v,u_{\mathrm{1}},u_{\mathrm{2}},\dots ,u_m)$ is singular at the value 0 of its variables.

We propose Jacobi-Gauss-Lobatto collocation approximation for the numerical solution of a class of fractional-in-space advection-dispersion equation with variable coefficients based on Caputo derivative. This approach has the advantage of transforming the problem into the solution of a system of ordinary differential equations in time this system is approximated through an implicit iterative method. In addition, some of the known spectral collocation approximations can be derived as special cases from our algorithm if we suitably choose the corresponding special cases of Jacobi parameters $\alpha$ and $\beta$. Finally, numerical results are provided to demonstrate the effectiveness of the proposed spectral algorithms.

We investigate solutions of the Helmholtz equation involving local fractional derivative operators. We make use of the series expansion method and the variational iteration method, which are based upon the local fractional derivative operators. The nondifferentiable solution of the problem is obtained by using these methods.

We study the existence of mild solution of a class of nonlinear nonautonomous fractional integrodifferential equations with nonlocal conditions in a separable Banach space $X$. Combining the techniques of operator semigroup, noncompactness measures, and the fixed point theory, we obtain new existence of mild solution without the assumptions that the nonlinearity $f$ satisfies a Lipschitz type condition and the semigroup $\{\text{exp}(-tA(s))\}$ generated by ${\{-A(s)\}}_{s\in [0,T]}$ is compact. An application of the abstract result is also given.

We consider a theory of thermoelasticity constructed by taking into account the heat conduction in deformable bodies which depends on two temperatures. The first one is the conductive temperature, the second is the thermodynamic temperature, and the difference between them is proportional to the heat supply.

We study the nonlinear $q$-difference equations of fractional order $(D_q^{\alpha }u)(t)+f(t,u(t))=\mathrm{0}$, , $(D_q^{i}u)(\mathrm{0})=\mathrm{0}$, $(D_q^{\beta }u)(\mathrm{1})=a(D_q^{\beta }u)(\eta )$, $\mathrm{0}\le i\le n-\mathrm{2}$, where $D_q^{\alpha }$ is the fractional $q$-derivative of the Riemann-Liouville type of order $\alpha$, , $\alpha >\mathrm{2}$, $\mathrm{1}\le \beta \le n-\mathrm{2}$, and $\mathrm{0}\le a\le \mathrm{1}$. We obtain the existence and multiplicity results of positive solutions by using some fixed point theorems. Finally, we give examples to illustrate the results.

We study a class of fractional stochastic dynamic control systems of Sobolev type in Hilbert spaces. We use fixed point technique, fractional calculus, stochastic analysis, and methods adopted directly from deterministic control problems for the main results. A new set of sufficient conditions for approximate controllability is formulated and proved. An example is also given to provide the obtained theory.

The traditional integer-order partial differential equation-based image denoising approaches often blur the edge and complex texture detail; thus, their denoising effects for texture image are not very good. To solve the problem, a fractional partial differential equation-based denoising model for texture image is proposed, which applies a novel mathematical method—fractional calculus to image processing from the view of system evolution. We know from previous studies that fractional-order calculus has some unique properties comparing to integer-order differential calculus that it can nonlinearly enhance complex texture detail during the digital image processing. The goal of the proposed model is to overcome the problems mentioned above by using the properties of fractional differential calculus. It extended traditional integer-order equation to a fractional order and proposed the fractional Green’s formula and the fractional Euler-Lagrange formula for two-dimensional image processing, and then a fractional partial differential equation based denoising model was proposed. The experimental results prove that the abilities of the proposed denoising model to preserve the high-frequency edge and complex texture information are obviously superior to those of traditional integral based algorithms, especially for texture detail rich images.

This paper is devoted to the study of the existence and uniqueness of solutions for $n$th order differential equations with nonlocal integral boundary conditions. Our results are based on a variety of fixed point theorems. Some illustrative examples are discussed. We also discuss the Caputo type fractional analogue of the higher-order problem of ordinary differential equations.

Recently, several authors demonstrated the usefulness of fractional calculus operators in the derivation of particular solutions of a considerably large number of linear ordinary and partial differential equations of the second and higher orders. By means of fractional calculus techniques, we find explicit solutions of second-order linear ordinary differential equations.

This paper is concerned with the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional $q$-difference equations with nonlocal integral boundary conditions. The existence results are obtained by applying some well-known fixed point theorems and illustrated with examples.

A spectral deferred correction method is presented for the initial value problems of fractional differential equations (FDEs) with Caputo derivative. This method is constructed based on the residual function and the error equation deduced from Volterra integral equations equivalent to the FDEs. The proposed method allows that one can use a relatively few nodes to obtain the high accuracy numerical solutions of FDEs without the penalty of a huge computational cost due to the nonlocality of Caputo derivative. Finally, preliminary numerical experiments are given to verify the efficiency and accuracy of this method.

We study boundary value problems for the following nonlinear fractional Sturm-Liouville functional differential equations involving the Caputo fractional derivative: ${}^C{D}^{\beta }(p(t{)}^C{D}^{\alpha }u(t))$ + $f(t,u(t-\tau ),u(t+\theta ))=\mathrm{0}$, $t\in (\mathrm{0,1})$, ${\hspace{0.17em}}^C{D}^{\alpha }u(\mathrm{0}){=\hspace{0.17em}}^C{D}^{\alpha }u(\mathrm{1})={({\hspace{0.17em}}^C{D}^{\alpha }u(\mathrm{0}))}^{\prime \prime }=\mathrm{0}$, $au(t)-b{u}^{\prime }(t)=\eta (t)$, $t\in [-\tau ,\mathrm{0}]$, $cu(t)+d{u}^{\prime }(t)=\xi (t)$, $t\in [\mathrm{1,1}+\theta ]$, where ${}^C{D}^{\alpha }$, ${\hspace{0.17em}}^C{D}^{\beta }$ denote the Caputo fractional derivatives, $f$ is a nonnegative continuous functional defined on $C([-\tau ,\mathrm{1}+\theta ],ℝ)$, , , , are suitably small, $a,b,c,d>\mathrm{0}$, and $\eta \in C([-\tau ,\mathrm{0}],[\mathrm{0},\mathrm{\infty }))$, $\xi \in C([\mathrm{1,1}+\theta ],[\mathrm{0},\mathrm{\infty }))$. By means of the Guo-Krasnoselskii fixed point theorem and the fixed point index theorem, some positive solutions are obtained, respectively. As an application, an example is presented to illustrate our main results.

A new generalized fractional subequation method based on the relationship of fractional coupled equations is proposed. This method is applied to the space-time fractional coupled Konopelchenko-Dubrovsky equations and Nizhnik-Novikov-Veselov equations. As a result, many exact solutions are obtained including hyperbolic function solutions, trigonometric function solutions, and rational solutions. It is observed that the proposed approach provides a simple and reliable tool for solving many other fractional coupled differential equations.

The fractional partial differential equations stand for natural phenomena all over the world from science to engineering. When it comes to obtaining the solutions of these equations, there are many various techniques in the literature. Some of these give to us approximate solutions; others give to us analytical solutions. In this paper, we applied the modified trial equation method (MTEM) to the one-dimensional nonlinear fractional wave equation (FWE) and time fractional generalized Burgers equation. Then, we submitted 3D graphics for different value of $\alpha$.

We apply the lower and upper solutions method and fixed-point theorems to prove the existence of positive solution to fractional boundary value problem $D_{0^{+}}^{\alpha }u\left(t\right)+f\left(t,u\left(t\right)\right)=0$, , , $u\left(0\right)=u^{\prime }\left(0\right)=0$, $D_{0^{+}}^{\alpha -1}u\left(1\right)=\beta u\left(\xi \right)$, , where $D_{0^{+}}^{\alpha }$ denotes Riemann-Liouville fractional derivative, β is positive real number, $\beta {\xi }^{\alpha -1}\ge 2\mathrm{\Gamma }\left(\alpha \right)$, and $f$ is continuous on $\left[\mathrm{0,1}\right]\times \left[0,\left.\infty \right)\right.$. As an application, one example is given to illustrate the main result.

We extend the application of the Galerkin method for treating the multiterm fractional differential equations (FDEs) subject to initial conditions. A new shifted Legendre-Galerkin basis is constructed which satisfies exactly the homogeneous initial conditions by expanding the unknown variable using a new polynomial basis of functions which is built upon the shifted Legendre polynomials. A new spectral collocation approximation based on the Gauss-Lobatto quadrature nodes of shifted Legendre polynomials is investigated for solving the nonlinear multiterm FDEs. The main advantage of this approximation is that the solution is expanding by a truncated series of Legendre-Galerkin basis functions. Illustrative examples are presented to ensure the high accuracy and effectiveness of the proposed algorithms are discussed.

A modification of the variational iteration method (VIM) for solving systems of nonlinear fractional-order differential equations is proposed. The fractional derivatives are described in the Caputo sense. The solutions of fractional differential equations (FDE) obtained using the traditional variational iteration method give good approximations in the neighborhood of the initial position. The main advantage of the present method is that it can accelerate the convergence of the iterative approximate solutions relative to the approximate solutions obtained using the traditional variational iteration method. Illustrative examples are presented to show the validity of this modification.

Two numerical algorithms are derived to compute the fractional diffusion-wave equation with a reaction term. Firstly, using the relations between Caputo and Riemann-Liouville derivatives, we get two equivalent forms of the original equation, where we approximate Riemann-Liouville derivative by a second-order difference scheme. Secondly, for second-order derivative in space dimension, we construct a fourth-order difference scheme in terms of the hyperbolic-trigonometric spline function. The stability analysis of the derived numerical methods is given by means of the fractional Fourier method. Finally, an illustrative example is presented to show that the numerical results are in good agreement with the theoretical analysis.

An inverse problem for the diffusion operator on a finite interval with discontinuities conditions inside the interval is studied. We have shown that the potential function of the diffusion operator can be established uniquely by a set of values of eigenfunctions at the midpoint of the interval $(\mathrm{0},\pi )$ and one spectrum.

This paper investigates the existence of nonnegative solutions for nonlinear fractional differential equations with nonlocal fractional integrodifferential boundary conditions on an unbounded domain by means of Leray-Schauder nonlinear alternative theorem. An example is discussed for the illustration of the main work.

We consider a system of boundary value problems for fractional differential equation given by $D_{0^{+}}^{\beta }{\varphi }_p(D_{0^{+}}^{\alpha }u)(t)={\lambda }_1{a}_1(t)f_1(u(t),v(t))$, $t\in (\mathrm{0,1})$, $D_{0^{+}}^{\beta }{\varphi }_p(D_{0^{+}}^{\alpha }v)(t)={\lambda }_2{a}_2(t)f_2(u(t),v(t))$, $t\in (\mathrm{0,1})$, where , $\beta \le 2$, , ${\lambda }_1$, ${\lambda }_2$ are eigenvalues, subject either to the boundary conditions $D_{0^{+}}^{\alpha }u(0)=D_{0^{+}}^{\alpha }u(1)=0$, $u(0)=0$, $D_{0^{+}}^{{\beta }_1}u(1)-{\mathrm{\Sigma }}_{i=1}^{m-2}{a}_{1\mathrm{i\hspace{0.17em}}}{D}_{0^{+}}^{{\beta }_1}u({\xi }_{1i})=0$, $D_{0^{+}}^{\alpha }v(0)=D_{0^{+}}^{\alpha }v(1)=0$, $v(0)=0$, $D_{0^{+}}^{{\beta }_1}v(1)-{\mathrm{\Sigma }}_{i=1}^{m-2}{a}_{2\mathrm{i\hspace{0.17em}}}{D}_{0^{+}}^{{\beta }_1}v({\xi }_{2i})=0$ or $D_{0^{+}}^{\alpha }u(0)=D_{0^{+}}^{\alpha }u(1)=0$, $u(0)=0$, $D_{0^{+}}^{{\beta }_1}u(1)-{\mathrm{\Sigma }}_{i=1}^{m-2}{a}_{1\mathrm{i\hspace{0.17em}}}{D}_{0^{+}}^{{\beta }_1}u({\xi }_{1i})={\psi }_1(u)$, $D_{0^{+}}^{\alpha }v(0)=D_{0^{+}}^{\alpha }v(1)=0$, $v(0)=0$, $D_{0^{+}}^{{\beta }_1}v(1)-{\mathrm{\Sigma }}_{i=1}^{m-2}{a}_{2\mathrm{i\hspace{0.17em}}}{D}_{0^{+}}^{{\beta }_1}v({\xi }_{2i})={\psi }_2(v)$, where , $\alpha -{\beta }_1-1\ge 0$ and ${\psi }_1$, ${\psi }_2:C([\mathrm{0,1}])\to [0$, $\mathrm{\infty })$ are continuous functions. The Krasnoselskiis fixed point theorem is applied to prove the existence of at least one positive solution for both fractional boundary value problems. As an application, an example is given to demonstrate some of main results.

For integer-order systems, there are well-known practical rules for RL sketching. Nevertheless, these rules cannot be directly applied to fractional-order (FO) systems. Besides, the existing literature on this topic is scarce and exclusively focused on commensurate systems, usually expressed as the ratio of two noninteger polynomials. The practical rules derived for those do not apply to other symbolic expressions, namely, to transfer functions expressed as the ratio of FO zeros and poles. However, this is an important case as it is an extension of the classical integer-order problem usually addressed by control engineers. Extending the RL practical sketching rules to such FO systems will contribute to decrease the lack of intuition about the corresponding system dynamics. This paper generalises several RL practical sketching rules to transfer functions specified as the ratio of FO zeros and poles. The subject is presented in a didactic perspective, being the rules applied to several examples.

The medium through which the groundwater moves varies in time and space. The Hantush equation describes the movement of groundwater through a leaky aquifer. To include explicitly the deformation of the leaky aquifer into the mathematical formulation, we modify the equation by replacing the partial derivative with respect to time by the time-fractional variable order derivative. The modified equation is solved numerically via the Crank-Nicolson scheme. The stability and the convergence in this case are presented in details.

This paper discusses the fundamentals of negative probabilities and fractional calculus. The historical evolution and the main mathematical concepts are discussed, and several analogies between the two apparently unrelated topics are established. Based on the new conceptual perspective, some experiments are performed shading new light into possible future progress.