Advances in Difference Equations

http://www.advancesindifferenceequations.com/

List of Papers (Total 3,024)

Numerical solution of Volterra partial integro-differential equations based on sinc-collocation method

We provide the numerical solution of a Volterra integro-differential equation of parabolic type with memory term subject to initial boundary value conditions. Finite difference method in combination with product trapezoidal integration rule is used to discretize the equation in time and sinc-collocation method is employed in space. A weakly singular kernel has been viewed as an...

Optimality conditions for fractional variational problems with Caputo-Fabrizio fractional derivatives

In this paper, we study the necessary and sufficient optimality conditions for problems of the fractional calculus of variations with a Lagrange function depending on a Caputo-Fabrizio fractional derivative. The new kernel of Caputo-Fabrizio fractional derivative has no singularity, which is critical to interpreting the memory aftermath of the system. This property was not...

A nonstandard finite difference scheme for a multi-group epidemic model with time delay

In this paper, we derive a discretized multi-group epidemic model with time delay by using a nonstandard finite difference (NSFD) scheme. A crucial observation regarding the advantage of the NSFD scheme is that the positivity and boundedness of solutions of the continuous model are preserved. Furthermore, we show that the discrete model has the same equilibria, and the conditions...

Multi-quasi-synchronization of coupled fractional-order neural networks with delays via pinning impulsive control

We investigate the collective dynamics of multi-quasi-synchronization of coupled fractional-order neural networks with delays. Using the pinning impulsive strategy, we design a novel controller to pin the coupled networks to realize the multi-quasi-synchronization. Based on the comparison principle and mathematical analysis, we derive some novel criteria of the multi-quasi...

Transmission dynamics of a Huanglongbing model with cross protection

Huanglongbing (HLB) is one of the most common widespread vector-borne transmission diseases through psyllid, which is called a kind of cancer of plant disease. In recent years, biologists have focused on the role of cross protection strategy to control the spread of HLB. In this paper, according to transmission mechanism of HLB, a deterministic model with cross protection is...

On fractional Hahn calculus

In this paper, the new concepts of Hahn difference operators are introduced. The properties of fractional Hahn calculus in the sense of a forward Hahn difference operator are introduced and developed.

Stationary distribution and extinction of a stochastic SIRS epidemic model with information intervention

In this paper, a new SIRS epidemic model which considers the influence of information intervention and environmental noise is studied. The study shows that information intervention and white noise have great effects on the disease. First, we show that there is global existence and positivity of the solution. Then, we prove that the stochastic basic production R s $\mathscr{R}_{s...

Existence results for hybrid fractional neutral differential equations

We discuss the existence of solutions of initial value problems for a class of hybrid fractional neutral differential equations. To prove the main results, we use a hybrid fixed point theorem for the sum of three operators. We also derive the dependence of a solution on the initial data and present an example to illustrate the results.

Approximate solutions of a sum-type fractional integro-differential equation by using Chebyshev and Legendre polynomials

We investigate the existence of solutions for a sum-type fractional integro-differential problem via the Caputo differentiation. By using the shifted Legendre and Chebyshev polynomials, we provide a numerical method for finding solutions for the problem. In this way, we give some examples to illustrate our results.

Existence and stability of solutions to non-linear neutral stochastic functional differential equations in the framework of G-Brownian motion

In the past decades, quantitative study of different disciplines such as system sciences, physics, ecological sciences, engineering, economics and biological sciences, have been driven by new modeling known as stochastic dynamical systems. This paper aims at studying these important dynamical systems in the framework of G-Brownian motion and G-expectation. It is demonstrated that...

Existence results for a class of generalized fractional boundary value problems

In this paper, we study a class of generalized fractional order three-point boundary value problems that involve fractional derivative defined in terms of weight and scale functions. Using several fixed point theorems, the existence and uniqueness results are obtained.

Source degenerate identification problems with smoothing overdetermination

We consider degenerate identification problems with smoothing overdetermination in abstract spaces. We establish an identifiability result using a projection method and suitable hypotheses on the operators involved and develop an identification method by reformulating the problem into a nondegenerate problem. Then we use perturbation results for linear operators to solve the...

Normalized Bernstein polynomials in solving space-time fractional diffusion equation

In this paper, we solve a time-space fractional diffusion equation. Our methods are based on normalized Bernstein polynomials. For the space domain, we use a set of normalized Bernstein polynomials and for the time domain, which is a semi-infinite domain, we offer an algebraic map to make the rational normalized Bernstein functions. This study uses Galerkin and collocation...

Finite-time synchronization of delayed complex dynamical network via pinning control

We investigate a finite-time synchronization problem of hybrid-coupled delayed dynamical network via pinning control. According to linear feedback principle and finite-time control theory, the finite-time synchronization can be achieved by pinning control with suitable continuous finite-time controller. Some sufficient conditions are given for finite-time synchronization of...

Synchronization of fractional-order and integer-order chaotic (hyper-chaotic) systems with different dimensions

By constructing two scaling matrices, i.e., a function matrix Λ ( t ) $\Lambda (t)$ and a constant matrix W which is not equal to the identity matrix, a kind of W − Λ ( t ) $W-\Lambda(t)$ synchronization between fractional-order and integer-order chaotic (hyper-chaotic) systems with different dimensions is investigated in this paper. Based on the fractional-order Lyapunov direct...

Solution of two-dimensional time-fractional Burgers equation with high and low Reynolds numbers

Burgers’ equation frequently appears in the study of turbulence theory, as well as some other scientific fields. High and low Reynolds numbers play important roles in both modeling and numerical simulation. In this paper, we apply a numerical scheme to solve a two-dimensional time-fractional Burgers equation. The key feature of the proposed method is formed by combining the...

Local stable manifold of Langevin differential equations with two fractional derivatives

In this paper, we investigate the existence of local center stable manifolds of Langevin differential equations with two Caputo fractional derivatives in the two-dimensional case. We adopt the idea of the existence of a local center stable manifold by considering a fixed point of a suitable Lyapunov-Perron operator. A local center stable manifold theorem is given after deriving...

On the method of finding periodic solutions of second-order neutral differential equations with piecewise constant arguments

This paper provides a method of finding periodical solutions of the second-order neutral delay differential equations with piecewise constant arguments of the form x ″ ( t ) + p x ″ ( t − 1 ) = q x ( 2 [ t + 1 2 ] ) + f ( t ) $x

Some properties of algebraic difference equations of first order

We prove that if g ( z ) $g(z)$ is a finite-order transcendental meromorphic solution of ( △ c g ( z ) ) 2 = A ( z ) g ( z ) g ( z + c ) + B ( z ) , $$\bigl(\triangle_{c} g(z)\bigr)^{2}=A(z)g(z)g(z+c)+B(z), $$ where A ( z ) $A(z)$ and B ( z ) $B(z)$ are polynomials such that deg A ( z ) > 0 $\deg A(z)>0$ , then 1 ≤ ρ ( g ) = max { λ ( g ) , λ ( 1 g ) } . $$1 \leq\rho(g)=\max...

Weak solutions for a coupled system of Pettis-Hadamard fractional differential equations

In this paper, by applying the technique of measure of weak noncompactness and Mönch’s fixed point theorem, we investigate the existence of weak solutions under the Pettis integrability assumption for a coupled system of Hadamard fractional differential equations.

Stochastic Volterra integral equations with a parameter

In this paper, we study the properties of continuity and differentiability of solutions to stochastic Volterra integral equations and backward stochastic Volterra integral equations depending on a parameter.