Wavelet operational matrix method for solving the Riccati differential equation

A Haar wavelet operational matrix method (HWOMM) was derived to solve the Riccati differential equations. As a result, the computation of the nonlinear term was simplified by using the Block pulse function to expand the Haar wavelet one. The proposed method can be used to solve not only the classical Riccati differential equations but also the fractional ones. The capability and the simplicity of the proposed method was demonstrated by some examples and comparison with other methods.     

Singleton sets random attractor for stochastic FitzHugh–Nagumo lattice equations driven by fractional Brownian motions

The paper is devoted to the study of the dynamical behavior of the solutions of stochastic FitzHugh–Nagumo lattice equations, driven by fractional Brownian motions, with Hurst parameter greater than 1/2. Under some usual dissipativity conditions, the system considered here features different dynamics from the same one perturbed by Brownian motion. In our case, the random dynamical system has a unique random equilibrium, which constitutes a singleton sets random attractor.     

Certain subclasses of p-valently analytic functions involving a generalized fractional differintegral operator

By making use of the principle of subordination between analytic functions and the generalized fractional differintegral operator, we introduce and investigate some new subclasses of p-valently analytic functions in the open unit disk. Such results as inclusion relationships, integral-preserving properties, convolution properties, subordination and superordination properties, and sandwich theorems for these classes are derived.     

Existence and regularity of solutions for evolution equations with Riemann–Liouville fractional derivatives

In this paper, we derive the existence and uniqueness of mild solutions for inhomogeneous fractional evolution equations in Banach spaces by means of the method of fractional resolvent. Furthermore, we give the necessary and sufficient conditions for the existence of strong solutions. An example of the fractional diffusion equation is also presented to illustrate our theory.     

[3, 3] Padé approximation method for solving space fractional Fokker–Planck equations

The fractional Fokker–Planck equation has been used in various areas of engineering and physics. In this paper, we proposed a novel numerical scheme for solving the space fractional Fokker–Planck equation with the help of the [3, 3] Padé approximation. It is proved that the numerical method is unconditionally stable in view of the matrix analysis method. Finally, a numerical example is proposed to prove the effectiveness of the numerical scheme.     

Existence of solutions of initial value problems for nonlinear fractional differential equations

In this paper, by using the Schauder fixed point theorem, we study the existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations and obtain some new results.     

Integral manifolds of impulsive fractional functional differential systems

In this paper, a class of impulsive Caputo fractional functional differential systems with variable impulsive perturbations is investigated. Sufficient conditions for the existence of integral manifolds are obtained. The main results are proved by means of piecewise continuous Lyapunov functions and the fractional comparison principle. The demonstrated techniques can be applied in studying properties of many applied problems of diverse interest.     

Approximate solution for some stochastic differential equations involving both Gaussian and Poissonian white noises

By combining the Kramers-Moyal expansion with fractional Brownian motion of order n, in a formal symbolic calculus, one can obtain an approximation for the solution of some stochastic differential equations involving both Gaussian and Poissonian white noises, in terms of rotating Gaussian white noises on the grid defined by the complex roots of the unity. Illustrative examples are outlined.