Existence of positive solutions for the nonlinear fractional differential equation Dsu(x) = f (x,u(x)), 0 < s < 1, has been studied (S. Zhang, J. Math. Anal. Appl. 252 (2000) 804–812), where Ds denotes Riemann–Liouville fractional derivative. In the present work we study existence of positive solutions in case of the nonlinear fractional differential equation: L(D)u= f (x,u), u(0)= 0, 0 < x < 1...

The State-Dependant Riccati Equation method has been frequently used to design suboptimal controllers applied to nonlinear dynamic systems. Different methods for local stability analysis of SDRE controlled systems of order greater than two such as the attitude dynamics of a general rigid body have been extended in literature; however, it is still difficult to show global stability properties of...

We study the motion of a heavy tracer particle weakly coupled to a dense interacting Bose gas exhibiting Bose-Einstein condensation. In the so-called mean-field limit, the dynamics of this system approaches one determined by nonlinear Hamiltonian evolution equations. We derive the effective dynamics of the tracer particle, which is described by a non-linear integro-differential equation with me...

This paper investigates the existence of solutions of the nonlinear fractional differential equation { Du(t) + f(t, u(t),Du(t)) = 0, 0 < t < 1, 3 < α ≤ 4, u(0) = u′(0) = u′′(0) = 0, u(1) = u(ξ), 0 < ξ < 1, where D is the Caputo fractional derivative, β > 0, α− β ≥ 1. The peculiarity of this equation is that the nonlinear term depends on the fractional derivative of the unknown function, compare...

We study a stochastic Schrödinger equation with quadratic nonlinearity and space-time fractional perturbation, in space dimension d ≤ 3 <mml:annotation encodin...

In this paper, we apply a new method for solving system of partial differential equations within local fractional derivative operators. The approximate analytical solutions are obtained by using the local fractional Laplace variational iteration method, which is the coupling method of local fractional variational iteration method and Laplace transform. Illustrative examples are included to demo...

We consider a nonlinear parabolic equation with fractional diffusion which arises from modelling chemotaxis in bacteria. We prove the wellposedness, continuation criteria and smoothness of local solutions. In the repulsive case we prove global wellposedness in Sobolev spaces. Finally in the attractive case, we prove that for a class of smooth initial data the L∞x -norm of the corresponding solu...

We analytically and numerically investigate the stability dynamics of plane wave solutions fractional nonlinear Schrödinger (NLS) equation, where long-range dispersion is described by Laplacian (??)?/2. The linear analysis shows that in defocusing NLS are always stable if power ??[1,2] but unstable for ??(0,1). In focusing case, they can be linearly any ??(0,2]. then apply split-step Fourier sp...

In this paper we propose a new solution technique for numerical solution of fractional Benney equation, a fourth degree nonlinear fractional partial differential equation with broad range of applications. The method could be described as a hybrid technique which uses advantages of both wavelets and operational matrices. Having applied the present method, fractional Benney equation is converted ...