This paper is concerned with the initial value problem to a nonlinear fractional difference equation with the Caputo like difference operator. By means of some fixed point theorems, global and local existence results of solutions are obtained. An example is also provided to illustrate our main result.

We consider a general class of discrete nonlinear Schrödinger equations (DNLS) on the lattice hZ with mesh size h > 0. In the continuum limit when h → 0, we prove that the limiting dynamics are given by a nonlinear Schrödinger equation (NLS) on R with the fractional Laplacian (− ) as dispersive symbol. In particular, we obtain that fractional powers 1 2 < α < 1 arise from long-range lattice int...

We consider a class of nonlinear integro-differential equations involving a fractional power of the Laplacian and a nonlocal quadratic nonlinearity represented by a singular integral operator. Initially, we introduce cut-off versions of this equation, replacing the singular operator by its Lipschitz continuous regularizations. In both cases we show the local existence and global uniqueness in L...

This paper presents a novel robust fractional PIλ controller design for flexible joint electrically driven robots. Because of using voltage control strategy, the proposed approach is free of problems arising from torque control strategy in the design and implementation. In fact, the motor's current includes the effects of nonlinearities and coupling in the robot manipulator. Therefore, cancella...

We undertake a detailed comparison of the results of direct numerical simulations of the soliton gas dynamics for the Korteweg – de Vries equation with the analytical predictions inferred from the exact solutions of the relevant kinetic equation for solitons. Two model problems are considered: (i) the propagation of a ‘trial’ soliton through a one-component ‘cold’ soliton gas consisting of rand...

In this paper, the ( / ) G G  -expansion method is extended to solve fractional differential equations in the sense of modified Riemann-Liouville derivative. Based on a nonlinear fractional complex transformation, certain fractional partial differential equations can be turned into ordinary differential equations of integer order. For illustrating the validity of this method, we apply it to fi...

in this paper a new form of the homptopy perturbation method is used for solving oscillator differential equation, which yields the maclaurin series of the exact solution. nonlinear vibration problems and differential equation oscillations have crucial importance in all areas of science and engineering. these equations equip a significant mathematical model for dynamical systems. the accuracy o...