In this paper, we apply fractional complex transform to convert the fractional nonlinear Schr"{o}dinger equations to the nonlinear Schr"{o}dinger equations.  

In this paper we consider the positive definite solutions of nonlinear matrix equation X + A X−δA = Q, where δ ∈ (0, 1], which appears for the first time in [S.M. El-Sayed, A.C.M. Ran, On an iteration methods for solving a class of nonlinear matrix equations, SIAM J. Matrix Anal. Appl. 23 (2001) 632–645]. The necessary and sufficient conditions for the existence of a solution are derived. An it...

Asymptotic dynamics of ordinary differential equations (ODEs) are commonly understood by looking at eigenvalues of a matrix, and transient dynamics can be bounded above and below by considering the corresponding pseudospectra. While asymptotics for other classes of differential equations have been studied using eigenvalues of a (nonlinear) matrix-valued function, there are no analogous pseudosp...

In this paper, numerical solutions of the linear and nonlinear fractional integrodifferential equations with weakly singular kernel where fractional derivatives are considered in the Caputo sense, have been obtained by Legendre wavelets method. The block pulse functions and their properties are employed to derive a general procedure for forming the operational matrix of fractional integration f...

The paper develops the method for construction of families of particular solutions to some classes of nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic matrix equations and PDE. Admittable solutions involve arbitrary functions of either single or several variables.

In the present study, the dynamic buckling of the graphene sheet coupled by a viscoelastic matrix was studied. In light of the simplicity of Eringen's non-local continuum theory to considering the nanoscale influences, this theory was employed. Equations of motion and boundary conditions were obtained using Mindlin plate theory by taking nonlinear strains of von Kármán and Hamilton's principle ...

A generalized dynamical robust nonlinear filtering framework is established for a class of Lipschitz differential algebraic systems, in which the nonlinearities appear both in the state and measured output equations. The system is assumed to be affected by norm-bounded disturbance and to have both norm-bounded uncertainties in the realization matrices as well as nonlinear model uncertainties. W...

This article concerns the dressing method for solving of multidimensional nonlinear Partial Differential Equations. In particular, we join hierarchy of matrix Burgers type equation with hierarchies of equations integrable by the Inverse Spectral Transform (IST). Example of resonance interaction of wave packets in (3+1)-dimensions is given.

The paper develops the method for construction of the families of particular solutions to the nonlinear Partial Differential Equations (PDE) without relation to the complete integrability. Method is based on the specific link between algebraic matrix equations and PDE. Example of (2+2)-dimensional generalization of Burgers equation is given. PACS numbers: 05.45.-a, 05.45.Yv

Most process models resulting from first principles consist of not only nonlinear differential equations but also contain nonlinear algebraic equations, resulting in nonlinear DAE systems. Since large-scale nonlinear DAE systems are too complex to be used for real-time optimization or control, model reduction of these types of models is a strategy that needs to be applied for online application...