sufficient conditions are established under which the zero solution x = 0 of equation (2) isunstable.

solitons are ubiquitous and exist in almost every area from sky to bottom. for solitons to appear, the relevant equation of motion must be nonlinear. in the present study, we deal with the korteweg-devries (kdv), modi ed korteweg-de vries (mkdv) and regularised longwave (rlw) equations using homotopy perturbation method (hpm). the algorithm makes use of the hpm to determine the initial expansio...

in this letter, the numerical scheme of nonlinear volterra-fredholm integro-differential equations is proposed in a reproducing kernel hilbert space (rkhs). the method is constructed based on the reproducing kernel properties in which the initial condition of the problem is satis ed. the nonlinear terms are replaced by its taylor series. in this technique, the nonlinear volterra-fredholm integr...

in this paper, we introduce a family of fractional-order chebyshev functions based on the classical chebyshev polynomials. we calculate and derive the operational matrix of derivative of fractional order $gamma$ in the caputo sense using the fractional-order chebyshev functions. this matrix yields to low computational cost of numerical solution of fractional order differential equations to the ...

In this study, nonlinear vibration of a composite cable is investigated by considering nonlinear stress-strain relations. The composite cable is composed of an aluminum wire as reinforcement and a rubber coating as matrix. The nonlinear governing equations of motion are derived about to an initial curve and based on the fundamentals of continuum mechanics and the nonlinear Green-Lagrangian stra...

In this paper, a collocation method for solving high-order linear partial differential equations (PDEs) with variable coefficients under more general form of conditions is presented. This method is based on the approximation of the truncated double exponential second kind Chebyshev (ESC) series. The definition of the partial derivative is presented and derived as new operational matrices of der...

The hydro turbine dynamics have a considerable influence on the dynamic stability of power system. In the study of dynamic stability, the system is modeled by the linear differential equations (small  signal analysis). Small signal stability of power systems is needed in all conditions and only is dependent on the conditions of power system performance before commotion occurrence. This paper pr...

In this paper, at first the elemantary and basic concepts of multiplicative discrete and continous differentian and integration introduced. Then for these kinds of differentiation invariant functions the general solution of discrete and continous multiplicative differential equations will be given. Finaly a vast class of difference equations with variable coefficients and nonlinear difference e...

this paper studies the existence of solutions for acoupled system of nonlinear fractional differential equations. newexistence and uniqueness results are established using banach fixedpoint theorem. other existence results are obtained using schaeferand krasnoselskii fixed point theorems. some illustrative examplesare also presented.