Solutions Of Nonlinear Delay Differential Equations Research Articles

On meromorphic solutions of nonlinear delay-differential equations

Using Cartan's second main theorem and Nevanlinna's theorem concerning a group of meromorphic functions, we obtain the growth and zero distribution of meromorphic solutions of the nonlinear delay-differential equation fn(z)+P(z)f(k)(z+η)=H0(z)+H1(z)eω1zq+⋯+Hm(z)eωmzq, where n,k,q,m are positive integers, η,ω1,⋯,ωm are complex numbers with ω1⋯ωm≠0, and P,H0,H1,⋯,Hm are entire functions of order less than q with PH1⋯Hm≢0. Especially for η=0, some sufficient conditions are given to guarantee the above equation has no meromorphic solutions of few poles.

Convergence and stability of extended BBVMs for nonlinear delay-differential-algebraic equations with piecewise continuous arguments

Delay-differential-algebraic equations have been widely used to model some important phenomena in science and engineering. Since, in general, such equations do not admit a closed-form solution, it is necessary to solve them numerically by introducing suitable integrators. The present paper extends the class of block boundary value methods (BBVMs) to approximate the solutions of nonlinear delay-differential equations with algebraic constraint and piecewise continuous arguments. Under the classical Lipschitz conditions, convergence and stability criteria of the extended BBVMs are derived. Moreover, a couple of numerical examples are provided to illustrate computational effectiveness and accuracy of the methods.

Galerkin Approximations for Higher Order Delay Differential Equations

In this work, Galerkin approximations are developed for a system of n first order nonlinear delay differential equations (DDEs) and also for an nth order nonlinear scalar DDE. The DDEs are converted into an equivalent system of partial differential equations of the same order along with the nonlinear boundary constraints. Lagrange multipliers are then introduced and explicit expressions for the Lagrange multipliers are derived to enforce the nonlinear boundary constraints. To illustrate the method, comparisons are made between the numerical solution of nonlinear DDEs and its Galerkin approximations for different parameter values.

An existence theorem for nonlinear delay differential equations

In this paper we prove a theorem on the existence of solutions of nonlinear delay differential equations, with implicit derivatives. The result is established using the measure of noncompactness of a set and Darbo′s fixed point theorem.

Some results on the stability of non-linear delay-differential systems

Tho problem of stability properties for the solutions of non-linear delay-differential equations ia considered. The approach used is to study the behaviour of the solution -of non-linear delay-differential equations with respect to solutions of a linear delay-differential equation. The principal technique employed is an extension of Liapunov1 s direct method. A series of theorems is obtained yielding criteria for the behaviour of the solutions in terms of existence of the Liapunov-type function with appropriate properties. The criterion is a condition on the roots of a certain ‘quasi-polynomial’ i.e. a polynomial in a variable and exponential of that variable.