Chapters authored
Analytical Solutions of Some Strong Nonlinear Oscillators
Oscillators are omnipresent; most of them are inherently nonlinear. Though a nonlinear equation mostly does not yield an exact analytic solution for itself, plethora of elementary yet practical techniques exist for extracting important information about the solution of equation. The purpose of this chapter is to introduce some new techniques for the readers which are carefully illustrated using mainly the examples of Duffing’s oscillator. Using the exact analytical solution to cubic Duffing and cubic-quinbic Duffing oscillators, we describe the way other conservative and some non conservative damped nonlinear oscillators may be studied using analytical techniques described here. We do not make use of perturbation techniques. However, some comparison with such methods are performed. We consider oscillators having the form x¨+fx=0 as well as x¨+2εẋ+fx=Ft, where x=xt and f=fx and Ft are continuous functions. In the present chapter, sometimes we will use f−x=−fx and take the approximation fx≈∑j=1Npjxj, where j=1,3,5,⋯N only odd integer values and x∈−AA. Moreover, we will take the approximation fx≈∑j=0Npjxj, where j=1,2,3,⋯N, and x∈−AA. Arbitrary initial conditions are considered. The main idea is to approximate the function f=fx by means of some suitable cubic or quintic polynomial. The analytical solutions are expressed in terms of the Jacobian and Weierstrass elliptic functions. Applications to plasma physics, electronic circuits, soliton theory, and engineering are provided.
Part of the book: Engineering Problems
Perspective Chapter: Families of Seventh-Order KdV Equations Having Traveling Wave and Soliton Solutions
In this paper, we consider the problem of finding traveling wave solutions to the generalized seventh-order KdV equation (KdV7). Solitons are non-linear waves that exhibit extremely unexpected and interesting behavior—solitary waves that propagate without deformation. We use different approaches in order to find one and multisoliton solutions. Soliton travels through liquid, solid, and gaseous media and even as electron waves through an electromagnetic field. Making use of a traveling wave transformation, we obtain a non-linear ode, which is solved using either hyperbolic or elliptic algorithm. We also use the Hirota method to get the bilinear form, and then we may obtain multisoliton solutions. In the end, we consider the forced KdV7.
Part of the book: Nonlinear Systems and Matrix Analysis - Recent Advances in theory and Applications [Working title]