Abstract
In this paper, we investigate the existence and approximation of the solutions of a nonlinear nonlocal three-point boundary value problem involving the forced Duffing equation with mixed nonlinearities. Our main tool of the study is the generalized quasilinearization method due to Lakshmikantham. Some illustrative examples are also presented. Mathematics Subject Classification (2000): 34B10, 34B15.
Highlights
1 Introduction The Duffing equation plays an important role in the study of mechanical systems
There are multiple forms of the Duffing equation, ranging from dampening to forcing terms. This equation possesses the qualities of a simple harmonic oscillator, a nonlinear oscillator, and has an ability to exhibit chaotic behavior
Chaos is defined as behavior so unpredictable as to appear random, allowing great sensitivity to small initial conditions
Summary
The Duffing equation plays an important role in the study of mechanical systems. There are multiple forms of the Duffing equation, ranging from dampening to forcing terms. The quasilinearization technique is a variant of Newton’s method This method applies to semilinear equations with convex (concave) nonlinearities and generates a monotone scheme whose iterates converge quadratically to a solution of the problem at hand. The nineties brought new dimensions to this technique when Lakshmikantham [7,8] generalized the method of quasilinearization by relaxing the convexity assumption This development was so significant that it attracted the attention of many researchers, and the method was extensively developed and applied to a wide range of initial and boundary value problems for different types of differential equations. There exist monotone sequences {an} and {bn} that converge in the space of continuous functions on J quadratically to a unique solution x(t) of the problem
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