Approximation of solutions for an initial and terminal value problem for the forced Duffing equation with non-viscous damping
type: Item
instance of: scholarly article
title: Approximation of solutions for an initial and terminal value problem for the forced Duffing equation with non-viscous damping
zbMATH Open document ID: 1213.65156
author: Bashir Ahmad
author: Juan J. Nieto
published in: Applied Mathematics and Computation
publication date: 2010-06-25T00:00:00Z
review text: This paper is concerned with the approximate solution of some special boundary value problems for second order functional equations of the type \[ L[u](t) \equiv u''(t) + a_0 u'(t) + f(t, u(t)) + \int_0^t K(t,s, u'(s) )\, ds =0, \quad t \in [0,T], \tag{1} \] together with the boundary conditions \( u(0)=a, u'(T)=0\). These problems, for the special functions \( f(t,u)= a_1 w^2 u + a_2 u^3 - a_3 \cos ( w t + a_4)\) with \( a_1, a_2, a_3, a_4, w \) constants, are here referred to as initial and terminal value problems for the forced Duffing equation with non viscous damping. The main result is that assuming the existence of a lower solution \( u= \alpha (t)\) and an upper solution \( u = \beta (t)\) of (1) that satisfy \( L[\alpha](t) \leq 0\), \(\alpha (0) \leq a\), \(\alpha'(T) \leq 0\) and \( L[\beta](t) \geq 0\), \(\beta (0) \geq a\), \(\beta'(T) \geq 0\) together with some conditions on the functions \( f(t,u)\) and \( K(t,s,u')\), it can be proved the existence of a monotonic sequence \( \{ w_n (t) \}_{n \geq 0}\), \( t \in [0,T]\) with \( \alpha = w_0 \leq w_1 \leq w_2\leq \dots \leq \beta\) that converges uniformly in \([0,T]\) to a solution of (1) that turns out to be unique. This proof based on a quasilinearization technique and provides a constructive algorithm to approach the unique solution. Finally, an example is presented to show the explicit calculation of some iterates with the proposed approach.
reviewed by: Manuel Calvo
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 5726779
zbMATH Keywords: boundary value problems
zbMATH Keywords: Duffing equation with an integral term of non-viscous damping
zbMATH Keywords: iterative solution
zbMATH Keywords: numerical examples
zbMATH Keywords: initial and terminal value problem
zbMATH Keywords: quadratic convergence
zbMATH Keywords: algorithm
MaRDI profile type: Publication
full work available at URL: j.amc.2010.03.046
OpenAlex ID: W2006468896
Wikidata QID: Q60199381
cites work: Closed-Form Solutions and the Eigenvalue Problem for Vibration of Discrete Viscoelastic Systems
cites work: Variational Principles in Irreversible Thermodynamics with Application to Viscoelasticity
cites work: Dynamics of Viscoelastic Structures—A Time-Domain, Finite Element Formulation
cites work: An investigation into the linear theory of ship response to waves
cites work: Fractional calculus - A different approach to the analysis of viscoelastically damped structures
cites work: ON A GENERAL MODEL FOR DAMPING
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: Q3996709
cites work: Q3736576
cites work: Q3851672
cites work: Q5514046
cites work: Quasilinearization and invariant imbedding. With applications to chemical engineering and adaptive control
cites work: Q4397695
cites work: A quadratic monotone iteration scheme for two-point boundary value problems for ordinary differential equations
cites work: Quasilinearization and rate of convergence for higher-order nonlinear periodic boundary-value problems
cites work: The Bellmann-Kalaba-Lakshmikantham quasilinearization method for Neumann problems
cites work: A quasilinearization method for a class of integro-differential equations with mixed nonlinearities
cites work: A quasilinearization method for a class of second order singular nonlinear differential equations with nonlinear boundary conditions
cites work: A quasilinearization method for elliptic problems with a nonlinear boundary condition
cites work: Approximation of solutions of the nonlinear Duffing equation involving both integral and non-integral forcing terms with separated boundary conditions
cites work: The generalized quasilinearization method for second-order three-point boundary value problems
cites work: Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions
cites work: Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs
cites work: Q5313551
cites work: Existence, uniqueness, and quasilinearization results for nonlinear differential equations arising in viscoelastic fluid flow
cites work: Q4147003
cites work: Q3807603
cites work: Q4872463
DOI: 10.1016/J.AMC.2010.03.046
description: scientific article; zbMATH DE number 5726779