Approximation of solutions for an initial and terminal value problem for the forced Duffing equation with non-viscous damping
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.
Duffing equation with an integral term of non-viscous damping
scientific article; zbMATH DE number 5726779
scientific article