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A linear recursive scheme associated with the love equation
Le Thi Phuong Ngoc
,
Nguyen Tuan Duy
,
Nguyen Thanh Long
Abstract
This paper shows the existence of a unique weak solution of the following Dirichlet problem for a nonlinear Love equation $$\left\{ \begin{aligned} &u_{tt}-u_{xx}-\varepsilon u_{xxtt}=f(x,t,u,u_{x},u_{t},u_{xt}), \quad 0 < x < L,~ 0 < t < T, \\ &u(0,t)=u(L,t)=0, \\ &u(x,0)=\tilde{u}_{0}(x),\qquad u_{t}(x,0)= \tilde{u}_{1}(x), \end{aligned} \right.$$ where $\varepsilon > 0$ is a constant and $\tilde{u}_{0}, \tilde{u}_{1}, f$ are given functions. This is done by combining the linearization method for a nonlinear term, the Faedo–Galerkin method and the weak compactness method.