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Answer by Denis Serre for Heat equation close to the steady state
Let $v:=\partial_xu$. It still satisfies the heat equation $\partial_tv=\partial_x^2v$, but with the Neumann boundary condition:$$\partial_xv(t,0)=\partial^2u(t,0)=\partial_tu(t,0)=\partial_t0=0,$$and...
View ArticleAnswer by Infojoe for Heat equation close to the steady state
Fortunately your boundary conditions are very simple, such that you can express the analytical solution of your problem explicitly. In fact, by using the ansatz of separation of variables one derives...
View ArticleHeat equation close to the steady state
Let $u(t, x)$ be the unique solution of the heat equation on the unit interval with Dirichlet boundary conditions and initial data $u_0$:$$\left\{ \begin{array}{l}\partial_t u(t, x) = \partial_x^2 u(t,...
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