Tagged Questions

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Examples of quasilinear wave equations

Consider a quasilinear wave equation equation of the form $\sum g^{ij}(u, Du)\partial_i\partial_j u = F(u, Du)$ on $R \times R^n$ subject to initial data $u(0,x)=g, \; \partial_t u(0,x)=h.$ Given ...
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47 views

Solution of a particular PDE in 4 variables with non-constant coefficients

I have come across the following equation while reading about the Unruh Effect in Black Hole Physics. . K is a function of $x,y,\rho,t$ i.e $K=K(x,y,\tau, \rho)$. $\omega, k,m$ are constants. ...
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One partial differential equation

Where can I find information about equation $$\frac{\partial u(x,t)}{\partial t}-\operatorname{div}\left(A(x)\nabla u(x,t)\right)=f(x,t),\text{where } A(x) \text{ is a matrix 2x2} ?$$ I would be ...
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When does a PDE solve a variational problem?

I understand that for a functional $J[f]$ on the space of differentiable functions $f$ on some domain, setting $\delta J[f]|_{f=f_0} = 0$ yields a (possibly nonlinear) partial differential equation in ...
270 views

1D Green's function: from interval to infinite line

Let's consider two problems for diffusion equation. The first one: $$u_t = a^2u_{xx},\qquad 0<x<l,\quad 0<t\leq T$$ $$u(x,0) = \phi(x), \qquad 0 \leq x \leq l$$ ...
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Why is it true that there are no resonances for Schrodinger operator when dimension is $\geq 5$

For a Schrodinger operator $H=-\Delta+V$, we say that the zero is a resonance of $H$ if the quation $Hu=0$ has a solution $u\notin L^2(\mathbb{R}^n)$ such that ...
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Green's function. Basic

Can anyone give some advice about books where I could find introductory information about Green's function. What are the methods of constructing Green's function. Actually, Green's function for 3D ...
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