# Tagged Questions

Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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### Euler-Lagrange equation [duplicate]

This is PDE Evans, 2nd edition: Chapter 8, Exercise 2: Find $L=L(p,z,x)$ so that the PDE $$-\Delta u + D\phi \cdot Du = f \quad \text{in }U$$ is the Euler-Lagrange equation corresponding to the ...
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### PDE - Feynman-Kac vs. finite difference methods

I've heard that in greater than three dimensions, it's more efficient to solve a second-order parabolic PDE using a Monte-Carlo method based on the Feynman-Kac formula that it is to use finite ...
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### Entropy Solution of the Burger's Equation

I am working on the following problem, which gives the Burgers' equation $u_t + uu_x=0$ with the initial data $g(x)=1, x < 0$, $g(x)=2, 0 < x < 1$, $g(x)=0, x > 1$. It then asks to ...
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### Finding the uniquely determined region of a PDE

(a) Solve the equation $yu_x+xu_y=0$ with the condition $u(0,y) = e^{-y^2}$. (b) In which region of the xy plane is the solution uniquely determined? I did the first part but I don't understand ...
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### Harmonic Function with linear growth

We want to find harmonic functions $w$ in (say) $\mathbb{R}^2$ that are zero on $\{y=0\}$ with the linear growth bound $$\sup_{\mathbb{B}_R} |w| \leq C(1+R)$$ where $C>0$...
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### Differential operators confussion

I want to solve this PDE: $$u_t-6uu_x+u_{xxx} = 0\,(1)$$ with the Inverse Scattering Method. This method is based on showing that the above equation can be expressed as $$L_t=LB-BL,\,(2)$$ where $L$ ...
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I am studying an article of Berestychi-Caffarelli-Niremberg - Monotonicity for elliptic equations in unbounded Lipschitz domains, and I don't understand a convergence in the demonstration of the lemma ...
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### Solve the partial differential equation $u_t + uu_x=0$ [duplicate]

Solve the following partial differential equation $u_t + uu_x=0$ with $u=u(x,t)$ and $u(x,0)=x$. I am having trouble in applying the SIDE CONDITION. The Characteristics are $dx/dt$=$u$, here u is ...
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### Does the following have a solution for f(x,y)?

I have the following equations: {1\over f(x,y)} {\partial f(x,y) \over \partial x} \alpha(x,y) + {1\over f(x,y)} {\partial f(x,y) \over \partial y} \beta(x,y) = \gamma(x,y) \end{...
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### Sequence of functions: Convergence

For each of the following, give an example of a sequence of functions $f_n$ that converges to f A. uniformly but not in the mean square sense. B. in the mean square sense but pointwise nowhere. I ...