# Tagged Questions

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

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### Solving system of PDEs

I am stuck on the following problem: Solve for $f(x,y)$, where: $\frac{\partial f}{\partial y} = y$, $\frac{\partial f}{\partial x} = \frac{1}{2}xy$ My original strategy was to integrate the first ...
1answer
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### Nodal lines of wave equation on rectangular membrane [closed]

I'd appreciate if someone could please let me know how to draw nodal lines. I don't really know how to do it and couldn't find much information with Google. I could find some info on nodal curves ...
1answer
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### Gradient of the solution for Poisson equation

Let $f(x)=(\nabla N *g)(x)$, where $N(x)=\frac{1}{|x|^{n-2}}$ for $n\geq 3$ is the Newtonian kernel, and $g\in L^1(\mathbb{R}^n)\cap L^{\infty}(\mathbb{R}^n)$. Then we can have that $f\in L^{\infty}$ ...
0answers
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### A $C^1$ function in Orlicz Sobolev space

How to prove that thi functional is $C^1$: $$I(u)=\int_{\mathbb{R}^N} \Phi(|\nabla u|)+\Phi(|u|) dx-\int_{\mathbb{R}^N} F(u) dx$$ Where $\Phi$ is an N-function and $F(t)=\int_{0}^t f(s) ds$ where ...
0answers
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### Complex version of Lax-Milgram Theorem

I'm trying to prove Lax-Milgram Theorem in the complex case, i.e. Let $X$ be complex Hilbert space and let $f\in X'$, its topological dual. If $a(\cdot,\cdot):X\times X\to \mathbb{C}$ is ...
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### Expressing spherical harmonics as a combination of other spherical harmonics

Spherical harmonics are a useful tool in physics, particularly in classic electrostatics and electrodynamics. Given an integer $l$, the spherical harmonic $Y_{l,m}$, where $-l\leq m\leq l$, solves the ...
0answers
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### Wave equation in a cube

Is it possible find a computable solution to the following homogeneous wave equation problem: Let $\mathcal{C}=\{(x,y,z)\in \mathbb{R}^3, 0< x,y,z < 1\}$ be the open unit cube. Find $u$ such ...
1answer
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### “Hessian” differential equation

In my homework, I'm given the following problem: Let $f: \mathbb{R}^n \to \mathbb{R}$ be a twice differentiable function. For an $\alpha \geq 2$, let: $$f(\lambda x) = \lambda^a f(x)$$ for ...