# Tagged Questions

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

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### Why is there no classification scheme for linear pde of third order or above

I am well aware of the classification of 2nd Order PDE into Elliptic, Parabolic or Hyperbolic type depending on the analogy with conic sections from analytic geometry. I have seen third order PDE ...
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### PDE: Classical Solution?

Part (a) is a standard problem which can be solved either using separation of variables or Fourier transform.I haven't seen a proper definition for 'classical solution' to answer (b). Any help is much ...
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### $-\Delta u = u^p$ in bounded domain

In my PDE lecture we had the following theorem and I am wondering how strong it is: Theorem Let $\Omega \subset \mathbb{R}^n$ be a bounded domain (with $\partial \Omega$ sufficiently smooth, lets ...
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### Fixed Point method and existence of entropy solution of a Nonlinear Parabolic PDE [on hold]

Could someone point me to literature that addresses this kind of problem? Thanks
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### Heat Equation : Commutation of partial derivatives and summation

I'am having a problem when checking the validity of the solution i found for the heat equation: \begin{cases} U_{t}(x,t)=U_{xx}(x,t),\ {(x,t)\in (0,1)\times(0,+\infty)} \\ U(x,0) = x^2 - x\\U(0,t)=0\...
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### Solving non-homogenous PDE with forcing function (which diappears!) dependent only on time

Applying the method of eigenfunction expansion to the PDE $$u_t -c^2u_{xx}=F(t)$$ $$0<x<L, t>0$$ $$u(x,0)=f(x)$$ $$u_x(0,t)u_x(L,t)=0$$ for the homogenous part of this equation ($L[v(x,t)]=0$...
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### Resonance in wave equation

I have solved the non-homogenous equation by the method of eigenfunction expansion $$u_{tt} - c^2 u_{xx}=F(x)\sin(\omega t)$$ $$0<x<L, t>0$$ $$u(x,0)=u_t(x,0)=0$$ $$u(0,t)=u(L,t)=0$$ and got ...
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### Singularities in an Equation

these might sound like extremely trivial questions but since my background is more in probability and statistics I'm not too sure what to do, or even what to read up on to understand what to do. I ...
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### Groundwater related features and terminology. [on hold]

Applying mass conservation to an element in a homogeneous, isotropic unconfined aquifer, show that the groundwater flow in the aquifer is given by Eq. 2. Assume a constant value of h (=ℎ0) for $𝑡≤0$. ...
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### PDE argument about passage to limit in Bochner space and weak derivative

I have a function $u \in L^2(0,T;H^1_0)$ with $u' \in L^2(0,T;H^{-1})$, all on some smooth bounded domain $\Omega$. In addition I have a sequence $u_n$ such that $u_n \to u$ in $L^2(0,T;H^1_0)$ and ...
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### Intuition for Fredholm operators?

Alot of the material I'm reading lately seems to mention Fredholm operators and the 'Fredholm alternative' and operators being 'Fredholm of index $0$'. Can someone give me a high level overview of ...
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### finding solution to $u_x + 2u_y + (2x − y)u = 2x^2 + 3xy − 2y^2$

I tried to solve this equation with the coordinate method but I got a bit different answer compared to the suggested one by the solution manual. Where I am making the mistake in my solution? My ...
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### finding solution to $au_x+bu_y=f(x,y)$ where $a \neq 0$

I can't get the solution in the required form of $$u(x,y)=(a^2+b^2)^{-\frac{1}{2}}\int_{L}fds +g(bx-ay)$$ "where g is an arbitrary function of one variable, L is the characteristic line segment from ...
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### What does 'dissipative PDE' means?

Can you give me an idea what is meant with dissipative partial differential equations? I am no phycist (and do not know the difference between initial energy to final energy), but wikipedia told me ...
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### Why this abuse of notation correctly solves the heat equation

Here's a stupid method I observed to solve the heat equation in $\mathbb R^d$, \begin{align*} \partial_tu=\Delta u,\quad u|_{t=0}=u_0. \end{align*} Pretend that $\Delta$ is a constant so this just ...
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### equation $u_x+u=e^{2y+x}$ (part of the solution to $u_x+u_y+u=e^{x+2y}$)

I solved/analyzed the below PDE $$\left\{\begin{matrix} u_x+u_y+u=e^{x+2y}\\ u(x,0)=0 \end{matrix}\right.$$ and have a question to the one of the steps involving the integration, see below ...
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### The eigenfunction of modified 1-laplace equation

Let $\Omega\subset \mathbb R^2$ be a bounded, smooth boundary domain. I am interested in the following operator $$-\operatorname{div}\left(\frac{\nabla u}{\sqrt{|\nabla u|^2+\epsilon}}\right)=0.$$ ...
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### The eigenvalue of Laplace problem on a domain with a line removed

Let $\Omega\subset \mathbb R^2$ be given such that $\Omega$ is open bounded with nice boundary. Let $\Gamma$ be a finite segment in $\Omega$. We also define $\Omega_1:=\Omega\setminus \Gamma$. ...
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### Modified Wave Equation: Bound $\int u^2 \, dx$

I'm studying for a qualifying exam and I can't figure this problem out: Suppose $B \subset \mathbb R^n$ is the unit ball centered at the origin and that $u$ is a smooth solution of \begin{align*} ...
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### Spectrum of the Laplacian for free space versus spectrum of Laplacian with boundary conditions?

In this question the spectrum of the Laplacian in free space is defined as $]-\infty, 0]$. So it seems the spectrum can be determined without regard to boundary conditions? If instead we consider the ...
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### Partial differential equation solvable as ordinary differential equation [closed]

Solve the following partial differential equation for $u=u(x, y)$: $$U_{yy}+6U_y+13U= 4e^{3y}.$$
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### Laplace Equation on Prolate Spheroidal Coordinates

I'm currently trying to solve Laplace's equation outside of a prolate spheroid. The scalar field has to vanish far from the spheroid. Because of the geometry I thought it might be convenient to use ...
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### Weak problem formulation for PDE and boundary conditions

Consider the following example: $$- \Delta u = f \mbox{ in } \Omega,$$ $$u = 0 \mbox{ on } \Gamma,$$ Here $\Gamma$ is boundary of $\Omega$. To produce weak formulation we multiply by arbitrary $v$ ...
I am looking for a necessary and sufficient condition on a function $\gamma: \mathbb{R} \to \mathbb{R}$ such that there is a solution $u \in C(\mathbb{R}^2)$ for the wave equation $$\partial^2_t u(t, ... 0answers 51 views ### How to get the idea of the formula for the mean value property for the heat equation From the mean-value property of the Laplace's equation, we have the following mean-value property:$$ u(x)=\frac{1}{a(n)r^n}\int_{B(x,r)}udy.  But for the mean-value property of the Heat equation, ...
Just as a bit of background, I'm working with the Black-Scholes PDE and I'm testing some things out by taking an initial condition for it as $\sin(S/50)$, where $S$ is the spot price (but that's ...