Tagged Questions

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

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Find wave equation initial condition such that solution consists of right-going waves only

Let $u(x,t)$ solve the wave equation $u_{tt}=c^2u_{xx}$ and let $u(x,0)=A(x)$ for some function $A(x)$. Find the function $B(x)=u_t(x,0)$ such that the solution consists of right going waves only. My ...
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An imbedding inequality in PDE.

u is a function of 3-dimension, I'm trying to prove this: $\|u\|_4^4 \leq C \|u\|^2_{H^1} \|u\|_{L^2}^2$ Anyone can shed light on this? Thanks very much.
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Inhomogeneous heat equation with source term orthogonality

This is a question on the lecture notes. Basically we have the usual heat equation: $$\frac{\partial y}{\partial t}(x,t)=k^2\frac{\partial^2 y}{\partial^2 x}(x,t)+F(x,t)$$ We also have the usual ...
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Numerical scheme for system of PDEs

I'm trying to solve the following coupled PDE system for my master thesis: \begin{align} \kappa_0\frac{\partial p}{\partial t}&=- \nabla \cdot v \\ \rho_0\frac{\partial v }{\partial t} &= (...
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Important ODE Solutions for Solving PDEs

Which ODEs pop up most often in the study of Partial Differential Equations such as the Heat Eq, Laplace Eq, Wave Eq, etc. At least in the homogeneous case. What are their solutions? I'm going to take ...
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Sturm-louville heat problem

Could I request for an example to the above question? I've read thru the regular sturm-liouville theory but have no idea how should the theory be applied to this problem. I understand that the method ...
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Partial Differential Equation with no solution - Transversality condition?

I have the following equation: $$x u_x + y u_y = \frac{2e^u }{xy } , x>0,y>0$$ with the initial condition (corresponding to $t=0$ ): $$\Gamma =\{ (s,s,0) | 0<s<\infty \}$$ By using ...
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Solve laplace equation for a semi-infinite plate. Where is my mistake?

The plate is semi-infinite. 2 Of its sides have $f=0$ and the bottom part satisfies $f=cos(x)$. Its width is $\pi$. The temperature distribution $f(x,y)$ satisfies the Laplace equation $\nabla^2 f=0$. ...
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Partial Differential Equation - The Chain Rule

$\displaystyle \sum_{i,j=1}^{n}\int_{U}a^{ij}u_{x_{i}}\zeta^{2}u_{x_{j}}dx$ $\displaystyle =\sum_{i,j=1}^{n}\int_{U}a^{ij}D_{i}u\zeta^{2}D_{j}u dx$ Can someone please explain to me how we use the ...
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HELP to solve - PDE First order

I have this equation $$u_x+uu_y=0$$ by the book "Handbook of First order Partial Differential Equations - page 290." The general solution is $F(ux-y,u)$, where $F$ is a arbitrary function. I try ...