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

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Semilinear Poisson Equation Using Direct Method of Calculus of Variations

The following problem comes from: http://people.physics.anu.edu.au/~gvn105/analyticMethPDE.pdf 12.9 Exercises 12.3: Let $\Omega$ be a bounded domain in the plane with smooth boundary. Let $f$ be ...
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1answer
26 views

PDE $ u_{x}+u_{t}+f(x)*u=0$

How would I solve this pde using characteristic line? $u_{x}+u_{t}+f(x)u=0$---arbitrary function f $u(x,0)=u_{0}(x)$---$u_{0}$ can be any value $u(0,t)=\varphi(t)$---non-homogeneous where $u(x,t)\ge ...
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32 views

Fourier transform for pde

I solved the following PDE: $u_{t}-(u_{t})_{x}+(b+1)uu_{x}=bu_{x}u_{xx}+uu_{xxx}$ numerically, using Fourier Transform method. For this i wrote it in the following way: ...
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27 views

Establishing a certain bound

If I know the bound of the certain derivatives: $$|\dfrac{\partial^k}{\partial x^{k_1}_1\cdots \partial x^{k_n}_n}V(x)|\leq M\epsilon^{-k} \exp(-m \epsilon^{-1}g(x)),$$ where $k=k_1+\cdots+k_n$ and ...
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40 views

Laplacian eigenvalue problem

I'm working through a PDE problem and we are given the eigenvalue problem $-\Delta u = \lambda u$ with $\frac{\partial u}{\partial n} = 0$ along the boundary given by the rectangle $\Omega = (0, \pi)$ ...
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1answer
79 views

Solution to Anisotropic Heat Equation

I am trying to find the solution to a 1-D anisotropic heat equation. The domain is a line segment of length L (i.e., it's a line segment extending from $x = 0$ to $x = L$). The form of the equation ...
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51 views

Solve Partial differential equation(geometric optics)

Solve $x^2((u_x)^2+(u_y)^2)=1$ , $u(x,0)=0$ Use the characteristic equation The solution is $u(x,y)=-\ln\dfrac{\sqrt{x^2+y^2}+y}{x}$ I drove $\dfrac{dx}{dt}=2x^2p$ $\dfrac{dy}{dt}=2x^2q$ $Z=2t$ ...
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25 views

Help with semilinear PDE problem

I need some help: Let $T>1$, $\Omega_T=\{(x,y)\in\mathbb{R}\>|\>x\in(1,T)\}$, $\Gamma=\{(x,y)\in\mathbb{R}\>|\>x=1\}$ and $g\in C^1(\mathbb{R})$. Prove that there exists $T>1$ such ...
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58 views

Fractional Sobolev spaces and weighted L2 spaces

For $s\in[0,1]$ define function spaces $H^s(\mathbb{R})=\{u\in L_2(\mathbb{R}): (1+|\cdot|^2)^{s/2}\mathcal{F}u\in L_2(\mathbb{R}) \}$ (where $\mathcal{F}$ denotes the Fourier transform) i.e. the ...
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59 views

Solving the Laplace equation in terms of exponential of hyperbolic trigonometric functions

I'm solving the Laplace equation $U_{xx}+U_{yy}=0$ subject to BC's: \begin{align} U(0,y) &= 0 \\ U(a,y) &= 0 \\ U(x,0) &= 0 \\ U(x,b) & = \left\{x \text{ for } x \in ...
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1answer
43 views

Wave Equation Descending from 2D to 1D

I am stuck deriving the familiar d'Alembert formula using the Method of Descent, going from 2D to 1D. After using Kirchhoff's Formula, writing the solution independently of $y$ and some manipulations, ...
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22 views

Conjugating an operator with a gauge transformation; how is the kernel affected.

For the differential operator $$ D := i I \frac{d}{dx} + A(x) \colon C^\infty_T([0,\beta]),\mathbb{C}^m) \to C^\infty_T ([0,\beta],\mathbb{C}^m) $$ where $A(x)$ is Hermitian and $C^\infty_T ...
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1answer
28 views

Extension of a function from the edge.

How can I extend the function $f\in W^{1-\frac{1}{p},p}(\mathbb R\times\lbrace0\rbrace)$ up to $g\in W^{1,p}(\mathbb R\times(0,\infty))$?
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57 views

Don't understand a PDE argument involving $L^p$ norms and inequalities

I'm reading this paper. I do not understand how the author proves Corollary 5.12 for the case $p=2$. He addresses everything except $p=2$ but claims it holds for all $p$. Can someone help me to see ...
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22 views

Proving an elementary inequality of real vectors related to the p-Laplacian [duplicate]

How would you prove the following inequality? $$ \left|\left|a\right|^{p-2}a-\left|b\right|^{p-2}b\right|\leq C_p\left|a-b\right|^{p-1} $$ where $a,b\in\mathbb{R}^{n}$ and $C_p>0$ is some constant ...
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1answer
48 views

Solve PDE with the Fourier transform

I have a problem with solving PDEs with the fourier trasform method when the function not depends only on x and t but also on the y variable. In particular, when I have to solve this equation ...
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28 views

Reference needed for the following sobolev inequalties

I'm reading a paper and the authors applied the following sobolev type estimates $$ ||(Dv)^{2}||_{H^{3k-2}(\Omega)}\leq C||v||_{H^{3k-1+\alpha}(\Omega)}^{2} $$ for $\alpha>\frac{1}{4}$, where $v$ ...
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29 views

Finding a minimization problem corresponding to a PDE

I was trying to find an equivalent minimization problem to the following PDE in $\Omega \subset \mathbb{R}^2$ $$ \Delta^2 u-\nabla \cdot (k(x,y) \nabla u)+\lambda u = f(x,y) $$ where $\lambda >0 $ ...
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25 views

Easy ode question.

I forget ODE. Please solve this. Thanks. $M=M(u,v))$ $M_{uu}+M=0$ I guess I need to take $D=\frac{d M}{d u}$ Then I need to write $D^2+1=0$ But I cannot remember properly.
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33 views

The infinitessimal generator of Brownian Motion

Background: I know, and can almost prove, that the infinitessimal generator of BM is the Laplacian/2. For me (and I have never heard it done differently) we have Feller Processes, of which BM is one ...
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25 views

$-\triangle \phi + u \cdot \nabla \phi = e^{\phi}$ and $\phi \in C(\overline{\Omega})$ implies $\phi \in C^2(\Omega)$

Suppose $U \subseteq \mathbb{R}^n$ is open, bounded, connected, $u$ is Lipschitz (or $C^1$ if it helps), and $$-\triangle \phi + u \cdot \nabla\phi = e^{\phi}.$$ If I know $\phi \in ...
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is there a way can dig out the criterion for initial viscosity and limit viscosity in navier-stokes equation?

here i give a method i find which is useful for the relation between initial viscosity and limit viscosity in n-s equation! The steady isentropic compressible navier-stokes equation in $R^3$ is: ...
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1answer
23 views

Find adjoint operator $L^*$ for the 3rd order operator $Lu(x,t)=u_t(x,t)-u_x(x,t)+\gamma^2u(x,t)$, with $wLu-uL^*w$ (1) is divergence expression.

I'm working out of the Zauderer PDEs book and am having some trouble on the adjoint operators section (3.6). Specifically this problem: "Obtain the adjoint operator $L^*$ for the third order operator ...
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24 views

Energy of heat equation goes to 0

Suppose $u_t=u_{xx}$ on $(0,1)\times(0,\infty)$ and $\int_0^1u(x,0)dx=0$ with Neumann boundary condition $u_x(0,t)=u_x(1,t)=0$. Show that $\int_0^1u^2(x,t)dx\to0$ as $t\to\infty$. The $t$-derivative ...
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1answer
149 views

Solving Wave Equations with different Boundary Conditions

Right now I'm studying the wave equation and how to solve it with different boundary conditions (i.e. $u(x,0);u(0,t);u_t(x,0);u_x(x,0);u(x,x);u_t(x,x)...$) I know how to solve it when the boundary ...
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29 views

characteristic curves tangent and gradient

In the PDE $aU_x+bU_y = 0$.This is equivalent to $\frac a{\sqrt {a^2+b^2}}U_x+ \frac b{\sqrt {a^2+b^2}}U_y = 0$. $\langle\frac a{\sqrt {a^2+b^2}}, \frac b{\sqrt {a^2+b^2}}\rangle$.$\nabla U=0$. Then ...
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Derivation of the advection equation

Is there a good derivation of the advection equation available online? By that I mean the equation $\partial_t u = -\nabla( \vec{v} u)$ I know a good explanation for the one-dimensional case ...
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53 views

variational derivative

Let $\Omega \subset \mathbb{R}^n,\ n=1,2 \mbox{ or } 3$. Define the following energy $$E=\int_{\Omega} \frac{1}{\varepsilon}\left[f(u)+\frac{\varepsilon^2}{2}|\gamma(n)\nabla u|^2\right]\,dx$$ ...
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1answer
48 views

Where to specify boundary conditions

The problem asks me where I need to specify boundary values for the linear PDE problem: $u_t + xu_x + yu_y = 0$ on the domain $\Omega = x^2 + y^2 \le 1$. Using characteristics I get that $u(x,y,t) = ...
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25 views

Self-Adjointness of beam stiffness operator

I think it is well known that the operator $\frac{EI}{\rho} \frac{\partial^4}{\partial x^4}$ which arises from a standard Euler-Bernoulli beam is self-adjoint in $H$, where $H = L^{2}$, given ...
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29 views

A useful criterion in dispersive PDE.

I would like to prove the following theorem: Consider two Banach spaces $X\hookrightarrow Y$ and $1<p,q\leq\infty$. Let $(f_n)_{n\geq 0}$ be a bounded sequence in $L^q(I,Y)$ and let $f:I\mapsto Y$ ...
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12 views

Prove that the Laplacian of the integral of a certain function is $0$

Let $f(x)$ be a continuous function. Define $$g(x,y)=\int_a^b\frac{yf(t)}{(x-t)^2+y^2}dt$$ Show that $\nabla^2g=0$
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1answer
45 views

Laplace's equation, integral, tends to steady state?

If $v(x,y)$ solves Laplace's equation $v_{xx} + v_{yy} = 0$ on a bounded domain $S$, and $u(x,y,t)$ solves $u_t = u_{xx} + u_{yy}$ on $S$, with $u=v$ on $\partial S$ for all $t$, one can show that ...
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36 views

Solving a parabolic PDE with boundary conditions given over ranges

How can one solve a Parabolic PDE (like the wave or diffusions equations) if the boundary conditions were given over ranges? Here is an example: How to solve the equation ...
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39 views

How the change of variables of a PDE affects the given condition(s)?

For example I have a PDE with the dependent variable $u$ and the independent variables $x$ and $y$ . Suppose I have the change of variables that $v=u_x$ , I know the condition of $u(x,a)=f(x)$ will ...
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25 views

Hitting time for a planar diffusion

Let $A$ be an open subset of $\Bbb R^2$, and let us consider a diffusion $\mathrm dX_t = f(X_t)\mathrm dt + g(X_t)\mathrm dW_t$ where $f$ and $g$ are globally Lipschitz continuous maps. Suppose I am ...
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1answer
43 views

What is the usual norm of $\mathbb{H_0^1}$?

What is the usual norm of product of sobolev spaces $\mathbb{H_0^1}=H_0^1 \times H_0^1=W^{1,2}\times W^{1,2}$? In my work i need to prove that the norm endowed by the inner product ...
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What does the solution guess for a PDE mean? - specific question

I got a partial differential equation, where a suggested solution is: $J(W,t)=\dfrac{g(t)^{\gamma}W^{1-\gamma}}{1-\gamma}$ Everything but $g(t)^{\gamma}$ is known. $g(t)^{\gamma}$ is being ...
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52 views

Solving method for basic PDEs

How is it called the method used in the second and in the fourth of the following steps? I don't understand it that well. Usually, everything you do on a term of the equation you must do it on the ...
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49 views

implications of convergence in sobolev spaces

If we are given that $O \subset \Omega$ is open and bounded and $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$. We have a sequence $\{u_{m}\}$ satisfying $$ u_{m} ...
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1answer
29 views

Differentiate Fourier cosine series

Suppose $f(x)$ and $\frac{df}{dx}$ are piecewise smooth. Prove that the Fourier cosine series of the continuous function $f(x)$ can be differentiated term by term. Can anyone help me with this ...
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1answer
38 views

Is there any Banach space $X$ that $L^2(\Omega)$ is compactly embedded into?

Let $\Omega \subset \mathbb{R}^n$. Is there a good (*) Banach space $X$ that $L^2(\Omega)$ is compactly embedded into: $$L^2(\Omega) \subset\!\subset X$$? If not compactly embedding, I at least would ...
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39 views

How to solve these two second-order coupled PDE?

I have two second-order equations governing the behaviour of two spatial function which are coupled: $$ 0 = A f(x,y) + B \frac{\partial^2 f(x,y)}{\partial x^2} + C \frac{\partial^2 f(x,y)}{\partial ...
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26 views

Could someone walk me through this pde?

$$\frac{\partial^2 y} {\partial x^2} = \lim_{\Delta x \to 0} \frac {1} {\Delta x} \left(\frac{\partial y} {\partial x}_x - \frac{\partial y} {\partial x}_{x+ \Delta ...
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61 views

Solve a nonhomogeneous wave equation PDE

$u_{tt} = c^2u_{xx}+sin(\alpha t)$ $u(0,t)=0=u(\pi,t)$ $u(x,0) = 0 = u_t(x,0)$ where $0<x< \pi $ and $t>0$ I know how to solve this problem using Fourier series, but I also encountered ...
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21 views

specific non linear pde

I would really appreciate to hear your insights or comments about the following problem: Consider the following non linear pde: let $\Omega$ be the unit square with vertices at (0,0),(1,0),(0,1) and ...
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66 views

Two PDE Questions

1. Solve the equation $$ u_x^3-u_y=0~, $$ with $u(x,0)=2x^\frac{3}{2}$ 2. Solve the equation $$ u=xu_x+yu_y+\frac{1}{2}(u_x^2+u_y^2)~, $$ with $u(x,0)=\dfrac{1}{2}(1-x^2)$ How I solve these ...
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33 views

PDEs along parametrized curves in $\mathbb R^2$

For Riemannian surfaces in $\mathbb R^n,n\geq 3$, the Laplace-Beltrami operator can be expressed in terms of the Riemannian metric and one can consider evolution equations (e.g. heat diffusion, wave ...
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2answers
50 views

Solving a Second Order PDE

I'm trying to solve the equation $u_t = \alpha^2 U_{yy}$ given u(y,t) bounded y $\rightarrow\infty$ and u(0,t) = $U_o e^{iw_ot}$. Initial is u(y,0) = 0. I have gotten both separations as Y'' - ...
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15 views

Hyperbolic Partial differential equation

I have the following problem: $$ u_{t}(x,t)+a(x,t)u_{x}(x,t)-f(x,t)=0\\ \ u(x,0)=0, \\$$ How to solve this equation? Thank you.