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

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some auxiliary results

I want to prove the following If $N\ge 3$ there exists a constant $c_0=c_0(\Omega)$ such that for all $\alpha\ge 1$ and $z\in H^1(\Omega)$ \begin{align} ...
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1answer
10 views

General solution to a PDE

Consider the equation $u_{xx}+2u_{xy}+u_{yy}=0.$ Write the equation in the coordinates $s=x$ and $t=x-y$ and find the general solution of the equation. We have that $x=s$ and $y=s-t,$ thus ...
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0answers
8 views

if and only if condition for the Weak solution of Burger's equation

Given the Burgers equation $$u_{y}+uu_x=0$$ with $y>0$. Suppose $u$ is continuous for all $y>0$ and $u_x$ has a jump discontinuity on the smooth curve $x=\xi(y)$ while $u$ is $C^1$ on either ...
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13 views

Fokker Planck PDE with a free boundary

Let's have this linear PDE: \begin{equation} \frac{\partial p(x,t)}{\partial t} = - a \frac{\partial p(x,t)}{\partial x} + D \frac{\partial^2 p(x,t)}{\partial x^2} \end{equation} a and D are constant ...
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5 views

Continuity of a function, after extending by oddness.

I'm blanking on how to prove a claim my professor stated in class, even though it should be really simple. We're working on deriving the homogeneous wave equation with clamped boundary conditions. So ...
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1answer
43 views

Techniques to solve such a PDE

I have the eigenvalues problem on $[0,\pi] \times [0,2\pi]$ $$\left(\frac{1}{\sin\theta}\frac{\partial}{\partial \theta} \left[\sin\theta \frac{\partial}{\partial \theta}\right] + ...
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2answers
33 views

How to solve this coupled linear differential equations?

$\partial_t f(x,t)= \alpha \partial_x^2f+\beta f + \gamma g \\ \partial_t g(x,t)= \alpha \partial_x^2g -\beta f - \gamma g$ With everything real. I tried to take the first equation and express ...
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1answer
26 views

weak solution of poisson equation

Consider the equation , with $u\in H^1$ $$\begin{cases}\Delta u = f & \text{in }\Omega\\ \displaystyle \frac{\partial u}{\partial \nu} = 0 & \text{in } \partial \Omega\end{cases}$$ where $\nu$ ...
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1answer
33 views

The general solution of PDE $u_{xx} +u_{yy}=0$

The general solution of PDE $u_{xx} +u_{yy}=0$. There are four options given (correct option is given as d): a) $ u=f(x+iy)-g(x-iy)$ b) $ u=f(x-iy)-g(x-iy)$ c) $ u=f(x-iy)+g(x+iy)$ d) $ ...
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1answer
13 views

Solve the PDE $xu_x-2yu_y+u=e^x,$ with the side condition $u(1,y)=y^2$

1b. $xu_x-2yu_y+u=e^x,$ side condition $u(1,y)=y^2$ My attempt: This has been a super endurance and I hope I got the whole thing right. So anyway, here it goes ...oh and one more thing... can someone ...
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50 views

a simple calculation

Can anyone see how (59) lead to (60)? \begin{align} ...
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1answer
18 views

Heat equation, initial-boundary value problem

Let $u (x, t)$ be a solution of the initial -boundary value problem $$\left\{\begin{array}{ll} U_t - U_xx = 0 & 0 < x < L, t > 0 \\ U (0, t) = U (L, t) = 0 & t > 0 \\ U (x, 0) ...
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0answers
12 views

Maximum Likelihood Estimation for function with several variables

I have a function in four variables and I need to find out the variable values where the function will have maximum value. $f_n = (1-v1\cdot v2\cdot v3\cdot v4)\cdot (1-v1\cdot v2\cdot v4)\cdot ...
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0answers
15 views

Existenence of the solution for a PDE-ODE system.

I have the PDE-ODE system below: $\frac{\partial c}{\partial t}= D \Delta c - \eta \nabla.(c\nabla v)+g(c,v)$ $\frac{dv}{dt}=-\alpha cv+\xi(c,v)$ with initial conditions and Neumann boundary ...
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2answers
40 views

Composition operator on Sobolev space

I am trying to consider hypothesis on $g$ such that the operator $$ H_0^1 (\Omega) \to H^{-1}(\Omega), \qquad v \mapsto g(v) $$ is $\mathcal C^1$. Of course, $g(0) = 0$. I believe that $g \in ...
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1answer
18 views

PDEs with Variable Coefficents: Solve $xu_x-xyu_y-u=0$ for all $ (x,y)$

Question: $xu_x-xyu_y-u=0$ for all $ (x,y)$ My attempt: Our characteristic curve is in the form of $\frac{dy}{dx}$. Since our $dy = -xy$ and $dx = x$ we have the following separable equation . ...
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12 views

Question about characteristic method

I am self-studying the book titled Partial Differential Equations by Jeffrey Rauch. I got stuck on the following problem.Chapter 1 Problem 2. I know that the charactersitc lines are straight ...
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26 views

Solving Black scholes PDE using Laplace transform

I'm trying to obtain the Laplace transform of Call option price with repect to time to maturity under the CEV process. The well known Black scholes PDE is given by $$ ...
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0answers
24 views

How do I solve an inhomogeneous Helmholtz boundary value problem in 2-D with a rectangular boundary?

I need to solve the following BVP: $\Delta u - 1/\delta * u = R(x,y)$, where $\Delta$ is the laplacian operator. and boundary conditions: $u(0,y)=u(L,y)=u(x,0)=u(x,L)=0$ where $L=1$, ...
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19 views

Karhunen Loeve representsiton of Ising Model

I was wondering if there is a Karhunen Loeve-type of representation for an Ising random process? In truth I would be really happy to learn about work in KL series for any type of spatial random ...
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0answers
10 views

Solve a PDE: $ x(y^2+z)p-y(x^2+z)q=(x^2-y^2)z$

Solve a PDE $ x(y^2+z)p-y(x^2+z)q=(x^2-y^2)z$ where, $ p=\displaystyle \frac{\partial z}{\partial x}$ and $ q=\displaystyle \frac{\partial z}{\partial y}$ My attempt: I start with Lagrange's ...
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1answer
17 views

Time-dependent Schrödinger equation for Heaviside function as initial condition

Consider the Schrödinger equation for a free particle: $$i\partial_t\psi(x,y)=-\partial_x^2\psi(x,t)$$ with initial condition $$\psi(x,0)=\theta(x)$$ and boundary conditions ...
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0answers
23 views

Denominator of a function

I have a function $S(x,y)$ which satisfies the following PDE $$\frac{\partial S(x,y)}{\partial y}=-H\left(x,\frac{\partial S(x,y)}{\partial x}\right)$$ where the known function ...
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38 views

How to select a strongly convergent subsequence from a weak convergent sequence in $L^2$?

Let $(p_n)_{n \in \mathbb N}$ be a sequence of probability density functions, which satisfies i)$$ \partial_t p_n (t,x) = \partial_{xx} (a_n (t,x) p_n (t,x)), $$ where $a_n$ is a sequence upper ...
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1answer
27 views

Two theorems about approximation by smooth functions

Let $U$ be an open subset of $\mathbb{R}^{n}$. The following are two theorems taken from the chapter about Sobolev Spaces of the Evans' book. Theorem 1 Assume $u\in W^{k,p}(U)$ for some $1\le ...
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0answers
28 views

Reference for the following equation

Can someone suggest me references about the following equation $$u_t+A\cdot\nabla u=i\Delta u$$ with $A$ a smooth vector field.
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0answers
23 views

mean curvature of a nonlinear parabolic PDE

I want to compute the mean curvature of a reaction diffusion equations of the form $$ u_t=u_{xx}-u^2 $$ Any suggestions is appreciated!
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8 views

How to draw characteristics for the Method of characteristics for the solution of a PDE

How to draw characteristics for the Method of characteristics for the solution of a PDE? What is the easiest way to draw it say for a function $u_x sin(u)+u_y cos(u)=0$ , $u(0,y)=u_0 (y)$? Many ...
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1answer
23 views

Show that u = 0 on the surface of V

Here is the uniqueness theorem in Evans PDE : I can show u is harmonic, however I don't know how to show it = 0 on the surface and the contradiction.
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1answer
13 views

Solving the wave equation bounded by one free end and one fixed end

Given that $\{\sin\left[\frac{(2n-1)\pi}{2L}x\right] : n\in\mathbb N\}$ is the complete set of eigenfunctions of a regular Sturm-Liouville with boundary points $0$ and $L$ and weight function $1$, and ...
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1answer
37 views

Heat equation convection

I want to solve the heat equation with convection $F_t = F_{xx} - F_x$ with initial condition $F(x,0) = f(x)$ So far what I've got is that if $F(x,t) = G(x-t,t)$ and G satisfies the heat equation ...
2
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1answer
64 views

Find the particular solution of $u_x+2u_y-4u=e^{x+y}$ satisfying the following side condition $u(x,-x) = x$

2.Find the particular solution of $u_x+2u_y-4u=e^{x+y}$ satisfying the following side condition $u(x,-x) = x$ I know that under that condition $y = -x$ which is the reflection of the $x$ graph. I ...
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3answers
48 views

General solution to PDE

Consider the equation $$xu_x+(1+y)u_y=x(1+y)+xu.$$ Find the general solution. Now assume an initial condition for the form $u(x,6x-1)=\phi(x).$ Find a necessary and sufficient condition ...
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1answer
26 views

Proof for Neumann boundary PDE

Here is the problem : Here is my attempt , Please have a look and point out any error:
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1answer
18 views

Solving an elementary PDE using characteristic equations.

I am trying to solve the equation $xU_x + yU_y = 0$. The characteristic equation is $\frac{dy}{dx} = \frac{y}{x}$. Hence $\frac{1}{y}dy = \frac{1}{x}dx,$ so $\ln y = \ln x + c_0$. This implies that $y ...
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0answers
14 views

First order partial differential equations in complex domain

Try to solve a first order linear partial differential equation $P(x,\partial)u(x)=f(x)$ in complex domain, while the operator is of the following form: $$ ...
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2answers
48 views

Improve Liouville's Theorem in Evans ' PDE

Here is Liouville's Theorem Suppose that $u \colon \mathbb{R}^n \to \mathbb{R}$ is harmonic and $u \geq 0$. Prove that $u$ is constant. (In this problem , instead of $u$ is bounded now $u \geq 0$ ...
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3answers
29 views

Please explain Evans 's PDE Liouville 's Theorem

Here is the proof : d. Liouville's Theorem. We assert now that there are no nontrivial bounded harmonic functions on all of $\Bbb R^n$. THEOREM 8 (Liouville's Theorem). Suppose $u:\Bbb ...
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0answers
49 views

numerical method (Implicit) for nonlinear pde

$\newcommand{\lbar}{\underline{\lambda}}$ I need a numerical method (implicit , backward difference or forward difference) for estimate $A$ in this nonlinear PDE: $$ A_t + \mu(\lambda -\lbar ) ...
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0answers
16 views

Differential Equation for brownian bridge?

For the brownian motion, we know that probability density of the particle's position at time $ t $, $ \rho(x,t) $ satisfies the diffusion equation pde: $ \partial_t \rho = d \; \partial_x^2 \rho $. Is ...
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0answers
18 views

The use of Schauder fixed point in ladyzehskaya

The book Linear and Quasilinear equations of parabolic type gives the uniform parabolic pde theory in the literature. Ladyzhenskaya use Leray-Schauder rather than Schauder fixed point theorem. why? ...
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1answer
22 views

How to prove that the L2 norm is a non-increasing function of time for a 2nd-order PDE?

I am having a test in few days and I saw an interesting question while I was skimming through the book problems. The problem is concerned about initial-boundary value problem of 2nd order PDEs. To ...
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1answer
63 views

Find a solution $g(x)$ that satisfies the PDE $u_x +3u_y-u = 1$.

Question: What form must $g(x)$ have in order that the following problem have a solution? $u_x+3u_y-u=1,u(x,3x)=g(x)$. If $g(x)$ has the required form, will there be more than one solution? My ...
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1answer
42 views

What can you say about f if g is harmonic?

Suppose that f : R → R is such that, whenever g : $R^n$→ R is harmonic, so is f(g(x)). What can you say about f? This is my attempt , and I think f is a linear function.
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1answer
63 views

Is it true that $L^2$ is compactly embedded in $(W^{1,2}_{0})^{\ast}$?

Is it true that $L^{2}(\mathbb R^{n})$ is compactly embedded in $(W^{1,2}_{0}(\mathbb R^{n}))^{\ast}$? If so, how can I prove it? Context I've just started to study Functional Analysis. I tried to ...
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1answer
16 views

PDEs Diffusion eq dirichlet BC

I want to find the $2l$ periodic solution of the diffusion equation: $u_t = u_{xx}$, $∀x∈ (0, l), ∀t ∈ ℝ$ with initial condition $u(x,0) = x$ and the Dirichlet boundary condition $u(0,t) = 0$ ...
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15 views

Smooth bounded domains are finite union of star-shaped smooth domains

Given a $C^k$ function $f:\mathbb{R}^{n-1}\to\mathbb{R}$, we can define the hyperplane $\mathcal{H}_f=\{(\bar{x},x_n):x_n>f(\bar{x})\}$. In a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, every ...
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0answers
20 views

boundary conditions and existence theorem

I am studying existence and uniqueness of the weeks elution of a system of nonlinear parabolic PDE subject to initial and boundary conditions. I wonder whether changing boundary conditions will lead ...
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1answer
77 views

What is wrong in solving this pde?

I solved the first order pde and I found it is impossible to express x and t using X and y, so I cannot get the solution u from z. But the right answer is pretty simple. It is (4x-y)^2/16. Can anyone ...
6
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1answer
150 views

$\nabla \cdot f + w \cdot f = 0$

Let $w(x,y,z)$ be a fixed vector field on $\mathbb{R}^3$. What are the solutions of the equation $$ \nabla \cdot f + w \cdot f = 0 \, ? $$ Note that if $w = \nabla \phi $, then the above equation is ...