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

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1answer
13 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 ...
2
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0answers
12 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 ...
1
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0answers
15 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 ...
3
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1answer
14 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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0answers
8 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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0answers
22 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
14 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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0answers
17 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
9 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 ...
2
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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 ...
1
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0answers
19 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 ...
1
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0answers
33 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 ...
1
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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
25 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.
2
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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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0answers
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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0answers
6 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 ...
1
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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
votes
1answer
62 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 ...
2
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3answers
45 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 ...
0
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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 ...
2
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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: $$ ...
0
votes
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 ...
0
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0answers
47 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 ) ...
2
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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? ...
0
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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 ...
1
vote
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 ...
0
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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.
2
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1answer
61 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 ...
0
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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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0answers
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 ...
4
votes
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
votes
1answer
146 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 ...
1
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1answer
57 views
+50

Using a multivariate chain rule to solve a partial differential equation

(a) Let $a$ and $b$ be some parameters for which $a^2 + b^2$ > 0 and use the substitution $u = bx−ay$ and $v = ax+by$, to rewrite the partial differential equation $af_{x} + bf_{y} = 0$, into a ...
2
votes
1answer
93 views

A solution of the PDE $au_x+bu_y=-u$ vanishes identically under certain boundary condition

It has bothering me for days. We learnt characteristic methods and I try to use but it doesn't work. The question is as above. Thank you. I think of a solution but I think it is somewhere wrong: 1
0
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1answer
48 views

For what value of n this relation holds $\frac{x}{\pi}+\sum_{n \geq 1}\frac{2}{n\pi}\cos(n\pi)\sin(\pi x)=0$

I want the value of $n$ for which this relation must satisfy: $$ \frac{x}{\pi}+\sum_{n \geq 1}\frac{2}{n\pi}\cos(n\pi)\sin(\pi x)=0 $$ How to solve this?
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2answers
31 views

A cute little system of nonlinear PDEs

I am wondering about the solutions to the following system of PDEs. Suppose we have functions a(x,y,z), b(x,y,z), and c(x,y,z) and the following equations: ...
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0answers
19 views

Weak derivatives and sobolev [closed]

Show that a function is weakly differentiable in a domain $\Omega$ if and only if it is weakly differentiable in a neighbourhood of every point in $\Omega$.
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0answers
17 views

Equation of continuity from maxwell's equation

Recently I've known that equation of continuity can be derived from Maxwell's equation. How?? I want to know how could I derive continuity equation from Maxwell's equation. I also want the details ...
1
vote
1answer
12 views

compactly supported eigenfunction

Does it true that there exists a compactly supported eigenfunction corresponding to the first positive eigenvalue $\lambda_1$ of hyperbolic Laplacian operator $\Delta$ on $L^2(S)$, $S$ is a hyperbolic ...
2
votes
1answer
32 views

Product of weakly differentiable functions

Let $ u,v \in W^{1,1}_\mathrm{loc}(\Omega) $ and assume that $ uv \in L^{1}_\mathrm{loc}(\Omega) $ and $ u\, Dv + v \,Du \in L^{1}_\mathrm{loc}(\Omega) $. I want to prove that $ uv \in ...
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0answers
53 views

Does compactness preserve continuity? [closed]

Are all compact set continuous in a Banach space? I am clear that continuity preserves compactness, but not clear about the reverse, i.e. does compactness preserve continuity? If not, can someone ...
0
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0answers
32 views

using separation of variables to solve $u_{tt}=16u_{xx}, u(x,t)$

Question: Find a nontrivial family of solutions of the following PDEs by the method of separation of variables. You need not find the most general solution obtainable in this way. 4c. ...
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0answers
23 views

Game Theory and Algorithms for Cournot Competition [closed]

Cournot's Game with n rms. Consider the Cournot game where the demand function is given by the demand function discussed in class: ...