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

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Find the general solution of the PDE

Find the general solution of the PDE $ xu_x-xyu_y-y=0 $ for all $u(x,y)$ and find the parametric form of the solution of the PDE which follows the side condition $ **u(s^2,s)=s^3** $ I got part ...
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Evans PDE: Chapter 5, Problem 9 - Clarification

I've been trying to work out the solution to Question 9 in Chapter 5 of Evans, and I'm having some difficulties. I've been looking at the solution posted here: question 9 - chap 5 evans PDE And I ...
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1answer
61 views

How to solve $f\frac{\partial f}{\partial x}+\frac{\partial f}{\partial t}=0$?

How to solve $f\frac{\partial f}{\partial x}+\frac{\partial f}{\partial t}=0$? (where $f(x,t)$ is assumed to be in $C^\infty(\mathbb{R}\times\mathbb{R^+}\rightarrow\mathbb{R})$) I can find a ...
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Application of the Fundamental Theorem of Calculus

I was wondering if someone could help me with a problem I'm having. I'm reading a paper 'Spatiotemporal dynamics of continuum neural fields' and on page 13 they authors derive a model for spatially ...
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+50

Does $\operatorname{div}\left(\nabla G +xG\right)=0\Longleftrightarrow \nabla G +xG=0$?

Let $G$ be a function of $\mathcal{C}^2(\textbf{R}^d$;$\textbf{R}^*_+)$ such that $G \in \operatorname{L}^1(\textbf{R}^d)$. I read on the Internet that one has the following equivalence ...
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1answer
305 views

Find all of the equilibrium points and describe the behavior of the $x' = sin(x), y' = cos(y) $.

Find all of the equilibrium points and describe the behavior of the $$x' = \sin(x), \quad y' = \cos(y) .$$ It has been a while since I took DE...Do we first need to set $x' = y'$ to solve for their ...
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1answer
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Proof of Mean-value for Laplace's equation.

$\textbf{ Statement of Theorem:}$ If $u \in C^2(U)$ is harmonic, then $$u(x) = \frac{1}{m(\partial B(x,r))}\int_{\partial B(x,r)} u dS = \frac{1}{m(B(x,r))}\int_{B(x,r)} u dy$$ for each $B(x,r) ...
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30 views

Where is surprising “heat PDE” *surprisingly* relevant?

Historically, the term 'heat' has value as it hearkens back to the context in which Fourier studied the heat equation. Pedagogically, the term 'diffusion' has value as it imparts a more general, ...
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Given the PDE $u_t = u_{xx}$, how does one show that $u(x, t) \leq \mathrm{max}_x\;u_0(x)$ for all $x$ and $t$?

We have the PDE $u_t = u_{xx}$ with initial conditions $u(x, 0) = u_0(x)$ given. How does one show that $u(x, t) \leq \mathrm{max}_x\;u_0(x)$ for all $x$ and $t$? I later have to show that a maximum ...
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26 views

Do we suppose that $y$ is the variable at which $f$ is differentiable in $\mathbb{R}$ ?

In my notes there is the following: The solution of the problem $$u_{tt}-c^2u_{xx}=0, x \in \mathbb{R}, t>0 \\ u(x, 0)=f(x), x \in \mathbb{R} \\ u_t(x,0)=g(x)$$ is ...
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1answer
23 views

Canonical form of the differential equation

In my notes there is the following: Find the canonical form of the differential equation $$4u_{xx}-12u_{xy}+9u_{yy}+u_{y}=0$$ $$\Delta=(12)^2-4^2 \cdot 3^2=0$$ The canonical form will be of ...
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1answer
27 views

Solve an initial value problem using the directional derivative

In my notes there is the following example of solving an initial value problem using the directional derivative. The problem is the following: $$u_t(x,t)=u_x(x,t), x \in \mathbb{R}, t>0 \\ ...
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PDE first order

I'm heading the book Elements Of Partial Differential Equations -Sneddon 1957. At chapter two exists this exercise "Eliminate the arbitrary function $f$ fron the equation $$ z= ...
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25 views

Find a solution of the Laplace equation $-\Delta u=1$ with boundary condition $u=0$ on a spherical shell

Let $n\ge 2$ $B_\varepsilon$ and $\overline{B}_\varepsilon$ be the open and closed ball around $0$ with radius $\varepsilon>0$ in $\mathbb{R}^n$, respectively $R>0$, $\rho\in (0,R)$ and ...
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5 views

What is the parameter of the Streamline-Diffusion Method?

Recently, I want to apply streamline-diffusion(SD) method to solve Burgers Equation. \begin{align} u_t + uu_x - \varepsilon u_{xx} &= 0,\\ u(0,t) = u(X,t) &= 0,\\ u(x,0) &= u_0(x),\ \ \ ...
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173 views

Holder estimates for the gradient of the solutions to the linear divergence form elliptic equation?

Now I'm considering the Drichlet problem \begin{align} (a_{ij}(x)u_{x_i})_{x_j}+b_i(x)u_{x_i}+c(x)u &= f(x),\quad x\text{ in }\Omega \\ u(x) &= g(x),\quad x\text{ on } \partial \Omega.\tag{1} ...
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Energy inequalities with negative sobolev number.

Let $\phi\in H^{s}$ such that the following energy inequality is true: $$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\cdot)\|_s \, dt $$ where $P$ is a strictly hyperbolic linear operator. For ...
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A physical model for pde

Do you know any mathematical model of a physical process such that it satisfies the following equations ? $$\mathrm{u_t=a(t)u_{xx}},\,\,\,0<x<1,\,\,\,t>0$$ ...
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Closed form solutions of 2D p-laplace equation.

While investigating a physics problem, I found the following PDE: $\vec\nabla. (|\vec\nabla P|^p \vec \nabla P) = 0 $ Where $\vec\nabla =(\dfrac{\partial }{\partial r},\dfrac {1}{r} ...
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Solving $du/dt=a \Delta u + b (\Delta u)^2$

Consider the function $u(\boldsymbol{x},t)$, where $u:\mathbb{R}^n \times \mathbb{R}_+ \rightarrow \mathbb{R}$ ($\mathbb{R}_+$ denotes nonnegative reals). My question is related to the PDE ...
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An interesting semi-linear PDE problem

Assume $u\in H^1(\mathbb{R}^n)$ has compact support, and assume that it is a weak solution of the semi linear equation $$ -\Delta u+c(u)=f\;\;\text{in}\;\mathbb{R^n} $$ where $f\in ...
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continuity in $H_0^{1,2}$

Asuume that $u \in H_0^{1,2}(\Omega)$ and $f \in L^2(\Omega)$, and $\int_\Omega \nabla u \nabla v \, dx=-\int_\Omega fv \, dx$ holds for all $v\in C_0^1(\Omega)$. Show that $\int_\Omega \nabla u ...
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How do we deduce the condition for the solution?

Suppose that we have the differential equation $$u_t(x,t)=k^2u_{xx}(x,t), x \in (0,l), t>0$$ $$u(x,t): \text{ heat of rod at the position } x \ (0 \leq x \leq l )$$ If we have Dirichlet ...
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A Problem in Evans' PDE

Problem 7 in §6.6 states as follows: Let $u\in H^1(\mathbb{R}^n)$ have compact support and be a weak solution of the semilinear PDE $$-\Delta u+c(u)=f\,\,\text{ in } \mathbb{R}^n,$$ where ...
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Sobolev Embedding and Uniform $C^1$ bound

I am currently reading a paper and am a little confused about the following, which for clarity, I distill into the following question: Suppose $\{w_i\} \subset C^2(\mathbb{R}^n)$ is a sequence of ...
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1answer
21 views

Functions belonging to $L^p(\mathbb{R}^n ; \mathbb{R}^m)$ if and only of their norm belongs to $L^p(\mathbb{R}^n)$

The definition I have of $f \in L^p(\mathbb{R}^n ; \mathbb{R}^m)$ is that we require each component function to be in $L^p(\mathbb{R}^n)$. Is is true that $f \in L^p(\mathbb{R}^n ; \mathbb{R}^m) ...
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Partial Differential Equations Black Scholes Problem

Part 1) Consider the Black-Scholes problem $$\frac{\partial A}{\partial t}+\frac{\sigma^2B^2}{2}\frac{\partial^2A}{\partial B^2}+rB\frac{\partial A}{\partial B}-rA=0 ...
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1answer
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Black-Scholes Problem

Consider the Black-Scholes problem $$\frac{\partial A}{\partial t}+\frac{\sigma^2B^2}{2}\frac{\partial^2A}{\partial B^2}+rB\frac{\partial A}{\partial B}-rA=0 \hspace{3mm}\textrm{and}\hspace{3mm} ...
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How do I apply this PDE as an image filter?

I'm trying to preprocess a height map image with a helmholtz-type equation as described in this paper. The equation is: $$ddx(h') + ddy(h') + y(h'-h) = 0$$ I solved for h and got: ...
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First order elliptic pseudodifferential operator and Sobolev space

Let $P$ be an elliptic pseudodifferential operator of order 1 on a compact manifold $M$. Also let the spectrum of $P$ lie inside $(0, \infty)$. I am trying to see how one could prove that $P : H^1(M) ...
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1answer
20 views

Deflection of String

I am trying to determine u(x,t) for a string of length L=1 and c^2=1 when the initial velocity is 0 and initial deflection with small k(.01) is as follows: ...
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1answer
27 views

Second order PDE

Kindly help me with this... $$U_{xy} + yU_{yy} + \sin(x+y)=0$$ Here $A =0$, so how to calculate the characteristic equations ? as $$ {dy\over dx} = {B^2 \pm \sqrt D\over2A} $$
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Separation of variables and basis of solutions?

If a PDE can be solved by separation of variables. Then the superposition of the solutions found via this method can form all other solutions to the PDE. Is this statement correct? If it is ...
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1answer
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Variant of Ladyzhenskaya’s inequality

I am trying to show that if $\Omega \subset\subset \mathbb{R}^2$ with $C^1$ boundary and $ u \in W^{1,2}(\Omega)$ then \begin{equation*} \int u^4 < C \left(\int u^2 \right)^2 + C \left(\int u^2 ...
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12 views

Probability density function for a PDE with random inputs

I am looking for a general method or alternatively few textbook examples of deriving a probability density function for a solution of partial differential equation with random inputs in the equation ...
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1answer
21 views

Show that this function is weakly differentiable

I need to show that the function \begin{equation} u(x,y) = 1-x_1^2 \quad x_1>0 \\ u(x,y)=1+x_1^2 \quad x_1 \leq 0 \end{equation} is weakly differentiable on the unit ball. It is clear what the ...
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Elliptic regularity in an unbounded domain

Let $\Omega \subset \mathbb{R}^N$ be an unbounded domain with nonempty boundary. Let $$\mathcal{D}^{1,2}(\mathbb{R}^N) = \{ u \in L^{2^*}(\mathbb{R}^N) : \nabla u \in L^2(\mathbb{R}^N ; \mathbb{R}^N) ...
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Hyperbolic PDE with strange condition.

The 1D wave equation part is not tricky, but I am having trouble dealing with the max condition. I was thinking of using d'Alembert's formula somehow but I am not sure how to use it in this case. ...
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Elliptic regulartiy for nonlinear elliptic equations

Here is the question: Let us consider the Schrodinger type equation $$ \left\{ \begin{aligned} &-\Delta u + u = |u|^{p-2}u \quad \text{in } \mathbb{R}^N \\ &u\in H^1(\mathbb{R}^N) \qquad ...
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1answer
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Neumann boundary conditions for PDE

I have a question about Neumann boundary condition for PDE. Suppose $\Omega$ is an open bounded set in $R^n$ with a smooth boundary $\partial \Omega $. Then, a homoegenous Neumann boundary condition ...
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1answer
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Existence and uniqueness for the Cauchy problem for the Laplace equation

Given a Laplace equation $u_{xx}+u_{yy}=0$ in $\mathbb{R}^2$ with initial conditions $u(x,0)=u_0(x)$, $u_y(x,0)=u_1(x)$. Is this problem uniquely solvable in general? Does someone have a ...
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Boundary conditions for the time-independent Schrödinger equation on the sphere

if you have a free Schrödinger operator on a sphere $-\Delta \psi(\theta,\phi) = E\psi(\theta,\phi),$ then the substitution $\psi(\theta,\phi) = f(\theta)e^{i n \phi}$ leaves you with the ...
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What are the main methods to solve an evolution PDE and how are they applied?

When one sees an evolution PDE, what are the reflexes that he should have in order to tackle it. What I mean is what are the main methods that have been developed so far and to which kind of PDEs ...
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Scaling of the Dirichlet Laplacian eigenvalue [on hold]

Let Ω be a smooth domain and let $λ_1(\Omega)$ be the first (or lowest) eigenvalue of the Dirichlet Laplacian in $\Omega$. How can I show that for $\beta>0$, $λ_1(\beta\Omega)=\frac{1}{\beta^2} ...
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How to use the crank-nicolson method

I'm going over my study questions for an exam I have tomorrow in Applied Numerical Methods and I know everything except for one thing. There's a sample question about using the Crank-Nicolson method, ...
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120 views

Prove difference quotient converges to weak derivative in $L^p$

I am trying to solve the following exercise: Let $U$ be an open set in $\mathbb{R}^n$ and let $V$ be a compact subset with Lipschitz boundary. Assume that $f$ is in $L^p(U)$ with $1<p<\infty$ ...
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1answer
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The mysterious $\dot{H}^{-1}$ notation.

I have encountered the $\dot{H}^{-1}$ notation in one of the SIAM Journal on Mathematical Analysis articles. It appears to be standard (or at least not uncommon) to use this one in the field, since ...
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What does the Sturm-Liouville theory say about separation of variables?

In this question Completeness of solutions and the separation of variables method one of the comments says that the condition for the solutions formed by separation of variables to be a basis for all ...
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Understanding of the Hopf-Lax formula

This is an exercise in the book Partial Differential Equations by Evans: Here $L^*(q)=\max_{y\in {\Bbb R}^n}\{q\cdot y-L(y)\}$ and $L$ is assumed to be such that it is convex and satisfies $$ ...
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1answer
50 views

Trouble understanding proof of Sobolev trace theorem

I am trying to understand the proof of the Sobolev trace theorem. I am stuck at the bit where the boundary is flattend out using partitions of unity. See the following text (from the book of James C. ...