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

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6
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1answer
488 views

Textbook for Partial Differential Equations with a viewpoint towards Geometry.

Though similar questions have been asked at Good 1st PDE book for self study and Good reference texts for introduction to partial differential equation? but none of them really answer my query, so I ...
0
votes
1answer
66 views

How to use second derivative test?

I read a proof in my book on pde which I find a bit strange. Let $\Omega$ be some bounded domain. For $f\in\mathscr C^2(\Omega)\cap\mathscr C^0(\Omega)$ satisfying $\Delta f\geq0$ it holds that ...
0
votes
1answer
180 views

Help with proof about sub harmonic function

I know how to prove it using strong maximum principle, but I need to show it using conditions for a relative maximum. Does this mean using second derivative test? I think if $p\in \Omega$ and ...
2
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1answer
57 views

A function satisfying the mean value property is harmonic

Here is the problem. I know that if $u$ is harmonic the equation holds, but I don't know how to prove it from the other direction.
1
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1answer
42 views

Help with the proof of Mean Value Inequality

I am a beginner of Elliptic PDE. This is really hard for me who do not have a sound foundation in Calculus III. I get stumbled in the following proof, especially the part in the red rectangle. I would ...
2
votes
0answers
77 views

Green's function for Dirichlet Laplacian

I am thinking of the Dirichlet boundary condition $u|_{\partial \Omega}$ for a domain $\Omega \subset \mathbb{R}^n$. Let $\Delta$ be the Dirichlet Laplacian, which accepts only functions with the ...
2
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1answer
96 views

Wave Equation with Constant Boundary Conditions

I need to find a formal solution to \begin{eqnarray} &u_{tt} &= c^2 u_{xx}, \;\;\;0<x<1, \mathrm{and \;}t>0\\ &u(x,0)&=x+1,\\ &u_t(x,0)&=x(1-x), \;\;\;\;0 \leq x \leq ...
1
vote
1answer
48 views

Heat equation: Why are these ratios of functions constant

One can solve the heat equation $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} $$ by a separation of variables such that $u(x, t) = f(x)g(t)$. Substituting this into the ...
1
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1answer
25 views

Using the method of characteristics with an extra term

How do you apply the method of characteristics to get the solution to the following PDE: $$xU_x-yU_y-xU=x^2y$$
2
votes
1answer
63 views

How to make a coordinate rotation to solve a first-order linear PDE?

I know that a first-order linear constant coefficient PDE, such as $au_x + bu_y = 0$, can be transformed to an ODE by rotating the coordinate system so the $x'$ axis points to $(a,b)$ where the ...
0
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1answer
34 views

Question about Cauchy-Kowalevsky theorem

I have the following Cauchy problem: $u_x+u_y=u^2$ $u(x,0)=h(x)$ The solution I got is: $u(x,y)=$$h(x-y)\over{1-yh(x-y)}$ If h is analytic, what can I conclude applying Cauchy-Kowalesky theorem? ...
0
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1answer
39 views

differentiating a summation series

How would one go about partially differentiating the following with respect to L; z = $\frac{T}{L}\Psi_{z} - \frac{T}{L}\sum_{n=1}^N X(n) sinh[\frac{2 \pi n}{L}(h + z)] cos(\frac{2 \pi n}{L}x)$ I ...
1
vote
1answer
24 views

Finding the general solution to $2u_x-3u_y+(U-x)=0$

The PDE I'm working on is: $$2u_x-3u_y+(U-x)=0.$$ Using the method of characteristics I obtained $c_1=2x+3y.$ Where I am stuck is on $c_2$; currently I'm exploring $$\frac{dx}{2}=\frac{du}{u-x}.$$ ...
2
votes
1answer
49 views

Solution transport equation

I have to solve the following equation $$\partial_t v+2A\cdot\nabla v-iB(x)\cdot Av=0$$ where $A$ is a constant vector and $B$ a smooth vector field. I can solve the transport equation $\partial_t ...
1
vote
1answer
136 views

Feynman-Kac representation for a PDE

I have the following PDE: $$ u_t + r x u_x + \frac{\sigma^2 x^2}{2} u_{xx} + h(t,x) u_y - ru =0 $$ $$ u(x,T,y) = y $$ I wanted to check whether the following representation is correct (I used ...
1
vote
1answer
72 views

A basic partial differential equation of heat transfer

Let $u(x,t)$ be the temperature along a 1-D rod, from $x=0$ to $x=L$. $\frac{\partial u}{\partial t} = A \frac{\partial^2 u}{\partial x^2}+Bu$, where A and B are constants. Initial condition is ...
0
votes
0answers
66 views

Characteristic equation of the PDE $u_t+b\cdot Du=f$

I am trying to find the characteristic equation of the following PDE (Evans Chapter 3 Problem 2): $$u_t+b\cdot Du=f$$ in $\mathbb{R}^n \times (0,\infty)$ where $b \in \mathbb{R}^n$, $f=f(x,t)$. The ...
3
votes
1answer
90 views

Gronwall type inequality

Is there a Gronwall-type inequality for bounding $u(t)$ such that $$\vert \partial_t u(t)\vert\leq C \{ u(t)+u(t)^\alpha\}$$ with $\alpha>1$ ?
3
votes
1answer
58 views

Characteristic curve

Consider the equation $yu_x-xu_y=0$ for $(x,y)\in \mathbb{R} \times (0,\infty)$ with $u(x,0)=x^2$ as the initial condition. I just need help solving for the characteristic curve. I have that ...
2
votes
0answers
199 views

Max Principle for Heat Equation with Neumann Boundary Condition

This is a homework problem. I am looking for an additional hint or reference to theorems/ideas that can help. Some of my thought process is presented below. Suppose \begin{align*} u_t - \Delta u ...
0
votes
1answer
33 views

Question about Maximum Principle

I'm just learning about the maximum principle in my PDE class. I thought I'd be able to apply it to something else I was learning about in electrostatics (namely, that the surface of a conductor is ...
0
votes
2answers
48 views

General solution for $U_{xy}+U_y=e^{-x}$

Let $$U_{xy}+U_y=e^{-x}$$ I followed the substitution mentioned here. Let $V_x+V=e^{-x}$. So now we have $(e^{x}V)_x=e^{-x}$. Integrating w.r.t $x$ we get $$V=-e^{-2x}+e^{-x}c_1(y).$$ Then ...
0
votes
1answer
74 views

Second order PDE with boundary conditions

Solve the equation $$u_t=17u_{xx}, \ 0<x<\pi, \ t>0,$$ with the boundary conditions $$u(0,t)=u(\pi,t)=0, \ t\ge 0,$$ and the initial conditions $$u(x,0)=\left\{ \begin{array}{l l} 0 ...
1
vote
2answers
141 views

Solving a PDE with mixed derivatives

Let $u_{xy}+u_{y}=e^{x}.$ To solve this, I attempted to use the following substitution. Let $V=u_x$. I tend to hit a roadblock as this is not a homogenous equation. Could someone get me started on ...
1
vote
1answer
60 views

Generalizing Method of Characteristics: Example

I have consulted other sources, but the relevant ones have used notation that I am not entirely comfortable with. I'm told that generalizing this procedure is straightforward, but I find myself ...
1
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1answer
69 views

Surface measure and the wave equation

I am having an annoying conceptual problem trying to solve problem 46 in Chapter 8 of Folland's "Real Analysis". I'll try to explain my problem as briefly as possible. Consider the wave equation ...
2
votes
0answers
212 views

Boundary Conditions for a Finite Difference Approximation of a Sixth Derivative

I am trying to use a finite difference scheme to numerically solve sixth order parabolic equations such as \begin{equation} u_t = u_{xxxxxx} \end{equation} with symmetry conditions \begin{equation} ...
1
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1answer
28 views

Solving a PDE given a specific curve and condition

Consider the following PDE: $$-3U_x+4U_y=0$$ where $U=3x$ on curve $y=x+1$. First we invoke the method of characteristics: $$\frac{dx}{-3}=\frac{dy}{4}=\frac{dU}{0}.$$ From this we get ...
1
vote
1answer
99 views

Showing that two functions are orthogonal on a rectangle

I was given the following question, and I think I'm nearly there, I just wanted to ask for some clarification in the last step. Derive the eigenvalues and functions of the SL problem $\phi_{xx} ...
0
votes
1answer
212 views

Jacobian and PDE

I am wondering how to compute the Jacobian in order to know if a given PDE satisfying an initial condition has a unique solution or not. If I consider the PDE, $u_x=1$, satisfying the initial ...
1
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0answers
23 views

System of PDE's with unknown functions

So by messing around with some stuff in my own research I came across this problem and I have no idea how to proceed. I suspect it may have something to do with solving systems of PDE's but I could be ...
0
votes
1answer
194 views

Integration by parts on all of $\mathbb{R}^n$ with $n>1$

So this came up as I was thinking about the uniqueness of solutions to the wave equation. I have seen proofs for uniqueness on all of $\mathbb{R}$ or on bounded subsets of $\mathbb{R}^n$, but never ...
3
votes
2answers
126 views

Solve the 3 Dimension first order PDE $xu_x+yu_y+zu_z=0$

This is a 3D Higher Dimension first order PDE and I am kind of confused because I am taking the partial derivatives and for some reason nothing is being cancelled out. Question: Solve the PDE ...
0
votes
1answer
42 views

Partition of Unity's Lemma

Let $V\subset\mathbb{R}^n$ compact, $\Omega\subset\mathbb{R}^n$ open, $V\subset\Omega$, $\delta:=\inf\{|x-y|\mid x\in V,y\notin\Omega\}$, $U:=\left\{x \mid |x-y|<\frac{\delta}{2}\,\,\text{for ...
1
vote
1answer
76 views

Solution of the heat equation

Let $u:\mathbb{R}^n\times(0,+\infty)\to\mathbb{R}$ solves the following heat equation: $$u_t(x,t)-\triangle u=0,\quad (x,t)\in\mathbb{R}\times(0,+\infty)$$ (a) Show that for each ...
0
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1answer
54 views

Inequality in Evans PDE section 5.7

I'm stuck in the proof of the Compactness Theorem in Evans PDE 2nd edition book. On page 287, last line, how do you get the inequality $$ \epsilon ...
0
votes
2answers
26 views

Solving a PDE via charateristics

I'm wondering if I am on the right track: Let $2U_x-3U_y+2(U-x)=0$ and $U(x,x^2)=f(x)$. We solve this by the using the following relationship: $$\frac{dx}{2}=\frac{dy}{-3}=\frac{dU}{U-x}.$$ This ...
2
votes
0answers
54 views

finding solution to a partial integro differential equation

I want to find a function (or a set of functions) such that $u(x,t)$ satisfies the following partial integro-differential equation with singular kernel \begin{eqnarray} &&u_x(0,t) = \int_0^t ...
0
votes
1answer
99 views

The proof of a Sobolev embedding inequality by a compactness argument

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} ...
0
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1answer
24 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 ...
0
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0answers
61 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 ...
1
vote
1answer
71 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] + ...
1
vote
2answers
124 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 ...
2
votes
1answer
147 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 $\nu$ ...
1
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1answer
49 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) $ ...
1
vote
1answer
51 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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0answers
64 views

a simple calculation

Can anyone see how (1) lead to (2)? \begin{align} ...
0
votes
1answer
58 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) ...
2
votes
0answers
24 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 ...
2
votes
1answer
183 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 ...