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

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5
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1answer
90 views

Regularity theorem for PDE: $-\Delta u \in C(\overline{\Omega})$ implies $u\in C^1(\overline{\Omega})$?

I stumbled over this question in the context of PDE theory: Let $U$ be connected,open and bounded in $\mathbb{R}^n$ and $u \in C^0(\overline{U}) \cap C^2(U)$ and $\Delta u \in C^0(\overline{U})$ with ...
0
votes
0answers
18 views

Solving equations of the form $\phi(D_{x},D_{y})u=0$

I am taking a course called partial differential equations. We are mostly learning about solving equations of the form $\phi(D_{x},D_{y})u=0$, for example $(D_{x}^{2}+D_{x}D_{y}+D_{y}^{2})u=0$ or ...
1
vote
0answers
32 views

Dirichlet problem for Laplace equation in $\Omega=\left(\overline{B\left(0,a\right)}\right)^{c}$

I'm having troubles with the proof of the following theorem: let $\Omega=\left(\overline{B\left(0,a\right)}\right)^{c}$ and $U\in \mathbb{R},U>0$, then there exists a unique function $\psi\in ...
3
votes
1answer
35 views

References for the operator $(I-\Delta)^{\alpha /2}$

I am studying PDEs involving fractional differential operators, and I have found a few properties for the operator $(I-\Delta)^{\alpha /2}$ scattered through scientific papers. I wonder if there is a ...
1
vote
1answer
45 views

Example 3, Sobolev space Evans

In the following example p.260 Evans. I think I understand everything except for one calculus fact in the second last equation: $$ \int_{\partial ...
0
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0answers
7 views

Partial Differential Equation with no solution - Transversality condition?? [migrated]

I want to continue posting on this : Partial Differential Equation with no solution - Transversality condition? but my previous account was deletd . Will someone link this account to my previous one ...
2
votes
1answer
30 views

spectral theory of Laplacian on $\mathbb R^n$ [duplicate]

Can you describe the spectrum of the Laplacian $ \Delta : H^2(\mathbb R^n) \subset L^2(\mathbb R^n) \rightarrow L^2(\mathbb R^n)$? I am interested for which values $z \in \mathbb C$ the equation ...
1
vote
1answer
44 views

PDE - Don't understand teacher's solution

I'm reading a solution to a problem in PDE class that the teacher gave, and I don't fully understand his solution. The problem is $\frac{dx}{1+\sqrt{z-x-y}}=dy=\frac{dz}{2}$ and what we want is to ...
0
votes
2answers
53 views

Can we show this inequality? (PDE question)

I am attempting to show that $$\int^t_0 \left[e^{-\lambda \kappa (t-s)} a(s)\right] ds \leq \int^t_0 \frac{\lambda}{2\kappa} a(s)^2 ds \tag{1}\label{1}$$ for any $t$, where $\lambda, \kappa >0$, ...
1
vote
1answer
17 views

How to find the cosine series when solving a PDE with Dirichlet conditions?

Suppose I have to solve $\sum_{n=0}^{\infty} A_n \cos(\frac{(n+1/2)\pi x}{L}) = x $ from $0$ to $L$. If I we want to find $A_n$ my professor uses the formula for a cosine series: ...
1
vote
1answer
24 views

Solve a second-order PDE with non-constant coefficients

Solve the following equation: $$\frac{\partial u}{\partial t}+ax^2(1-x)^2\frac{\partial^2u}{\partial x^2}=0$$ with the boundary condition $u(1,x)=(.5-x)^2$, $u(t,0)=u(t,1)=.25$. Domain is $t\geq0, ...
2
votes
2answers
96 views

Partial Differential Equation with no solution - Transversality condition?

I have the following equation: $$ x u_x + y u_y = \frac{2e^u }{xy } , x>0,y>0 $$ with the initial condition (corresponding to $t=0$ ): $$ \Gamma =\{ (s,s,0) | 0<s<\infty \} $$ By using ...
0
votes
1answer
22 views

Solve laplace equation for a semi-infinite plate. Where is my mistake?

The plate is semi-infinite. 2 Of its sides have $f=0$ and the bottom part satisfies $f=cos(x)$. Its width is $\pi$. The temperature distribution $f(x,y)$ satisfies the Laplace equation $\nabla^2 f=0$. ...
0
votes
1answer
24 views

Partial Differential Equation - The Chain Rule

$\displaystyle \sum_{i,j=1}^{n}\int_{U}a^{ij}u_{x_{i}}\zeta^{2}u_{x_{j}}dx$ $\displaystyle =\sum_{i,j=1}^{n}\int_{U}a^{ij}D_{i}u\zeta^{2}D_{j}u dx$ Can someone please explain to me how we use the ...
0
votes
0answers
46 views

HELP to solve - PDE First order

I have this equation $$ u_x+uu_y=0$$ by the book "Handbook of First order Partial Differential Equations - page 290." The general solution is $F(ux-y,u)$, where $F$ is a arbitrary function. I try ...
1
vote
1answer
17 views

Transforming pde to nicer form?

I have a second order differential equation for $u$ $$\frac{d^2u}{dx^2} + \frac{d^2u}{dy^2} + 5u = 0$$ I am looking for a transformation $u(x,y) \rightarrow v(x,y)$ that gives $$\frac{d^2v}{dx^2} + ...
2
votes
1answer
16 views

Transformed pde but my answer doesn't match solution?

$$\frac{d^2u}{dx^2} + \frac{d^2u}{dy^2} + \frac{du}{dx} + 2\frac{du}{dy} + 3u = 0$$ Let $u = ve^{ax + by}$ and find $a, b$ such that we can transform to the following equation $$\frac{d^2v}{dx^2} + ...
1
vote
0answers
14 views

Understand Dankwerts boundary conditions in plug flow pde

I'm trying to model the advection-diffusion-reaction plug flow equation in mathematica: $\frac{\partial C_a}{\partial t}$=$D_s$$\frac{\partial^2 C_a}{\partial z^2}-V\frac{\partial C_a}{\partial ...
1
vote
1answer
40 views

How can I solve numerically this partial differential equation?

I am reading this paper and came across a system of differential equations with 4 ODEs and 1 PDE. The PDE is given below. My question is how to solve this numerically in MATLAB , Python or ...
1
vote
1answer
14 views

Convolution of derivatives

When transforming nonlinear PDE to its Fourier space, I encounter the following problem: Consider the equation $u_t=(u^3-u)_{xx}$. Then, when transforming to Fourier space we get \begin{equation*} ...
1
vote
0answers
23 views

solubility of nonlinear overdetermined PDE for holonomic scaling of a frame

This question is essentially a tweak of under what conditions can orthogonal vector fields make curvilinear coordinate system? . Suppose we have a frame of vector fields $\nu_i$ on $\mathbb{R}^n$. As ...
2
votes
1answer
37 views

How to Separate Quasi-Linear PDE

I'm attempting to solve the non-homogenous quasi-linear PDE below: $$z\frac{\partial z}{\partial x} - \alpha y\frac{\partial z}{\partial y} = \frac{\beta}{x^3y}$$ From what I've read in texts, the ...
0
votes
1answer
19 views

Equilibrium temperature in a heat equation

To find the equilibrium temperature distribution for a heat equation, $$U(x,t)$$ it is critical to note that the second partial derivatives WRT the space variables is zero. Why is this so?
1
vote
1answer
30 views

Extremizing the boundary value problem $I[y]=\int_0^1y'^2(x)\,dx+y^2(0)-2y^2(1)$

Extremizing the boundary value problem $$I[y]=\int_0^1y'^2(x)\,dx+y^2(0)-2y^2(1)$$ My Thought: First, we use Euler-Lagrange equation and solving we get , $y(x)=C_1x+C_2$. Then we put it in ...
2
votes
2answers
35 views

Change of variables for heat equation

How to make a change of variables to turn the equation $$\frac{\partial{u}}{\partial{t}}=D\frac{\partial^2{u}}{\partial{x}^2}+cu$$ back to the heat equation? Where can I read about change of ...
2
votes
0answers
23 views

How to use Duhamel's principle to solve wave equation

Let $x\in\mathbb{R}$,$t>0$, $$\frac{\partial u}{\partial t}+5\frac{\partial u}{\partial x}=e^{-t}sinx,$$ $$u(x,0)=(1-x^2)_{+}.$$ Using Duhamel's principle to solve it. By Duhamel's principle, the ...
0
votes
1answer
26 views

Solve the Poisson equation $\Delta f = x_2$ in the unit disk

Let $D$ be the disk of radius 1 centered at (0,0). Find a formula for the solution of $\Delta u=f$ in $D$ $u=1$ on $\partial D$ In the case where $f(x)=x_2$ In polar coordinates ...
1
vote
1answer
37 views

$\log|x|\in\text{BMO}(\mathbb R^n)$

Lemma. $A$ is fixed. For any ball $B$, there exists a constant $c_B$, which satisfies $$\frac{1}{B}\int_B|f-c_B|\leq A$$ then we have $f\in\text{BMO}(R^n)$. I want use the lemma to prove the ...
0
votes
1answer
22 views

Integral over a ball.

Let for $0<t<t_0$ $$e(t)=\int_{B(x, R(t_0-t))}v(x,t)dx$$ Given that $v(x,t)$ is a differentiable function prove that $$e'(t)=\int_{B(x, R(t_0-t))}v_t-R\int_{\partial B(x, ...
0
votes
0answers
15 views

Well-posedness of semilinear elliptic equations

I am trying to understand the well-posedness of some semilinear elliptic problems, such as: $$-\Delta u +F(x,u)=0, \ \Omega$$ $$u=f \in C^{2,\alpha}, \ \partial \Omega, $$ or $$-\Delta u ...
2
votes
0answers
24 views

The kernel $k(x,y)=\frac{y}{y^2+x^2}$ is a solution of which equation?

The kernel $$k(x,y)=\frac{y}{y^2+x^2}$$is a solution of (A) Heat equation (B) Wave equation (C) Laplace equation (D) Lagrange equation Which are correct ? I tried through ...
0
votes
0answers
36 views

Homogenization problem in Evans, Chapter 4

I would appreciate some help with the following problem, please. It it from Evans's PDE book, problem 4.8. I am quite stuck on it and any kind of help would be great. Let $n=1$ and suppose that ...
0
votes
0answers
23 views

Prove one property of harmonic function

Let $u(x)$ be a harmonic function defined in the square $[0,1]\times[0,1]$. Suppose that $u(x_k)=0$, where $x_k=(1/k,1/k).$ Prove that $u(x)=0$ everywhere in $[0,1]\times[0,1]$.
0
votes
1answer
24 views

$g(x,y,t) = f(x+yt,y,t)$ what is $\partial_t g$?

This is a simple question in multivariable calculus, but it confuses me. Say that $f \in C^2\left(\Bbb{R}^d \times \Bbb{R}^d \times [0,T]\right)$ and it satisfies a PDE, such as \begin{equation*} ...
0
votes
0answers
12 views

difficult* Is there more than 1 lamda solution for this Sturm-Lioville problem with derivative conndition

I've found solutions only for $$\lambda>0$$ My general solution is $$u_\text{(x,t)=}\sum _{n=1}a_n\text{Cos(}B_n\text{x)}e^{-\text{kB}_nt}$$ Have I missed out anything?
1
vote
0answers
19 views

Find an explicit formula for the solution of the boundary value problem

Let $D$ be the disk of radius $1$ centered at $(0,0)$. Find an explicit formula for the solution of the boundary value problem $\Delta u=0$ in $D$ $u=x_1^2$ on $\partial D$ I just don't even know ...
2
votes
1answer
25 views

PDE Manipulation - Calculus

I need help for this question, its a lot of calculus but I'm confuse. let $$ u= \dfrac{(x-b)^{2}+y^{2}-q^{2}}{(x-b-1)^{2}+y^{2}-q^{2}-1} $$ I need show that $$ u_{x}^{2}+u_y^{2}= ...
1
vote
1answer
40 views

Fundamental solution and Green's function

I am currently dealing with Poisson's equation $- \Delta u= f $ on some open domain $U$ and $u =g$ on the boundary $\partial U.$ Now a fundamental solution is a solution to $- \Delta u(x) = ...
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0answers
18 views

Searching non-homogeneous linear PDE solution (w/ non-homogeneous BCs) by Green's function

I'd like to know if this linear non-homogeneous PDE can be solved using Green's function ...
1
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0answers
23 views

Heat equation with heat source in form of delta function

Consider the problem \begin{equation} \left\{\begin{array}{cc}u_t-u_{xx}=\delta_0,&0<x<1,\ t>0\\ u_x(0,t)=u_x(1,t)=0,&t>0,\\ u(x,0)=0,& 0\leq x\leq 1.\end{array}\right. ...
0
votes
1answer
18 views

Semi-Linear First Order PDE (with non-linear reaction term)

I've been trying to figure out this problem for a while and was wondering what you thought. I have the PDE: $\partial c \over \partial t$ + $K \over r^2$ $\partial c \over \partial r$ + $Da \ c \over ...
0
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0answers
42 views

Elliptic W^{2,p}-estimates for a Neumann problem.

Consider the simplest elliptic-Neumann problem in $\Omega\subset \mathbb{R}^n$: $$ -\Delta u+u=f\quad \text{in } \Omega, \quad \frac{\partial u}{\partial \nu}=0\quad \text{on } \partial \Omega. ...
1
vote
1answer
44 views

Extremizing the following boundary value problem

Consider the functional $$J(y)=y^2(1)+\int_0^1y'^2(x)\,dx$$ with $y(0)=1$ , where $y\in C^2[0,1]$. If $y$ extremizes $J$ then find the value of $y(x)$. I tried through Bolza problem. Firstly ...
0
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0answers
8 views

Finite-Difference Scheme for a Non-Linear PDE?

I have the following non-linear PDE and I have no idea how to go about solving it using a finite difference scheme in Python. Can someone get me started and/or point me to an algorithm for doing this? ...
2
votes
1answer
44 views

Reduce PDE to ODE

Maybe you don't want to check all the details, but could look at a few equations here. Would you mind leaving a comment that you at least some part looks okay?- This way, I know that at least somebody ...
4
votes
1answer
44 views

Can we derive the PDE followed by a marginal transition probability density?

A pair of correlated stochastic processes follow the SDEs \begin{align} dX_t&=a(t,X_t)\,b(t,Y_t)\,dt+c(t,X_t)\,d(t,Y_t)\,dW_t, &&X_0=\bar{x}\\ dY_t&=f(t,Y_t)\,dt+g(t,Y_t)\,dZ_t, ...
0
votes
0answers
20 views

Question about heat equations?

I have to solve the heat equation with $u(0,t) = 0$, the end at $x=2$ insulated $\forall t \gt 0$ and initial condition $u(x,0)= 20\sin\frac{\pi x}{4}$. I interpreted the 2nd b.c to mean that ...
0
votes
0answers
7 views

Find a condition on $\lambda(k)$ such that the normal mode functions satisfy the given equation.

Let $k=(k_1,k_2)$, $x=(x_1,x_2)$ and $\alpha>0$. Find a condition on $\lambda(k)$ such that the normal mode functions $u(x)=a(k)e^{i(k_1x_1+k_2x_2)+\lambda(k)t}$ satisfy the following equation ...
1
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0answers
25 views

Partial Differential Equations, how to find a change of variables

I am trying to understand how to find variable changes for partial differential equations. I know characteristics method gives you a valid variable change when you have a condition given on a ...
3
votes
0answers
58 views

Is this derivative well defined?

Let $\Omega$ be an open bounded domain in $\mathbb{R}^n$. Let $z_1,z_2,\dots,z_k\in \bar \Omega $ be $k$ distinct points. Let $\bf Z$ denote the k-tuple $Z = (z_1,z_2, \dots, z_k)$ Define $R_i = ...