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

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2
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2answers
97 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
28 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
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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
45 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
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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
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1answer
31 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
40 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
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0answers
39 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
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0answers
24 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 $[0,1]\times[0,1]$.
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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?
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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
19 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
19 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
45 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
46 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
46 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
46 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
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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 ...
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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
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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 = ...
10
votes
1answer
86 views

Theoretical Basis for Eigenvalue transformation on Bessel's Equation

The method I've been taught for finding all of the eigenvalue solutions to Bessel's operator $$b(f)\equiv f''(x)+\frac{1}{x}f'(x)$$ goes as follows. Let $g(a)=f(\sqrt{\lambda}x)$. Then $$b(g)=\lambda ...
0
votes
1answer
25 views

Unsure of 2d finite difference method with second order term?

I have the following equation for the price of Black Scholes Euro option - (1) $$-\frac{\delta C}{\delta t} = -\alpha\frac{\delta^2 C}{\delta x^2} + [r - \delta + \frac{\sigma^2}{2}]\frac{\delta ...
2
votes
3answers
53 views

Find the Fourier Transform of $2x/(1+x^2)$

I tried doing this the same way you would find the Fourier transform for $1/(1+x^2)$ but I guess I'm having some trouble dealing with the 2x on top and I could really use some help here.
0
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1answer
60 views

Evans PDE problem 9,Chapter 6

Let $U\subset \mathbb{R}^n$ be an bounded domain with smooth boundary. Assume $u$ is a smooth solution of $$ Lu=-\sum_{i,j=1}^na^{i,j}u_{x_ix_j}=f \ \text{in} \ U, \ \ u=0 \ \text{on} \ \partial ...
8
votes
0answers
198 views

Overview of nonlinear analysis, differential equations (ODE and PDE), dynamical systems, and mathematical physics, and their relationships

The fields of (i) nonlinear analysis, (ii) ODE and PDE, (iii) dynamical systems, and (iv) mathematical physics are very huge, fertile, and, in a sense, unorganized (see Open problems in ...
0
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1answer
17 views

are those boundary $ C^1$

I am studying PDE and my question is that if we have a open unit disk with $ [0,1)$ on x axis removed, is the boundary of this set $C^1$ ? And on the other hand, is the boundary of a open rectangle ...
3
votes
0answers
28 views

Relationship of SDE and Feynman-Kac PDE

I am struggling with this problem: Given a stochastic differential equation $$ dX_t = b(X_t) dt + \sigma (X_t) \,dW_t $$ where $W$ is a Brownian motion and the functions $b$ and $\sigma$ are ...
1
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1answer
25 views

Intuition behind estimates on derivatives of a harmonic function

In Evans' PDE book he gives the following theorem. Assume $u$ is harmonic in $U$. Then, $$ |D^{\alpha}u(x_0) | \le \frac{C_k}{r^{n+k}}||u||_{L^1(B(x_0,r))}$$ When asking my professor for some ...
2
votes
1answer
44 views

Integral Inequality for Bound on Gradient of Solution to Heat Equation

My overall aim is to show that, for a bounded solution $u(x,t)$ to the heat equation in $\mathbb{R}^n \times [0,T]$ with boundary condition $u(x,0) = f(x)$, $$\max |\nabla u(x,T) | \leq \frac ...
1
vote
1answer
22 views

Verifying Duhamel Principle for Heat Equation

From separation of variables, we get a solution to the homogeneous problem for the heat equation $$u_t - u_{xx} = 0$$ $$u(0,t) = u(L,t) = 0$$ $$u(x,0) = f(x)$$ of the form $$u(x,t) = \int_0^L f(y) ...
3
votes
0answers
56 views

How to solve this boundary value problem?

I'm struggling with the following Boundary Value Problem for some time. The problem is to solve the biharmonic equation $\nabla^4\psi = 0$ with $\psi$ dependent not just on the coordinates on the ...