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

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4
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165 views

Unbounded Entropy Solution to Burger's Equation

I need to deduce that $ u(x) = \left\{ \begin{array}{lr} \frac{-2}{3}(t+\sqrt{3x+t^2}) & t^2+4x>0\\ 0 & t^2+4x<0 \end{array} \right. $ is an unbounded ...
1
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1answer
36 views

Separation of variables in the PDE $u_{tt}=c^2 u_{xx}$.

I'm stuck trying to work with the constants to the solution to the SOV problem Given the following equation: $u_{tt}=c^2 u_{xx}$ and the following conditions: $u(0,t)=0=u(\pi,t)$, $u(x,0)=0$, $u_t(...
2
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1answer
334 views

How to use D'Alembert formula for Neumann boundary conditions on a finite interval?

I have a PDE to solve that I am not sure how to do. I know how to solve this using D'Alembert's formula for Dirichlet boundary conditions but I do not know how to solve it for Neumann boundary ...
0
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1answer
62 views

Voltage Distribution Inside a Cylinder [closed]

I was assigned this problem, and quite honestly I do not know where to begin. If I could get some help and an explanation of the Bessel function, also? Thank you. I know my conditions are: \begin{...
1
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1answer
76 views

Solving a PDE equation

Could you please help me to solve this equation: $$\frac{\partial^2}{\partial x^2}E(x,t)-LC_1 \frac{\partial^2}{\partial t^2}E(x,t)+LC_2 \frac{\partial^4}{\partial x^2 \partial t^2}E(x,t)=0 \qquad \...
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0answers
33 views

Second order PDE with initial and boundary conditions

I'm trying to solve the following PDE: $u=u(x,y)$ $\left\{ \begin{array}{1 1} \partial_x ^2 u-6\cdot\partial_x \partial_yu+9\cdot\partial_x^2u=x^2+y^2\ \\ \ u(0,y)=0 \\ \partial_x u(0,y)=y ...
2
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1answer
78 views

d'Alembert's Solution: Should anything more be added to it?

I have this initial value problem: $$ u_{tt}- c^2u_{xx} = 0; \,0 < x < 1; \,t > 0; $$ $$ u(x, 0) = 0; \,u_t(x, 0) = 1; 0 <= x <= 1; $$ $$ u(0, t) = u(1, t) = 0; t >= 0: $$ ...
2
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1answer
118 views

Charge distribution on an arbitrarily shaped conductor

From physics we know that given a charged conductor put in vacuum (no external electric fields), the charge distribution on its surface is approximately proportional to the curvature of the surface on ...
0
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1answer
208 views

Dirichlet problem on rectangle with nonhomogeneous boundary conditions

I want to solve the following problem in Dirichlet Problem: Let $D = \{(x, y) \in \mathbb R^2 : 0 < x < a; 0 < y < b\}$ be a rectangle with boundary $\partial D$. I want to solve ...
0
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1answer
93 views

Physical interpretation of the integral formula for the solution of Laplace equation with Dirichlet/Neumann boundary condition

Suppose we have a bounded domain $D$ with smooth boundary, with $G(x,y)$ being the Green's function for the Poisson equation on $D$, i.e. $G(x,\cdot)=0$ on $\partial D$ and $\Delta_y G(x,y)=\delta_x(y)...
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1answer
62 views

Positivity of Poisson kernel

Let $K(x,y)$ be the Poisson kernel for the Dirichlet problem of Laplace equation on a bounded domain $D$ with smooth boundary, i.e. for a harmonic function $u$ on $\bar D$ with $u|_{\partial D}=g$, we ...
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1answer
670 views

PDE: Green's function and the method of images

I am stuck on a problem with the method of images. The formulation is rather simple; Solve the for green's function given by $\nabla^2 G = \delta( \underline{x} - \underline{x}_0)$ in the wedge ...
2
votes
1answer
103 views

Semi-infinite plate problem when 2 edges are equal temperature and one edge is a function of position

Here is the problem posted: Now here is my solution for a) I knew to discard the $e^{ky}$ and $\sin(kx)$ due to $T\to 20$ when $y\to\infty$ and $T= 20$ at $x = 0$. Which leaves me with $\cos(kx)...
0
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1answer
84 views

D'Alembert Formula where PDE has only one boundary condition

Solve the initial boundary value problem $$u_{tt}=4u_{xx}, \ x>0,t>0$$ $$u(x,0)=\frac{x^2}{8}, \ u_t(x,0)=x, \ x\ge0$$ $$u(0,t)=t^2, \ t\ge0.$$ I used D'Alembert forula and got $$u(x,t)=\...
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2answers
78 views

Characteristics method applied to the PDE $u_x^2 + u_y^2=u$

I am trying to solve: $u_x^2 + u_y^2=u$ with boundary conditions: $u(x,0)=x^2$. Unfortunately it leads to equations that makes no sense (sum of squares is $0$ and all constants are $0$). I would be ...
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0answers
26 views

What conditions must an operator meet, to have only real eigenvalues?

Given the problem $Lu = \lambda u$, what properties must $L$ have, for all its eigenvalues to be real? An answer in the context of (partial) differential equations would be appreciated.
3
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3answers
61 views

EigenFunction for $\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x} =\frac{2f^2}{x}$

When studying a computer vision problem I end up with a function $f(x,t)$ that satisfying $\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x} =\frac{2f^2}{x}$. My question includes two ...
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1answer
60 views

Change of variables - PDE

I was just wondering how do I use change of variables to obtain a more suitable equation to solve for the following PDE? If I know how to do that then I am sure I can solve the rest. $$u_t=Du_{xx}...
3
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1answer
104 views

Laplace's Equation in Polar Coordinates - PDE

Find the bounded solution of Laplace's equation in the region $\Omega=\{(r,\theta):r>1,0<\theta<\pi\}$ subject to the boundary conditions $u(r,\pi)=u(r,0)=0$ for $r>1$ and $u(1,\theta)=1$ ...
0
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1answer
77 views

Holder continuity using Sobolev imbedding

We assume for any $V\subset \subset U$ and $1<p<\infty$ $||u||_{W^{2,p}(V)}\le C(||\Delta u||_{L^p(U)}+||u||_{L^1(U)})$ for some $C=C(V,U,p)$. Given, $B=\{x∈R^3,|x|<1/2\}$ and we suppose $...
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1answer
40 views

Laplace Equation Boundary Problem

Solve the boundary value problem $$u_{xx}+u_{yy}=0, \ 0<x,\ y<1$$ $$u_x(0,y)=0, \ u(1,y)=0, \ 0<y<1$$ $$u(x,0)=1,u_y(x,1)=0, \ 0<x<1.$$ I have, $$\frac{\phi''(x)}{\phi(x)}=-\frac{h'...
2
votes
1answer
42 views

Fourier Series Coefficient

I am trying to review the basics. Find the Fourier series for the function $$f(x) =\left\{ \begin{array}{l l} 2x & \quad -\frac{\pi}{2}<x<\frac{\pi}{2}\\ 0 & \quad -\pi<...
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1answer
191 views

MATLAB, 1st order 2d hyperbolic equation, problem with convergence.

Follow up to my previous question: MATLAB: solving 1st order hyperbolic equation in 2 spacial dimensions The equation I'm solving has the form: $$f_t + A y f_x - B x f_y =0$$ I wrote the following ...
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0answers
33 views

Continuum Approach to Modeling Cell Proliferation and Differentiation (PDE)

I'm new here so I hope this post is appropriate. I recently read in a bioengineering textbook about an approach to model cell proliferation and differentiation. They proposed the following partial ...
4
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0answers
99 views

General form of a connection with zero curvature

I am looking for proofs of the following two theorems: Theorem 1. On a connected and simply-connected open set $\Omega\subset\mathbb{R}^3$, functions $L^p_{ij}\in C^1(\Omega)$ are given that satisfy $...
0
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1answer
37 views

D'Alembert Solution Formula

I have a test tomorrow and the only thing holding me back from getting a good grade is D'Alembert formula for boundary conditions. I have this example that I am trying to figure out. Find $u(\...
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0answers
409 views

Laplace equation on infinite strip

I'm trying to solve the following problem using the Fourier transform: $$u_{xx}+u_{yy}=0$$ on the domain $\;0\lt y\lt b$ , $-\infty\lt x \lt \infty \;$ with the following conditions: $$ u(x,0)= \...
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0answers
48 views

Solving the PDE $\frac{\partial u}{\partial t}=a\frac{\partial^2 u}{\partial x^2}+b\frac{\partial u}{\partial x}$

I am trying to solve the PDE $\frac{\partial u}{\partial t}=a\frac{\partial^2 u}{\partial x^2}+b\frac{\partial u}{\partial x}$ for constants $a$ and $b$ with conditions $\frac{\partial u}{\partial x}...
1
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1answer
71 views

Laplace equation in polar coordinates

Solve the Laplace equation in polar coordinates $u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0$ within the domain $0<\theta<\pi, 1<r<2$ subject to boundary conditions $$u(r,0)=0=u(...
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0answers
88 views

Measurability of solution of diffusion equation in sub-sigma algebra

I want to solve the following problem: Get $\omega \in \Omega \subset \mathbb{R}$, $x \in D \subset \mathbb{R}^2$ and $0<a_i\leq a(.,.)\leq a_x<\infty$. Let $a( x;. )$ and $f(x;.)$ be $\...
2
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0answers
69 views

Green function for gradient

Does a Green function for the gradient exist? Specifically, consider the equation $$\vec{\nabla}_x\, G(\vec{x},\vec{x}') = \vec{\delta}(\vec{x}-\vec{x}'),$$ where $\vec{\delta}(\vec{x}-\vec{x}')$ is ...
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1answer
55 views

Energy functional in Sobolev Space

Let $E[u]:=\int_W |\nabla u|^2 +V(x)u^2 dx$ be the energy functional on functions $u\in H^1_0(W)$ and $V\in L^\infty(W)$. Can someone help me show that $E$ is bounded below for all $u\in H^1_0(W)$ ...
2
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2answers
47 views

Range conditions on a linear operator

While reading though some engineering literature, I came across some logic that I found a bit strange. Mathematically, the statement might look something like this: I have a linear operator $A:L^2(\...
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1answer
37 views

Intermediate Forms Between Parabolic and Hyperbolic PDE (numerically)

Greetings MSE community, I have recently conducted some rudimentary experiments in matlab coding of PDE's. I have explicit and implicit numerical solutions to both the heat and the wave equation, for ...
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2answers
607 views

What is an alternative book to oksendal's stochastic differential equation: An introduction?

What is an alternative book to oksendal's stochastic differential equation: An introduction? But also An alternative that is over 300 pages and at the same level? Some professor refer that book as a ...
0
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1answer
31 views

Analytical solution to a second order PDE

Hey all here is my equation in a 2D system. $$\nabla^2u(x,y) = -\sin(\pi x)\sin(\pi y)$$ I haven't done anything like this in a while so could use a bit of guidance, how do I go about solving this ...
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1answer
74 views

One dimensional heat equation with multiple eigenfunctions

To solve the heat equation $u_{t}=u_{xx} \,\, (!)$ which is defined for $x\in[0,1]$ and $t>0$. I want to find the solution which satisfies the boundary conditions $u_{x}(0, t)=u_{x}(1,t)=0$ and ...
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1answer
91 views

Will this well enough to serve as a prerequisite to oksendal's book?

Will this well enough to serve as a prerequisite to oksendal's stochastic differential equations: an introduction with applications book? I refer to shiryeav's probability, but i guess it still miss ...
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1answer
18 views

Does anyone know how to solve this differential equation?

here is the equation: $\frac{\partial\alpha(r,\phi)}{\partial r}=\beta\sin\alpha(r,\phi)\cos\alpha(r,\phi)$ $r$, and $\phi$ are cylindrical coordinates. $\phi$ is the angle off the x-axis. So it ...
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1answer
436 views

MATLAB: solving 1st order hyperbolic equation in 2 spacial dimensions

According to wikipedia, the equation $$\psi_t + \mathbf{u} \cdot \nabla \psi =0$$ is hyperbolic. However, when I want to solve it in MATLAB using the pde toolbox (link), the general formula for a ...
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1answer
56 views

What happens to other eigenvalue? - steady state heat equation

A circular plate is bounded by circles of radii $r=2$ and $r=4$, its surface is insulated and temperatures along boundaries are given by $u(2,\theta)=10\cos\theta + 6\sin\theta$ and $u(2,\theta)=17\...
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0answers
104 views

Tricomi equation canonical form and solution.

Consider the Tricomi equation $$u_{xx}+xu_{yy}=0$$ à find thé canonical form but i did not solve it $$\left(v_{qq}+v_{rr}+\dfrac1{3r}v_r\right)=0.$$
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1answer
34 views

Calculus Problem___Prove $\lim_{x \rightarrow 0^+} u(x,t)=g(t)$ for any $t>0$

Given $g \rightarrow R$ continuous and bounded, let $$u(x,t)=\frac{x}{\sqrt{4 \pi}}\int_{0}^t \frac{1}{(t-s)^{3/2}}e^{\frac{-x^2}{4(t-s)}}g(s)ds$$. Prove that $\lim_{x \rightarrow 0^+} u(x,t)=g(t)$ ...
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0answers
39 views

How to obtain a general solution of a linear homogeneous first-order PDE?

Writing the equation as $$u_x+a(x,y)u_y+u(x,y)=0$$ I am wondering how to solve this generally. Is there any general method or solution exist?
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2answers
142 views

what is scattering theory?

I often read the the words "scattering theory", "scattering data", "scattering matrix", scattering XXX ... in my math lecture, but I realised that I am not able to define it correctly. A short search ...
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2answers
57 views

Intuition behind the definition of the d'Alembert operator

The d'Alembert operator is defined as $$\square^{2}=\frac{\partial^{2}}{\partial t^{2}}-\nabla^{2}$$ My question is what is the intuition behind this definition? My intuition tells me that we should ...
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0answers
81 views

Simple partial differential equation

I know this is a shame but I can't solve this simple partial differential equation. Can someone help me? $$\frac{\partial ^2F(x,y)}{\partial x \partial y}=h(x,y) F(x,y)$$
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1answer
44 views

Vibration on a rectangular Plate

I am trying to solve a problem that has been set for me. I haven't come across a problem like this like, so i need some help getting through it. It is used to model the vibrations of a rectangular ...
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0answers
90 views

Neutron density PDE

On Mathews and Walker's book exercise (8-2) We are given that the neutron density n inside $U_{235}$ obeys the differential equation $$\nabla ^2u+\lambda u=\frac{1}{k}\frac{\partial{n}}{\partial{t}} ...
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0answers
123 views

d'Alembert Solution of fixed end

in d'Alembert Solution for fixed end semi infinite string problem with wave equation $u_{tt} = c^2u_{xx}$,we get $0= \frac{f(ct)+f(-ct)}{2} + \frac{\int_{-ct}^{ct}g(s)ds}{2c}$ where $f$ and $g$ are ...