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

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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, ...
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20 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 ...
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45 views

Can you help me with this Non-homogeneous PDE Problem

The Problem: Solve the wave equation with time-independent sources if an equilibrium exists. Analyze the limit as t approaches $\infty$. If no equilibrium exists, explain why and reduce the problem to ...
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42 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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14 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 ...
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66 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 ...
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1answer
18 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 ...
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20 views

Suppose $\Omega\subset R^n$ is strictly convex. There exists a barrier function for each $\xi\in\partial\Omega$.

Suppose $\Omega\subset R^n$ is strictly convex. Then $\partial\Omega$ is regular in the sense there exists a barrier function for each $\xi\in\partial\Omega$. Basically, barrier function means there ...
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44 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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34 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 ...
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15 views

Two questions regrading to Laplace equation, the Green's Reconstruction Formula

All the following we use Evans notation. By Green's reconstruction formula, we could represent $u$ by $$ u(x)=\int_\Omega-\triangle u(y)G(x,y)dy-\int_{\partial \Omega}u(y)\partial_\nu ...
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1answer
18 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 ...
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69 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 ...
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12 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
40 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 $u\in H^1_0(W)$ ...
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28 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 ...
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1answer
29 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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52 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 ...
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1answer
22 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
29 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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44 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
15 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
36 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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28 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 ...
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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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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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21 views

Harnack Inequality in Evans PDE

I'm reading Evans' PDE Chapter 6 and have a question in the proof of Harnack inequality of second order elliptic equation. In page 352, Evans draws a conclusion (26): ...
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27 views

Solution of Laplace equation for a regular hexagon

I am trying to analytically solve Laplace equation in a regular hexagon. My equation is $\nabla^2 \phi=0$ and boundary conditions are : $\phi = 0$ (at the base of hexagon) $\phi= \phi_1$ $\phi= ...
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25 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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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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24 views

Existence of strong solutions to parabolic p-Laplace equation

Can I find a reference to where the existence of strong solution $u \in L^2(0,T;W^{1,p})$ with $u_t \in L^2(0,T;L^2)$ is proved to the equation $$u_t - \Delta_p u = f$$ $$u(0) = u_0$$ ...
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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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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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13 views

d'Alembert Solution of wave equation for open end problem

in d'Alembert Solution of wave equation for open end problem,i saw boundary condition as $u_x(0,t)=0$,but i didn't understand why this is like that,then i saw in one book as force at free end is ...
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58 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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22 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 ...
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1answer
42 views

D’Alembert Formula - PDE

Find $u(2,1)$ and $u(3.5,0.5)$ if $u$ solves $$u_{tt}=u_{xx}, \ 0<x<2, t>0$$ $$u(0,x)=x^2(2-x)^2, \ u_t(0,x)=x(2-x), \ 0\le x\le 2$$ $$u(t,0)=u(t,2)=0, \ t\ge0.$$ I can use D’Alembert ...
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24 views

$2\operatorname{D}$ heat equation with non-zero initial condition

I cant find anything about $2\operatorname{D}$ heat equation with non-zero initial condition, need to find the temperature of the rectangle.
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23 views

Does $a\ln(x^2 +y^ 2 )+b$ satisfy Laplace’s equation?

I can't verify that $F(x,y) = a\ln(x^2 +y^ 2 )+b$ satisfies Laplace’s equation ($F_{xx}+F_{yy}=0$). Here is what I did: \begin{align*} F_x &= \frac{2ax}{x^2 + y^2} &F_y &= ...
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1answer
36 views

Interesting counterexample of Strong Maximum Principle

Suppose $U=\Omega \cup (0,T)$ where $\Omega$ is a bounded domain. Let $u \in C_1^2 (U) \cap C(\overline U)$ satisfy $$u_t \le \Delta u+cu$$ in $U$ where $c \le 0$ is a constant. If $u \ge 0$, then ...
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26 views

Order of Integrating a partial derivative

I have some questions about the process involved when integrating higher order partial derivatives. I was going through a textbook on engineering mathematics on PDEs. If $\frac{\partial^2 ...
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34 views

What do the operator semi-groups have to do with PDE's?

Can anybody please help me to understand what does the semi-group do with partial differential equations? We started this subject very recently and we are now in the proof of Hille-Yosida Theorem, ...
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24 views

How to transform parabolic equation into heat equation?

Consider the parabolic equation: $$u_t-k(\Delta u+\sum\limits_{i=1}^n a_i\frac{\partial u}{\partial x_i}+bu)=0$$ where $a_i,b,k$ are constants and $k>0$. How this equation can be transformed to the ...
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70 views

Technical question about Strichartz estimate's proof.

I was studying the proof of Strichartz estimates from the book "Semilinear Schrödinger equation" of T. Cazenave. The proof is divided in several steps. Here we can assume ...
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Riemann-Green representation theorem

I'm trying to finish the proof of the following theorem: If $\Omega$ is an open, bounded set of $\mathbb{R}^n$, with a smooth boundary and $u \in C^2(\Omega)\cap C^1(\overline{\Omega})$ with $\Delta ...
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Solution of an IBVP (nonlinear McKendrick PDE)

Consider the following initial boundary value problem: $$u_t(x,t) + [g(x,P(t))u(x,t)]_x = -\mu(x,P(t))u(x,t)~~(1.1)$$ $$B(t)=g(0,P(t))u(0,t) = \int_0^{x_\infty} \beta(x,P(t)) u(x,t)\, dx~~(1.2)$$ ...
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Weak solution in Hilbert Space

How would you show that $u(x)=log|x|$ is a weak solution of $-\Delta u+cu=0$ for some $c(x)\in L_{weak}^{3/2}(B)$ and u is not bounded? I did take the derivative of u(x) and then its second ...
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38 views

$\frac{\partial^2 u}{\partial t^2}-c^2\frac{\partial^2u}{\partial x^2}=0\text{ implies } \frac{\partial^2 u}{\partial z \partial y}=0$

I have the following question: Show that if $z=x+ct$ and $y=x-ct$ then: $$\frac{\partial^2 u}{\partial t^2}-c^2\frac{\partial^2u}{\partial x^2}=0\text{ implies } \frac{\partial^2 u}{\partial z ...
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101 views

Explicit traveling wave solution for the diffusion equation

Find explicit formulas for $v$ and $\sigma$ so that $u(x,t)=v(x-\sigma t)$ is a traveling wave solution of the nonlinear diffusion equation $$u_t-u_{xx}=f(u)$$ where $$f(z)=-2z^3+3z^2-z$$ and ...
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1answer
67 views

Green's function of Dirichlet problem

Let $G(y,x)$ be the Green's function for the Dirichlet problem for Laplace equation on domain $\Omega$ with smooth boundary. Show that $$K(y,x):=\frac{\partial}{\partial \textbf{n}_y}G(y,x)\geq 0, ...