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

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Active contours and Chan-Vese algorithm

In the paper Chan, T. F., & Vese, L. A. (2001). Active contours without edges. Image processing, IEEE transactions on, 10(2), 266-277., the author minimizes the object function $$ F_\varepsilon(...
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quasi linear second order PDE.

I'm trying to solve $$\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} +4 u +2 \sin^2x,\quad x\in (0,\pi)$$ With the codition $ \frac{\partial u}{\partial x}(0,t)=\frac{\partial ...
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Is the naive solution of this PDE unique?

Problem statement Suppose I have a 2D or 3D equation of the form: $\vec{\nabla} \cdot \left[ \vec{\vec{a}}\left(\vec{x}\right) \cdot \vec{\nabla} f\left(\vec{x}\right) \right] = \vec{\nabla} \cdot \...
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Question concerning Gauss' Theorem

$\begin{cases}-\Delta u=0 & in\ B(0,2)\\u(x,y)=3xy+2 & on \ \partial B(0,2)\end{cases}$, then compute $u(0)$ I have the solution (skip the first $4$ lines, it is the justification of the ...
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Neumann boundary condition, spherical shell.

The velocity of a fluid $\mathbf{u}$ is assumed to have the velocity potential $\Phi$ such that $\mathbf{u}= \nabla \Phi$. The fluid is contained in a rigid shell, of radius $a$, which is moving with ...
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Solve first order partial diferential equation

Consider the Cauchy problem: $$\left\{\begin{array}{lll} x^2\partial_x u+y^2\partial_yu=u^2\\ u(x,2x)=1 \end{array}\right.$$ It is easy to show that the characteristic equations are given by: $$\frac{...
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Why is $\int_{0}^{r}\left(\int_{\partial B(x,s)}u\ dS\right)ds=u(x)\int_{0}^{r}n\alpha(n)s^{n-1}ds$

$\int_{B(x,r)}u\ dy=\int_{0}^{r}\left(\int_{\partial B(x,s)}u\ dS\right)ds=u(x)\int_{0}^{r}n\alpha(n)s^{n-1}ds=\alpha(n)r^nu(x)$ This appears in the proof of a theorem in the book of Evans PDE. I ...
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Solve non-linear pde

so i wonder the next thing : if i consider a pde like heat equation Fourier transform works very well. Now, if i consider this equation : $\frac{\partial u(t,x)}{\partial t}-k(u(t,x))\frac{\partial^2 ...
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domain of the fractional laplacian operator

Let $L$ be an operator on $C^2(\mathbb R)$, defined by $$L \phi (x) = \int_{|y|<1} (\phi(x+y) - \phi(x) - \phi'(x) \ y)\ \nu(dy), \quad \forall x\in \mathbb R$$ for a measure $\nu(dy) = |y|^{-2} dy$...
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Is it possible to find $g(\kappa)$ in this equation

I have ran into the following integral equation as part of my research. For $\xi = (\alpha\theta)^{1/\alpha}$ and for all $\theta>0$. I have the following equality $$\int\limits_0^\infty g(\...
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Proof - If one domain $D$ is contained in another domain $D'$, then $\lambda_n \leq \lambda_n'$

I would like to understand the proof of the theorem $4$ of the book page $326$. In fact, after a good while of trying to understand that proof, I am not even sure how it works. In being clear, does ...
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71 views

Classification of solution of non-linear first order partial differential equation

A solution of the PDE $xu_x+yu_y+(u_x)^2+(u_y)^2-u=0$ represents an ellipse in the $x$-$y$ plane. an ellipsoid in the $xyu$ plane. a parabola in the $u$-$x$ plane. a hyperbola in the $u$-$y$ plane....
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If $u_{k|_{\Omega}} \to u$ in $W^{1,p}(\Omega)$ with $(u_k) \subset C_c^{\infty}(\mathbb{R}^n)$ then $u_k \to u $ in $W^{1,p}(\bar{\Omega})$

I have already proven that for every $u \in W^{1,p}(\Omega)$ with $1 \leq p < \infty$ and every open set of class $C^1$ there exists a sequence $(u_k) \subset C_c^{\infty}(\mathbb{R}^n)$ such that ...
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What is the definition of $f \in C^{\infty}(\overline{U})$?

I am reading Evans' Partial Differential Equations, Chapter $5$. I am concerned about the notation $f \in C^{\infty}(\overline{U})$, for an open set $U$. Does it mean $f$ is $C^{\infty}(U)$ and that $...
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Solution of the transport equation is of exponential order

Consider the transport equation $\begin{cases}u_{x}(x,y)+u_{y}(x,y)=0,\ x\geq{}y\geq0\\ u(x,0)=f(x),\ x\geq0,\end{cases}$ where $f\in{}C^{1}([0,\infty),\mathbb{R})$. Q1. Without solving the ...
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How to solve this partial differential equation $\partial_s^2 y + \partial_ty + y = 0$?

I'm working on the following exercise on multiple-scale perturbation of $\varepsilon y'' + y' + y = 0$ with $y(0)=0, y'(0)=1$ and the following PDE came up (we consider $y = y(t,s)$ because of ...
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86 views

Is there a way to solve this heat equation IBVP

I have ran into the following Heat equation IBVP, but I am not quite sure how to solve it as it has these time dependent boundary conditions $$ v_t = kv_{xx} \ \ \ \ \ \ ( 0 \le x \le \infty, \ \ ...
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What does the solution to the transport equation describe?

I'm studying the transport equation $$u_t + cu_x = 0$$ Where c is a constant. What does the solution, $u(x,t)$, describe? Is it ... the height of the wave? The location of the wave? Does the ...
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Is there any other method to solve this Cauchy problem for wave equation in 3D?

I want to solve this Cauchy problem for wave equation in 3D: \begin{align*} &u_{tt}=4\Delta_3 u(x,t), \qquad x=(x_1,x_2,x_3)\in\mathbb{R}^3, \quad t>0,\\ &u(x,0)=(x_2x_3)^2, \quad u_t(x,0)...
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General solution of a linearized PDE of second order

$$\frac{\partial u}{\partial t}=\Delta u+\gamma f(u,v)\text{ for }x\in\Omega, t\geq 0$$ $$ \frac{\partial v}{\partial t}=d\Delta v +\gamma g(u,v)\text{ for }x\in\Omega,t\geq 0 $$ with Neumann ...
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60 views

An identity related to Poisson's equation in 3D

$r'=\sqrt {(x'-x)^2+(y'-y)^2+(z'-z)^2}$ $\nabla = \frac {\partial}{\partial x'}+\frac {\partial}{\partial y'}+\frac {\partial}{\partial z'}$ I was studying Poisson's equation in 3D see this link ...
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Laplace transform to solve a pde

i have to find the laplace transform of this : $$f(u(x,t))\frac{\partial u(x,t)}{\partial t}$$ I have no idea how to solve this... Thank you for your help.
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Why Fuchs index $-1$ always there during singularity analysis for ODEs?

During singularity analysis of ODE/PDE I have seen that $-1$ always occur as default resonance, someone told me that this is actually Fuchs index and Fuchs index is always $-1$ for ODE/PDE. Can anyone ...
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25 views

Solving A Partial Differential Equation Using Separation of Variables

I am having issues with this problem. I am asked to use the method of separation of variables to solve this: $u^\prime(t) = k u^{\prime\prime} (x)$ for $0 \leq x \leq L$ and $t > 0$ $u^\prime(x)(...
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Explicit heat kernels

For quite general domains, the Dirichlet heat kernel has a representation via the eigenfunctions of the corresponding Dirichlet problem. This form is usually not easy to analyse so I was wondering - ...
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the heat equation for mappings between closed Riemannian manifolds

Let $M$ be a closed (smooth) Riemannian manifold. Then we have the following existence and uniqueness theorem for the heat equation on $M$, which is considered more or less a standard result: Let $0&...
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Heat Equation IBVP on the Quarter plane

I have come across the following Heat equation IBVP but I am not quite sure how to solve it: $$v_t = kv_{xx} \ \ \ \ \ \ ( 1 < x < \infty, \ \ 0 < t < \infty) ,$$ $$ v(x,0) = \delta (x ...
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Imposing boundary conditions AND self-similarity on a PDE

I have a PDE in the form $$u_t=F(u,u_x)$$ where the unknown is $u(x,t)$ on say $\mathbb R\times[0,\infty)$. $F$ is very nonlinear so I was told to assume self-similarity in the form $$u(x,t)=t\tilde u(...
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Dirichlet-problem in one eights of the plane

I would like to solve this problem: Let $\Omega = \lbrace{ (x,y) \in \mathbb{R}^2: 0<x<y\rbrace }, f \in C_{c}(\Omega) $. Find the solution of $ \Delta u =f \text{ in } \Omega\\ u=0 \text{ on ...
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Boundary perturbation (wave equation)

I have the following problem, \begin{equation} u_{tt} - \Delta u = 0, \, \, \text{with } x \in R \subset \mathbb{R}^N, \end{equation} \begin{equation*} u = 0 \, \, \text{at } \partial R. \end{equation*...
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Showing $|I(\lambda)|\le C\lambda^{-N}$

Let $\lambda\in\mathbb R$ and $I(\lambda)=\int_{\mathbb R^n}e^{i\lambda\phi(\xi)}a(\xi)d\xi$, where $a\in C_c^{\infty}(\mathbb R^n)$ and $\phi\in C^{\infty}(\mathbb R^n)$, and assume $D\phi$ does not ...
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How Coulomb Gauge guarantees uniqueness in regard to Lax-Milgram Lemma, curl-curl problem

The Lax-Milgram lemma gives insight on existence and uniqueness of a PDE of the type $$ a(u,v)=f(v) $$ Positive definiteness and coercivity are required for the bilinear form $a(u,v)$. In the curl-...
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Are harmonic functions ($\Delta u=0$) with compact support unique?

Does anyone how to solve the problem $$ \Delta u=0$$ with $u \in R^n$ and $u(x) \to 0$ as $|x| \to \infty$? Is $u=0$ the only solution? Many thanks.
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Why does the coordinate transformation from Cartesian coordinates leads to an additional term in the biharmonic operator in spherical coordinates

I am trying to solve a problem in physics where the biharmonic operator is involved. I think that the bihahmonic operator can be obtained by taking twice the Laplace operator, such that $\nabla^4 f = \...
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How to solve $(\partial_xu)^2\partial_{xx}u+2\partial_xu\partial_yu\partial_{xy}u+(\partial_yu)^2\partial_{yy}u=0$

How to solve $(\partial_xu)^2\partial_{xx}u+2\partial_xu\partial_yu\partial_{xy}u+(\partial_yu)^2\partial_{yy}u=0$ in $\mathbb R^2$ If I write $(\partial_x+\partial_y)^2u=\partial_{xx}u+2\partial_{...
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Problem solving a PDE using separation of variables and Fourier expansion

I am trying to solve the heat equation: $$\frac{\partial \theta}{\partial t}=\frac{\partial^2 \theta}{\partial x^2}$$ The boundary conditions are: $$\frac{\partial \theta}{\partial x}(x=0)=0$$ $$\...
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Change of variables in Power series

Hello StackExchange community, this is probably a dumb question, but suppose you have a function that have the following power series $f= \sum_{i = 0}^{\infty} f_i r^i$. where $f:\mathbb{R} \...
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Method of Characteristics for $(1+x^2)u_x+2u_y=y\cdot(1+u^2)$

Suppose I had a problem like $$(1+x^2)u_x+2u_y=y\cdot(1+u^2),\;\; u(1,y)=1$$ This is of the form $$a(x,y)u_x+b(x,y)u_y+c(x,y,u)$$ So I would use the Method of Characteristics: $$\frac{dx}{dt}=(1+...
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Solution of an initial value problem (MCQ) (CSIR DEC 2015)

The solution of the initial value problem $ (x-y) u_{x} + (y-x-u) u_{y} = u $ with the initial condition $u(x,0) = 1$ satisfies $ u^2(x-y+u) + (y-x-u) = 0$ $ u^2(x+y+u) + (y-x-u) = 0$ $ u^2(x-y+u) -...
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Method of mirror charges applied to diffusion equation

The equation $\frac{\partial f}{\partial t} = \frac{\partial^2f}{\partial x^2}$ has the fundamental solution (in one dimension) $f(x,t) = \frac{1}{2\sqrt{t}}\exp (-x^2/4t)$ if there are no boundary ...
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Help with integral from Boltzmann equation

I have a function $$ g\left(x,v,t\right) = u\left(x,t\right)\cdot v + \theta\left(x,t\right)\frac{1}{2}\left(\left\lvert v\right\rvert^2 - 5\right) $$ where $g(x,v,t),\theta(x,t)$ are scalars and $u(...
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Dealing with the logarithm of absolute value when solving $u_t+xu_x=1$ by method of characteristics

I was trying to come up with a solution to the PDE IVP problem $u_t + x u_x =1$, $u(x,0) = f(x)$. Here's how I went about it: We can the following relations by applying the Method of Characteristics: ...
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Eigenvectors of matrix equation $AX+XB^\text{T}=\lambda (CX+XD^\text{T})$

We have the matrix eigenvalue problem $$AX+XB^\text{T}=\lambda (CX+XD^\text{T})$$ Where $\lambda$ is the eigenvalue, $X$ is $m\times n$ and plays the role of the eigenvector, and $A$, $B$, $C$, and $...
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How can one tell if a PDE describes wave behaviour?

I have been looking at a lot of different non-linear PDEs which describe waves lately and have come to the realisation that I don't know what it is about these PDEs that make them behave like waves. ...
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Solving inhomogeneous PDEs when you can't separate variables

$$4+U_y - U_{xy} =0, \quad U(x,0)=0, \quad U_y(0,y)=3y^2 .$$ Usually I can solve these kind of problems with separation of variables, so I tried $$ U=XY, \quad U_y=XY', \quad U_{xy}=X'Y' $$ $$ \...
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weak solution via Galerkin approximation

I am reading Evan's approach to Existence of weak solution via Galerkin method in some PDE notes. I have difficulty understand the following assumption i.e. Let "$\{w_k\}_{k=1}^\infty$ be an ...
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Hadamard counterexample Dirichlet-problem

Good afternoon, I try to understand the follwoing counterexample of Hadamard: The classical solution of $ \begin{cases} \Delta u = 0 ~~\text{ in }\Omega\\ u=g \text{ auf }~~ \partial \Omega \end{...
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$L^2$ and $C$ solutions of an initial-boundary value problem for 4$^th$ order equation

I study the initial-boundary value problem \begin{equation} \alpha^2\frac{\partial^4 w^0}{\partial x^4}+\frac{\partial^2 w^0}{\partial t^2}=P(t)\delta\left(x-\xi\right),~~ 0<x,\xi<1,~ t>0, \...
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Determine the equilibrium temprature [closed]

By solving the heat equation determine the equilibrium temperature distribution for the circular ring $\theta\in[0,2\pi]$ by both (a) directly setting $u_t=0$, and finding the equilibrium solution, ...
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Stuck trying to solve a PDE by method of characteristics

I've been trying to solve the inhomogeneous PDE IVP $u_t+c u_x = e^{2x}$; $u(x,0)=f(x)$, but got stuck. Would appreciate some help. Here's what I did by trying to use the method of characteristics: $...