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

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1answer
18 views

characteristics in a simple region

For the system of equations describing a compressible fluid $\frac{\partial \rho}{\partial t} + u \frac{\partial \rho}{\partial x} + \rho \frac{\partial u}{\partial x} = 0\\ \frac{\partial ...
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0answers
7 views

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 ...
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10 views

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 ...
3
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0answers
31 views

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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18 views

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. ...
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20 views

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 ...
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7 views

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 ...
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2answers
44 views

Scale invariant definition of the Sobolev norm $\|\|_{m,\Omega}$ for $H^m(\Omega)$

I learned the following from Constantin and Foias's Navier-Stokes Equations (Chapter 4): We say that a function of a bounded open set $\Omega\subset\mathbb{R}^n$, $c(\Omega)$, is scale invariant ...
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18 views

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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1answer
21 views

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 ...
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1answer
29 views

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: ...
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0answers
23 views

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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0answers
18 views

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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1answer
24 views

Factoring $R(ry')'-y(rR')'=[r(Ry'-R'y)]'$

In a problem this formula was used and I'm not seeing how this factor using the chain rule was derived. Other than calculating the derivative of the two that someone else already solved and showing ...
2
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1answer
696 views

Numerical techniques for solving systems of first-order semi-linear hyperbolic PDEs in two variables

I need to solve such systems of PDEs numerically and I'm wondering what the "standard" method (if there is one) is. The systems I'm interested in have the form: $\frac{\partial F_i}{\partial t} + ...
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1answer
74 views

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) ...
2
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2answers
89 views

Solving a Second Order PDE

I'm trying to solve the equation $u_t = \alpha^2 U_{yy}$ given $u(y,t)$ bounded $y \rightarrow\infty$ and $u(0,t) = U_o e^{iw_ot}$. Initial is $u(y,0) = 0$. I have gotten both separations as $Y'' - ...
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FitzHugh–Nagumo system with diffusion

I was studying the FitzHugh-Nagumo model with diffusion and I quite do not understand the meaning of it. If we consider the system without diffusion, \begin{equation}\label{FHN}\begin{cases} ...
2
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1answer
67 views
+200

Solving or knowing something about a non-linear PDE which is “almost” linear?

Let $a>0$ be fixed. I have the following PDE: $u=u(t,x)$, $t\in [0,1]$, $x\in \mathbb{R}$, $$-\partial_t u = |\partial_x u| + \frac{1}{2}\partial_x^2 u, \quad ...
3
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2answers
44 views

How can I actually solve this kind of partial differential equations?

$$ x \frac{\partial z}{\partial x}+t \frac{\partial z}{\partial t}+y \frac{\partial z}{\partial y}=xyt$$ I can see that one soultion for this equation is $$z=(1/3)xyt+ C$$ however how can one solve ...
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1answer
15 views

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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1answer
23 views

First order linear pde

Let $xyu=C_{1}$ and $ x^{2}+ y^{2}-2u= C_{2}$, where $c_{1}$ and $c_{2}$ are arbitary constants be the first integrals of the pde $$ x(u+y^2)\frac{\partial u}{\partial x}- y (u+x^{2})\frac{\partial ...
2
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2answers
91 views

Find a solution that satisfies Laplace's equation in polar coordinates

How may I find a solution that solves Laplace's equation in polar coordinates, subject to the boundary conditions? In particular, I need to find one solution that satisfies $$\Delta u = 0,$$ subject ...
2
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1answer
42 views

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: ...
2
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1answer
488 views

Show $\nabla^2g=-f$ almost everywhere

let a continuous function $f(x,y,z)$ be absolutely continuous over every bounded region and let it be in $L^1$ that is $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} ...
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1answer
17 views

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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15 views

Help with integral from Boltzmann equation

I have a function $$g(x,v,t) = u(x,t)· v + θ(x,t)\frac{1}{2}(|v|^2 - 5)$$ where $g(x,v,t),\theta(x,t)$ are scalars and $u(x,t),v∈ \Bbb R^N$, $N=2,3$. I also have a matrix valued function $X=X(v)∈\Bbb ...
3
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1answer
32 views

Compute the characteristic strip and conoid solution of a geometric PDE

This is a problem from Fritz John's Partial Differential Equations, which I'm working through for self-study. Given a family of spheres of radius $1$ with centers in the $xy$-plane ...
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20 views

How to solve this mixed pde/finite-difference equation?

I have the following mixed pde/finite-difference equation for $f(t,x,y)$: $a x^2 f_{xx} + bxf_x + f_t - bxy + ce^{d\delta}-re^{-s\delta} = 0$ subject to $f(T,x,y)=0$, $x>0,\ t\geq 0,\ ...
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0answers
26 views

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 ...
0
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1answer
26 views

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 ...
2
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2answers
44 views

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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32 views

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. ...
3
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2answers
79 views

Solution of partial differential equation

Solve the differential equation, $$ z=\frac{\partial z}{\partial x}x + \frac{\partial z}{\partial y}y+ (\frac{\partial z}{\partial x})^2 + \frac{\partial z}{\partial x}\frac{\partial z}{\partial y}+ ...
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1answer
37 views

Confusion with changing variables in second order DE

So in my physics assignment, we're given Schrodinger's equation: $$\frac{-\hbar^2}{2m}\frac{d^2\psi}{dx^2}+e\xi x\psi=E\psi$$ We're asked to substitute a function of the form $w(x)=Ax+B$ to arrive at ...
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0answers
26 views

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 ...
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363 views
+50

Change in neumann boundary conditions through coordinate transformation of elliptic PDE, weak formulation

The standard weak formulation of the Neumann problem for the Poisson equation is to find $u \in H^1 ( \Omega)$ such that for every $v \in H^1 ( \Omega)$: $$ \int_{\Omega} \nabla u \nabla v d x = ...
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22 views

Compact resolvent proof

I want to prove the following proposition: We consider the operator $$A=A_V=-\Delta+\frac{1}{4}|\nabla V|^2-\frac{1}{2}\Delta V$$(where $V$ is a polynomial) with domain ...
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7 views

$L^2$ 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 [on hold]

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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2answers
222 views

Why is the integral of $\|\nabla f\|^2$ called the energy of $f$?

Let $\Omega$ be a region in $\mathbb{R}^2$ with $f:\Omega \to \mathbb{R}$ a smooth function. Why is the quantity, $$ \tfrac{1}{2} \iint_{\Omega} \|\nabla f\|^2 $$ Called the "energy" of $f$? I am ...
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0answers
16 views

Method of characteristics - integral surface formation

On page 2 of this PDF from Standord, which describes the Method of Characteristics for first-order PDEs, it is written at the end of the page: "In doing so, we see that $z(x,t)$ is constant along ...
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1answer
18 views

Particular integral of PDE.

The PDE $$\frac{\partial ^{2}u}{\partial x^{2}}+2\frac{\partial^{2}u}{\partial x\partial y}+\frac{\partial^{2}u}{\partial y^{2}}=x$$ Has $1.$ Only one particular integral. $2.$ a particular integral ...
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15 views

Necessary condition for local existence of solution for system of two first order PDEs

Let $f,g$ be two smooth functions on $\mathbb{R}^2$ and $U,V$ be two smooth vector fields on $\mathbb{R}^2$. What is the sufficient condition for local existence of a solution $\phi$ of equations ...
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1answer
18 views

Nonlinear first order pde with both IC and BC

I am trying to solve the first order problem, namely: $$ \begin{cases} u_{t}+uu_{x} = 0, \hspace{0.5cm} x>0, t>0 \\ u(0,t)=t, \hspace{0.5cm} t>0 \\ u(x,0)=x^2, \hspace{0.5cm} x>0 ...
1
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1answer
36 views

Solution of a sublinear elliptic problem.

In a lecture notes, the author showed the problem $\tag{$P$}$ $\begin{cases} -\Delta u = |u|^{q-2}u \textrm{ in } \Omega, \\ u(x)= 0 \textrm{ in } \partial\Omega, \end{cases}$ where $\Omega ...
2
votes
2answers
77 views

Cauchy's problem. Equation of mathematical physics

$$U_{tt} = \Delta U + x^3 - 3xy^2$$ $$U|_{t=0} = e^x \cos y$$ $$U_t|_{t=0} = e^y \sin x$$ Help me, please, with solution of this equation. Can you prompt me algorithm to find the ...
1
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1answer
39 views

First eigenvalue of laplacian

I know the laplacian $\Delta$ has only positive eigenvalues, but why there is a first one? Assume $\Delta$ is acting on an appropriate set of real valued functions on the bounded domain $\Omega ...
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0answers
10 views

Is the derivative product rule true for Bochner spaces?

If $u\in L^{2}(0,T; L^{2}(\Sigma))$ with $u_{t}\in L^{2}(0,T; H^{-1}(\Sigma))$ and $v\in L^{2}(0,T; H^{1}_{0}(\Sigma))$ with $v_{t}\in L^{2}(0,T; L^{2}(\Sigma))$ is it true that $$ ...
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0answers
27 views

pde of probability density function [on hold]

I have a question about the time derivative of the probability density function, assuming a density function,$\rho(S,T|S_t,t)$, if I calculate the derivative with respect to T, that is the Fokker ...