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

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4
votes
2answers
70 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 = ...
0
votes
0answers
9 views

Why Fuchs index $-1$ always there 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 ...
0
votes
0answers
7 views

Wave equation on an infinite domain with piecewise forcing term

Problem definition: PDE: $$u_{tt}-c^{2}u_{xx}=Q(x,t),\quad-\infty<x<\infty,\;t>0$$ BC: $$u(x,t)\rightarrow0\;as\;x\rightarrow\pm\infty,\quad t>0$$ IC: ...
0
votes
1answer
16 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 ...
1
vote
1answer
19 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 $t > 0$ ...
2
votes
1answer
498 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} ...
4
votes
1answer
89 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 ...
0
votes
0answers
25 views
+100

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 ...
0
votes
1answer
33 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 ...
0
votes
0answers
7 views

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 - ...
0
votes
0answers
9 views

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
votes
0answers
28 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 ...
2
votes
1answer
455 views

Conservation of momentum for nonlinear Schrödinger equation

I am having trouble proving the following momentum conservation law. Given smooth compactly supported solution $u(x,t)\in \mathbb{C}$ where $(x,t)\in \mathbb{R}^n\times \mathbb{R}$ of $iu_t+\Delta u= ...
5
votes
1answer
100 views

Connection of Fourier's work with Fredholm's

Im trying to formulate for myself in what sense Fredholms work on the Dirichlet problem is connected to Fouriers work on the heat equation. Fourier idea seems to have fundamental problems with ...
1
vote
1answer
15 views

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 ...
1
vote
1answer
19 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 ...
3
votes
0answers
9 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 ...
1
vote
0answers
22 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. ...
0
votes
0answers
12 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 ...
2
votes
2answers
46 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 ...
0
votes
0answers
25 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.
1
vote
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: ...
1
vote
0answers
25 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: $$\frac{\partial \theta}{\partial x}(x=0)=0$$ ...
0
votes
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} ...
0
votes
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
votes
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} + ...
1
vote
1answer
85 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
votes
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'' - ...
1
vote
0answers
23 views

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} ...
3
votes
2answers
46 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 ...
1
vote
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 ...
0
votes
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
votes
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
votes
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: ...
0
votes
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: ...
3
votes
1answer
33 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 ...
0
votes
0answers
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,\ ...
2
votes
0answers
32 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
votes
1answer
28 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
votes
2answers
48 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 $$ U=XY, \quad U_y=XY', \quad U_{xy}=X'Y' $$ $$ ...
7
votes
0answers
33 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
votes
2answers
82 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}+ ...
2
votes
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 ...
2
votes
0answers
28 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 ...
3
votes
0answers
366 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 = ...
1
vote
0answers
23 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 ...
0
votes
0answers
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, ...
1
vote
0answers
19 views

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, ...
7
votes
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 ...
1
vote
0answers
17 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 ...