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

learn more… | top users | synonyms (2)

1
vote
1answer
13 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 ...
3
votes
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 ...
0
votes
0answers
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 ...
1
vote
0answers
21 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
21 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 ...
0
votes
0answers
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 ...
0
votes
0answers
21 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.
3
votes
0answers
34 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
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 ...
1
vote
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: $$\frac{\partial \theta}{\partial x}(x=0)=0$$ ...
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} ...
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
1answer
75 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) ...
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
0answers
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 ...
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: ...
2
votes
0answers
28 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 ...
7
votes
0answers
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. ...
2
votes
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 $$ U=XY, \quad U_y=XY', \quad U_{xy}=X'Y' $$ $$ ...
0
votes
1answer
27 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
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 ...
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
18 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, ...
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: ...
1
vote
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 ...
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,\ ...
3
votes
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}+ ...
1
vote
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 ...
1
vote
0answers
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 ...
0
votes
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 ...
0
votes
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 $$ ...
1
vote
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 ...
0
votes
0answers
9 views

solution of the Heat equation on a closed manifold

Let $\rm(M,g_0)$ be a closed Riemannian manifold and let $(g(t),\phi(t))$ is the solution of the so-called List's flow, i.e. the following system of equations $$\left\{\begin{array}{11} ...
1
vote
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 ...
1
vote
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 ...
0
votes
0answers
34 views

Cauchy Problem for PDE $\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0$

Consider the Cauchy Problem of finding $u(x,t)$ such that $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0,x\in\mathbb{R},t>0$$ Which choices of the following functions of $u_{0}$ ...
1
vote
0answers
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 ...
0
votes
0answers
22 views

Solving large set of PDEs and verfication of solution

Please comment on solution of following system of PDEs, I have checked solution several times but could not find any error but there are problems when I proceed with solution obtained as below. I ...
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
68 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
votes
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 ...
0
votes
0answers
18 views

Resource of learning Fokas Method for linear PDEs

I want to study Fokas Method for solving linear PDEs, but I don't seem to find a good resource for that topic (Found some papers by Deconinck, which are difficult to follow). Is there maybe a resource ...
1
vote
1answer
15 views

Inequalities with negative Sobolev

In this paper I am read, it say that $||\triangle u||_{H^{-2}} \leq c||u||_{L^2}$, where $u$ solves the heat equation with zero boundary conditions on the boundary. I am still getting use to negative ...
0
votes
0answers
35 views

How do I get these equations ($n$th-order deformation equation and its initial/boundary conditions)?

Earlier, I asked a question What does 'equating the like-power of $q$ ' mean? and I already got the answer about the meaning of a particular phrase in a certain context. However, my real question ...
0
votes
0answers
14 views

Applying Boundary Condition to Finite Element Matrix

Several times now I have seen the following done without justification and I cannot figure out why it can be done: Consider the 1 dimensional "pde" $-u'' = f, u(0) = a, u(1) = b$ over $[0,1]$. We ...
0
votes
1answer
26 views

Concerning the proof of regularity of the weak solution for the laplacian problem given in Brezis

I am reading Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim brezis, and I am a bit confused about the proof. The theorem is stated as: Let $\Omega \subset ...
1
vote
0answers
19 views

computing the heat kernel for small times

The heat kernel on a two-dimensional manifold $M$ has the well-known expression $$H(p,q,t) = \sum_{i=1}^\infty e^{\lambda_i t}\phi_i(p)\phi_i(q)$$ where $\phi_i, \lambda_i$ are the eigenfunctions and ...
0
votes
0answers
30 views

On Atiyah-Singer Index theorem and Atiyah-Bott fixed point theorem.

I would like to understand the statement of Atiyah-Singer and Atiyah-Bott theorems enough to see lots and lots of applications they have. My background is pretty thin with some basic algebraic ...
0
votes
0answers
19 views

Showing the continuity of $\partial_{x_ix_j}v$ and $\partial^2_tv$

How to show that if $\begin{cases}\partial^2_tv(v,t;s)-\Delta v(x,t;s)=0\quad \text{for}\ (x,t)\in\mathbb R^2\times\mathbb R_{>s}\\v(x,s;s)=0,\quad\partial_tv(v,s;s)=f(x,s)\quad \text{for}\ ...
0
votes
1answer
31 views

Subsequence and diagonal process

We consider a sequence of functions défined on $\mathbb R^n$ by $f_m(x)=f(\frac{x}{m}),\ \forall m\in \mathbb{N}$ such that : 1) $f=1 $ in $B(0,1)$ 2) $\mathrm{supp\,} f\subset B(0,2)$ 3) $f ...