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

learn more… | top users | synonyms (2)

2
votes
1answer
39 views

Solutions of nonlinear system of PDE's.

During my work, I've faced with the following system of differential equations: \begin{cases} \frac{\partial^2f(x,y)}{\partial x^2}=-(x-y)(\partial_xg)^2\\\frac{\partial^2f(x,y)}{\partial y^2}=-(x-y)(...
2
votes
0answers
50 views

If $f=f(x,y)$, are there any theorems on $\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}$?

I am solving a bunch of differential equations to find the most general form of some functions satisfying the differential equations. I have arrived at my last differential equation which is: $$\...
2
votes
0answers
57 views

spectral theory question

Consider the operator $A=D_p^2+ip$, where $Dp=-i ∂_p$, and the domain of A is $$D(A)=\{u \in L^2(R,dp) : Au \in L^2(R,dp)\}.$$ Using the fact that $$ \|Au\|^2=\|D_p^2u\|^2 +\|pu\|^2 +2\langle u,...
2
votes
1answer
38 views

What is abstract finite element method?

Just curious if someone knew this answer. I know for conformal finite element method that the basis functions and the hat functions are from the same function space. I would conjecture that abstract ...
2
votes
0answers
34 views

Application of maximum principle for heat equation

I have the heat equation: $$u_t (x,t) -ku_{xx}(x,t)=0 \quad 0<x<L,0<t$$ $$u(x,0)=\phi(x) \quad 0 \leq x \leq L$$ I want to show that if $\phi(x)=0$ then using the maximum/ minimum principle ...
2
votes
1answer
48 views

Solving a Neumann problem in a disk

$\Delta u(x,y) = x^2 $ $x^2 + y^2 <9$ On the boundary $x^2 + y^2 = 9$, $\frac {\delta u}{\delta n} = y$. I've seen similar problems solved in texts (Strauss, Bleecker, Stavroulakis), but where $...
2
votes
1answer
21 views

Solving an Inhomogeneous $1$st Order PDE using Method of Characteristics

I wanna solve the equation $$u_x+u_y+u=\exp(x+2y), \quad u(x,0) = 0$$ I have just learned method of characteristics. But I don't know how to deal with $u$ term and inhomogeneous term simultaneously. ...
2
votes
1answer
24 views

Cauchy Problem (Waves with a Source)

Solve: $$u_{tt}=c^2u_{xx}+x t,\quad u(x,0)=0, \quad u_t(x,0)=0$$ The final answer should be $u=xt^3/6$. I keep getting $xt^3/2$. How I did the problem: 1/2[phi(x)+phi(-x)]+1/2c int(x+ct,x-ct, 0) dy ...
2
votes
0answers
23 views

Lebesgue Space/Bochner Space interpolation Theorem

I need the embedding, for $I\subset\mathbb{R}$ is a bounded intervall and $\Omega\subset\mathbb{R}^n$ is a bounded domain, $$L^{q_1}(I;L^{p_1}(\Omega))\cap L^{p_2}(I;L^{p_2}(\Omega))\hookrightarrow L^...
2
votes
0answers
43 views

An inverse problem for a parabolic equation and fixed-point theorem

I'm reading the paper "The determination of a parabolic equation from initial and final data" http://www.ams.org/journals/proc/1987-099-04/S0002-9939-1987-0877031-4/S0002-9939-1987-0877031-4.pdf. A ...
2
votes
1answer
37 views

d'Alemberts's solution on a semi infinite domain with a strange boundary condition

I want to use d'Alembert's solution of the wave equation to find the solution of $$\frac{\partial ^2 u}{\partial x^2} = \frac{\partial ^2 u}{\partial t^2} \qquad (0\leq t, 0\leq x)$$ $$\frac{\partial ...
2
votes
0answers
46 views

How proper is it to turn an ODE into a wave equation?

I have seen the following method used a few times for finding solutions of wave equations. Take an ODE with a known solution, of the form $y''(x) + g(y(x), x) = 0$ Switch it to a wave equation of ...
2
votes
0answers
19 views

Method to Linearise PDE

I have a Monge-Ampere-type PDE I wish to solve using a finite difference method: $$(1-u_{xx})(1-u_{yy}) -u_{xy}^2 = f(x,y).$$ Is there generally a preferred method for linearising the system after ...
2
votes
2answers
63 views

$u_m\rightharpoonup u$ in $L^2(0,T;H)$ and $u'_m\rightharpoonup v$ in $L^2(0,T,H^*) \longrightarrow u'=v$

Asuume $H$ a Hilbert space, $u_m\rightharpoonup u$ in $L^2(0,T;H)$, and $u'_m\rightharpoonup v$ in $L^2(0,T, H^*)$. Does this imply $v=u'$? This is something I've been wrestling with and some online ...
2
votes
1answer
18 views

Crank Nicolson scheme approximating $\frac{\partial V}{\partial t}$

I am looking at Crank Nicolson scheme for solving the Black-Scholes equation. I have that $V$ is an option price. When I approximate $\frac{\partial V}{\partial t}$ I get some weird result that is not ...
2
votes
0answers
24 views

Hilbert space and traces

Let $\Omega$ be the open unit ball in $\mathbb{R}^n$, and $\Gamma := \Omega \cap \{x_n=0\}$. Let $\Omega_1 = \{ x \in \Omega: x_n > 0 \}$ and $\Omega_2 = \{ x \in \Omega: x_n < 0 \}$. Define $...
2
votes
0answers
69 views

Visualizing the evolution of a Riemannian metric

I'm doing some reading into Riemannian geometry and PDEs and I have the following question. When we evolve a Riemannian metric (by say the Ricci flow) we are evolving a bilinear form on a manifold $\...
2
votes
0answers
22 views

Parabolic Regularity [closed]

A have a following problem. $u_t-\Delta u=f$ where $f\in L^{\infty}(\Omega\times(0,T))$ and $\Omega\times(0,T)$ is a limited domain. I want to know why $u\in C^{\infty}$. Actually result which ...
2
votes
0answers
28 views

Minimization problem associated with $\int_0^1 u'(x)v'(x)\,dx + \int_0^1 {{u(x)v(x)}\over{x^2}}\,dx = \int_0^1 {{f(x)v(x)}\over{x^2}}\,dx$?

See here. Set$$V = \{v \in H^1(0, 1) : v(0) = 0\}.$$Given $f \in L^2(0, 1)$ such that ${1\over x}f(x) \in L^2(0, 1)$, does there exist a unique $u \in V$ satisfying$$\int_0^1 u'(x)v'(x)\,dx + \...
2
votes
2answers
166 views

An exercise on Linear algebra in PDE

I'm struggling to show some exercise given in the PDE book of Krylov(Lecture s on Elliptic and Parabolic Equations in Sobolev Spaces. Exercise 1.4.7. Let $A=(a^{ij})$ and $U=(u_{ij})$ be $2\times ...
2
votes
2answers
32 views

Diffusion equation on the positive half-line with Dirac delta-function boundary condition.

The diffusion equation $u_{t} = k^{2}u_{xx}$ has initial boundary conditions $u(x,0) = A \delta(x - x_{0})$, $u(0,t) = 0$ where $A\neq0$ and $x_{0} > 0$ are given constants, and $\delta(\cdot)$ is ...
2
votes
2answers
43 views

Solve eigenvalue problem $(\frac{u'}{x})'+\frac{\lambda}{x}u=0$

Consider the eigenvalue $(\frac{u'}{x})'+\frac{\lambda}{x}u=0$,$x\in (1,2)$. And $u(1)=u(2)=0$. I want to determine the sign of the eigenvalues first. But since it is not a standard eigenvalue ...
2
votes
1answer
69 views

What is the intersection of all Sobolev spaces of square integrable functions?

Let $U \subset \mathbb{R}^n$. If $$H^k=\{f: U \rightarrow \mathbb{R}: D^\alpha f \in L^2(U)\ \forall \alpha \in \mathbb{N}^n \ \text{with} \ \vert \alpha \vert \leq k \}$$ then how do I show that $$...
2
votes
0answers
54 views

Does continuity imply weak differentiability?

I have recently been reading about weak derivatives. I have found few examples of only weakly differentiable functions and they were all continuous. Is there an example of a continuous function which ...
2
votes
0answers
39 views

A priori Estimates of PDE (burgers equation)

How can I find a-priori estimates for u, $u_x$ and $u_{xx}$ which do not depend on time? The two independent PDEs that I would like to find these estimates for are provided below: $u_t + u^2u_x - \...
2
votes
0answers
43 views

Summation of series involving $\sinh$ of a square root

In a physics problem that I assigned in one of my classes recently, the following series arises: $$ S = \sum_{\text{odd } n} \sum_{\text{odd } m} \frac{(-1)^{(n+m)/2}}{nm} \frac{\sinh( \pi \sqrt{n^2 + ...
2
votes
0answers
19 views

If the source term in an elliptic PDE is piecewise polynomial, does that mean that the solution and it's derivatives are also piecewise polynomials?

Let $\Omega\subset \mathbb{R}^d$ be an open bounded (possibly convex) domain. Suppose that $\Omega$ is divided up unto $N$ non-overlapping domains $(K_i)_{i=1}^N$ Suppose $f:\Omega\to \mathbb{R}$ is ...
2
votes
0answers
29 views

Uniqueness of weak solutions and the Riesz Representation Theorem

A common technique to show existence of weak solutions to the problem $Lu=f$ is to obtain an energy estimate of the form: $$\| u\|\leqslant\| L^{*}u\| $$ and the define the functional $$k(L^{*}v)=\...
2
votes
0answers
47 views

Neumann Problem - how to prove that a weak solution is also a classic one?

I think a similar problem appears in Evans' book: For a given Neumann problem, i.e. -$\nabla^2 u=f$ in $\Omega$, $\partial{u}/\partial{\nu}=0$ on $\partial \Omega$ where $\Omega$ is a bounded domain ...
2
votes
1answer
71 views

Solutions to the wave equation can be represented by a sine function?

Consider the one dimensional wave equation: $$\frac{\partial^2 f(x, t)}{\partial t^2} - c^{2}\frac{\partial^2 f(x, t)}{\partial x^2} = 0. $$ I understand that one may find "wavy" solutions to this ...
2
votes
0answers
64 views

The solutions of second-order ODE $y''+ay=0$ with a negative coefficient are hyperbolic functions

I am trying to solve a partial differential equation where $$u(a,\theta,z)=0,\quad -\pi \leq \theta \leq \pi, \quad 0\leq z \leq b$$ $$u(r,\theta,b)=0,\quad 0 \leq r < a,\quad -\pi\leq \theta \...
2
votes
1answer
27 views

Turing criteria for Sel'kvo glycolysis model

I have the Sel'kov reaction diffusion model for glycolysis as follows: \begin{eqnarray} u_t=D_uu_{xx}-u+av+u^2v\\ v_t=D_vv_{xx}+b-av-u^2v \end{eqnarray} How can I obtain the values for $D_u$ and $...
2
votes
0answers
33 views

Methods for first order PDEs in higher dimensions

What are the possible known methods for solving first order PDEs in higher dimensions? Is there anything else besides the method of characteristic curves? In particular, I have four first order, ...
2
votes
0answers
40 views

PDE that are unchanged under all axis-rotations

It is exactly the same question as Partial Differential Equation about Rotation question. Sadly, I gain nothing useful from the above post. Or I should say I am not familiar with the terms in the ...
2
votes
0answers
33 views

Step 2 of Strichartz's Estimate Proof.

I am stuck with Step 2 of the Strichartz's estimate. My question is actually a continuation of a topic which has been raised some time ago and it could be seen here Technical question about Strichartz ...
2
votes
0answers
39 views

Poincare Inequality in n-dimensions

I am trying to prove the Poincare Inequality on a n-dimensional box. That is a domain $ \Omega = (0,1)^n$ for $f(x) \epsilon H^{1}_{0}(\Omega) $, show there exists a constant $C$ such that $||f||_{L^{...
2
votes
0answers
27 views

PDE - three restrictions, wave equation (1 dimension)

I'm not very good at PDEs but this particular problem seems... Strange. It requires that the answer be "continuous (!!)" all in bold. \begin{align} u_{tt}&=9u_{xx},\quad x>0,\, t>0, \\ u(x,...
2
votes
0answers
20 views

Where does the name “tracking type problem” come from?

In PDE-constrained optimization problems, the distributed constrol problem $$ \begin{array}{ll} \displaystyle \min_{y,u} & J(y,u) = \frac{1}{2}\|y-y_d\|_{L^2(\Omega)}^2 + \frac{\alpha}{2}\|u\|_{...
2
votes
0answers
25 views

fourier transform for pde equation

I was solving the pde using fourier transform: $u_{tt}-u_{xx}+m^2u=0$ with initial values $u(0,x)=f(x)$ and $u_t(0,x)=g(x)$. I have received the answer $$U(t,k)=Ae^{-it \sqrt {k^2+m^2}}+Be^{it \sqrt {...
2
votes
0answers
58 views

Verify function is a Solution to Wave Equation in 3D Spherical Coordinates

The wave-equation is given by $$\nabla^2E=\frac{1}{c^2}\frac{\partial ^2E}{\partial t^2} $$ And I'm trying to prove that this wave $ E(r,\theta,\phi)= \frac{A_o}{r}sin(\theta) cos(\omega t - \frac{\...
2
votes
1answer
46 views

how to solve a pde whose coefficient is the function itself

I am studying differential geometry, Walker metric in three dimension. I try to find the geodesic equations of a Walker manifold and I need to solve the following PDE. Unfortunately, I didn't take any ...
2
votes
0answers
41 views

Numerical solution of $k\nabla^2 p(\vec{x}) = \nabla(\vec{f}(\vec{x})p(\vec{x}))$

I have the following equation: $$k\nabla^2 p(\vec{x}) = \nabla(\vec{f}(\vec{x})p(\vec{x}))$$ where the constant $k>0$ and vector field $\vec{f}(\vec{x})$ are both known. I wish to numerically ...
2
votes
0answers
100 views

How can we desribe a particle whose motion is perturbed by a random forcing using a stochastic partial differential equation?

Let $d\in\left\{2,3\right\}$ and $\mathcal V_t$ be the bounded set occupied by a fluid at time $t\ge 0$. Let $x_0\in\mathcal V_0$ be a particle and $$[0,\infty)\to\mathbb R^d\;,\;\;\;t\mapsto X_t(x_0)\...
2
votes
0answers
42 views

Brownian motion, harmonic functions and the Dirichlet problem

I am having trouble understanding one detail of the standard use of Brownian motion to solve the Dirichlet problem, I will write the statement and proof and then point to the detail I don't ...
2
votes
0answers
31 views

smoothness up to the boundary and the compatibility conditions

Let us say $\Omega$ is a smooth bounded domain. I have a baby quesiton: when we can say a solution $u$ of a heat equation is in $C^{2, 1}(\overline\Omega\times (0, T])\cap C(\overline\Omega\times [...
2
votes
0answers
21 views

What can we say about the number of linearly independent solutions of a PDE

What can we say about the number of linearly independent solutions to a PDE? Firstly, in what function space are we talking about linearity? Secondly, if there is not a general conclusion, what if ...
2
votes
1answer
52 views

Physical meaning of the various types of boundary conditions for a vibrating string

I wonder what is the physical meaning of Dirichlet, Neumann and Robin boundary conditions for a vibrating string? Or link to other applications?
2
votes
0answers
36 views

Existance and Uniqueness Theorem for PDE's

I am looking for a general uniqueness theorem for linear second order PDE's. I would like to have general conditions for the boundary conditions. For example, a very general formulation might look ...
2
votes
1answer
41 views

PDE - $y^2 \frac{\partial^2 u}{\partial x^2}=x^2 \frac{\partial^2 u}{\partial y^2}$ - how to derive the general solution

$\mathbf{y^2 \frac{\partial^2 u}{\partial x^2}=x^2 \frac{\partial^2 u}{\partial y^2}}$ is a hyperbolic PDE where $\xi =y^2+x^2$ $\eta =y^2-x^2$ which gives $u_{xx}=2(u_{\xi}-u_{\eta})+4x^2(u_{...
2
votes
1answer
88 views

Proving $\|u\|_{L^\infty(0,T;H)}\leq C$ from a given hint.

My question concerns to the problem 6, chapter 7, of Evans PDE book (2nd edition). In the book a hint is given but I couldn't get a solution from it. On the other hand, I got a solution without ...