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

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3
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0answers
58 views

Is this derivative well defined?

Let $\Omega$ be an open bounded domain in $\mathbb{R}^n$. Let $z_1,z_2,\dots,z_k\in \bar \Omega $ be $k$ distinct points. Let $\bf Z$ denote the k-tuple $Z = (z_1,z_2, \dots, z_k)$ Define $R_i = ...
10
votes
1answer
86 views

Theoretical Basis for Eigenvalue transformation on Bessel's Equation

The method I've been taught for finding all of the eigenvalue solutions to Bessel's operator $$b(f)\equiv f''(x)+\frac{1}{x}f'(x)$$ goes as follows. Let $g(a)=f(\sqrt{\lambda}x)$. Then $$b(g)=\lambda ...
0
votes
1answer
28 views

Unsure of 2d finite difference method with second order term?

I have the following equation for the price of Black Scholes Euro option - (1) $$-\frac{\delta C}{\delta t} = -\alpha\frac{\delta^2 C}{\delta x^2} + [r - \delta + \frac{\sigma^2}{2}]\frac{\delta ...
2
votes
3answers
53 views

Find the Fourier Transform of $2x/(1+x^2)$

I tried doing this the same way you would find the Fourier transform for $1/(1+x^2)$ but I guess I'm having some trouble dealing with the 2x on top and I could really use some help here.
0
votes
1answer
62 views

Evans PDE problem 9,Chapter 6

Let $U\subset \mathbb{R}^n$ be an bounded domain with smooth boundary. Assume $u$ is a smooth solution of $$ Lu=-\sum_{i,j=1}^na^{i,j}u_{x_ix_j}=f \ \text{in} \ U, \ \ u=0 \ \text{on} \ \partial ...
8
votes
0answers
221 views
+200

Overview of nonlinear analysis, differential equations (ODE and PDE), dynamical systems, and mathematical physics, and their relationships

The fields of (i) nonlinear analysis, (ii) ODE and PDE, (iii) dynamical systems, and (iv) mathematical physics (e.g., electromagnetism, general relativity, gravitation, etc,...) are very huge, ...
0
votes
1answer
17 views

are those boundary $ C^1$

I am studying PDE and my question is that if we have a open unit disk with $ [0,1)$ on x axis removed, is the boundary of this set $C^1$ ? And on the other hand, is the boundary of a open rectangle ...
3
votes
0answers
29 views

Relationship of SDE and Feynman-Kac PDE

I am struggling with this problem: Given a stochastic differential equation $$ dX_t = b(X_t) dt + \sigma (X_t) \,dW_t $$ where $W$ is a Brownian motion and the functions $b$ and $\sigma$ are ...
1
vote
1answer
25 views

Intuition behind estimates on derivatives of a harmonic function

In Evans' PDE book he gives the following theorem. Assume $u$ is harmonic in $U$. Then, $$ |D^{\alpha}u(x_0) | \le \frac{C_k}{r^{n+k}}||u||_{L^1(B(x_0,r))}$$ When asking my professor for some ...
2
votes
1answer
44 views

Integral Inequality for Bound on Gradient of Solution to Heat Equation

My overall aim is to show that, for a bounded solution $u(x,t)$ to the heat equation in $\mathbb{R}^n \times [0,T]$ with boundary condition $u(x,0) = f(x)$, $$\max |\nabla u(x,T) | \leq \frac ...
1
vote
1answer
22 views

Verifying Duhamel Principle for Heat Equation

From separation of variables, we get a solution to the homogeneous problem for the heat equation $$u_t - u_{xx} = 0$$ $$u(0,t) = u(L,t) = 0$$ $$u(x,0) = f(x)$$ of the form $$u(x,t) = \int_0^L f(y) ...
3
votes
0answers
60 views

Boundary value problem with biharmonic equation on the plane

I'm struggling with the following Boundary Value Problem for some time. The problem is to solve the biharmonic equation $\nabla^4\psi = 0$ with $\psi$ dependent not just on the coordinates on the ...
1
vote
0answers
20 views

Heat on a wedge

I'm trying to solve the PDE given by $$\begin{array}\ u_t = \nabla^2 u, & 0 \lt r \lt 1, & 0 \lt \theta \lt \alpha \lt 2\pi\end{array} \\ u(r,\theta, 0) = f(\theta) \\ u(1,\theta,t) = u(r,0,t) ...
2
votes
1answer
23 views

How can I prove this Bessel function relation

Prove $$J_{s \pm 1}(z) = \frac{s}{z}J_s(z) \mp J_s'(z)$$ from $$J_s(z) = \sum^\infty_{j=0} \frac{(-1)^j}{\Gamma(j+1)\Gamma(j+s+1)}\Big(\frac{z}{2}\Big)^{2j+s}$$ I proved $J_{s - 1}(z) = ...
1
vote
0answers
23 views

How to get first order term in wave equation

Like in heat equation $U_{t} + U_{xx} + bU_{x} + U = 0$. Just let $v(t,x) = e^{t}u(t,x+bt)$, then we turn the problem into a standard one. Now the question is how to turn the wave equation $U_{tt} - ...
1
vote
0answers
19 views

A PDE question using variation of parameters

Variation of parameters: Consider IBVP \begin{align} u_t − u_{xx} = f(x, t) \qquad & \text{on } \Omega = (0, \pi) \times \Bbb R^+\\ u(x, 0) = \varphi(x) \qquad & \text{on } (0, \pi)\\ ...
1
vote
1answer
23 views

Laplace Equations with Neumann boudary-value problem

The problem is that, Assume U is connected, use the maximum principle to show that the only smooth solutions of $-\Delta u=0$ in U and $\frac{\partial u}{\partial \nu}=0$ on $\partial U$ are $ u ...
1
vote
0answers
22 views

Does the differential operator in the heat equation have a name?

Does the operator $$\frac {\partial}{\partial t} - k\nabla^2$$ have a name?
1
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1answer
28 views

How do I turn this Crank-Nicolson type equation into three vectors representing the middle, upper, and lower diagonals in a tridiagonal matrix?

I have the following homework problem: I have calculated the Crank-Nicolson equation to be Equation 1 $$ -200.05u_{m-1}^{n+1}+400.9995u_{m}^{n+1}-199.95u_{m+1}^{n+1} = ...
0
votes
0answers
23 views

Find the supremum of the function

Hi I'm trying to figure out for which values of $w$ the absolute value of the supremum of $u(x,t)$ is infinity. The function $u(x,t)$ is the following. According to my calculation is going to be all ...
1
vote
2answers
43 views

Orthogonality lemma sine and cosine

I want to know how much is the integral $\int_{0}^{L}\sin(nx)\cos(mx)dx$ when $m=n$ and in the case when $m\neq n$. I know the orthogonality lemma for the other cases, but not for this one.
3
votes
1answer
68 views

How to Separate Charpit Equations

I've been attempting to solve this non-linear PDE $$4\Omega x^2 y^2 \frac{\partial z}{\partial y} -x^2 y (\frac{\partial z}{\partial x})^2 + 2x^2 y^2 E-N^2=0$$ using Charpit's method. The ...
0
votes
0answers
33 views

find the supremum

Hi I'm trying to figure out for which values of $w$ of $u(x,t)$ the absolute value of the supremum of $u(x,t)$ is infinity. The function $u(x,t)$ is the following. According to my calculation is ...
1
vote
1answer
31 views

nonhomogenious partial differential equation

How to solve this transport equation? $\dfrac{u_t}{u}-\dfrac{t}{x}\dfrac{u_x}{u}=-\dfrac{fx^2+2g}{ax^2+2b}$ in $(0,\infty)\times \mathbb{R}-\{0\}$ $u(t_0,x)=u_{t_0}(x) $ on $\{t=t_0\}\times ...
0
votes
1answer
19 views

The expansion of harmonic function at infinity

If $u$ is a harmonic function on $\mathbb R^n$ outside some compact set such that $u$ goes to $1$ at infinity. Then does $u$ have the following expansion $$ u=1+\frac{a}{|x|^{n-2}}+O(|x|^{1-n})\quad ? ...
1
vote
0answers
14 views

Reference Request for Penalty Method for Optimal Control?

Is there a good book or review article to read about the methods like penalty method, method of duality and method of relaxation in problems of calculus of variations and their relations to optimal ...
0
votes
1answer
20 views

Observe approximation order of numerical solution of a Partial Differential Equation

When solving a Partial Differential Equation numerically, I estimated the approximation orders theoretically as follows, $$ u(x,t)= u_{h,k} + C_1 h^{p} +C_2 k^{q}, $$ where $ u_{h,k} $ is a numerical ...
-1
votes
1answer
60 views

Why are nodes and nodal sets called this way?

Nodes of standing waves are points where they are zero. Generally, nodal sets of Laplacian eigenfunctions are the sets of points where they are zero. Why is this the name for them (that is, why is ...
1
vote
1answer
15 views

What is the dy/ds characteristic equation for this PDE?

$$u_t+\text{yu}_x+\frac{1}{2}\text{(u(u-1)})_y\text{=0}$$ The initial condition is given as $$\text{u(x,y,0)=}u_0\text{(x,y)}$$ I know what $$\frac{\text{dt}}{\text{ds}}\text{ ...
1
vote
1answer
48 views

How do I solve this PDE (diffussion equation) using the sepration of variables method?

$$\frac{\partial u}{\partial t} =\nu\large(\frac{\partial^2u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} \large), 0 < r < a, t >0.$$ Subject to the conditions $$\frac{\partial ...
1
vote
0answers
23 views

Ideas on how to improve stability in solving PDE

I am solving a non-linear second order system of PDEs in two variables. The equations are too complicated to write out here, but an essential feature is that there is a propagating wave which then ...
1
vote
1answer
26 views

Difficult boundary conditions for the PDE $U_{xx}=U_t$

I am given $U_{xx}=U_t$, $U_x(1,t)=U(1,t)$, and $U_x(0,t)=0$. I use separation of variables and set $U(x,t)=X(x)T(t)$, then $$X''T=XT'$$ $$\frac{X''}X=\frac{T'}T=-\lambda$$ for some $\lambda$. And ...
1
vote
0answers
22 views

Laplace's Equation with Neumann B.C. on disc

I am trying to solve Laplace's equation with the following boundary conditions: $On\hspace{0.1in}the\hspace{0.1in}disc\hspace{0.1in}of\hspace{0.1in}radius\hspace{0.1in}1,\hspace{0.1in} \triangledown ...
2
votes
1answer
54 views

Quasi linear partial differential equation

I am trying to solve the pde given by: $$\rho_t + (\rho v)_x = 0$$ Subject to conditions $v = f(x-vt)$ and $\rho(x, 0) = g(x)$. I have used the product rule on the second term to observe that ...
0
votes
2answers
33 views

Can we integrate from $0$ to $x$ / $t$?

I want to find all the solutions of the following problem: $$w_x(x,t)=3x^2t+t, x,t \in \mathbb{R}\\w_t(x,t)=x^3+x, x, t \in \mathbb{R}$$ I have tried the following: $$w_x(x,t)=3x^2t+t \Rightarrow ...
0
votes
1answer
22 views

What are the solutions to $\partial_{x_i x_j} F(x) = 0$?

Let $F\in C^2(\mathbb{R}^d, \mathbb{R})$. I'm looking for solutions to $\frac{\partial^2 F}{\partial x_1 \partial x_2}(x) = 0$ or $\frac{\partial^2 F}{\partial x_1 \partial x_1}(x) = 0$. My guess is, ...
1
vote
1answer
23 views

Boundary condition not compatible with initial condition…why?

Equation $$\frac{\partial^2 p}{\partial t^2} = \frac{\partial^2 p}{\partial x^2} + \frac{\partial^2 p}{\partial y^2}$$ Boundary conditions: $p = 0$ for $x = 0, \pi$ $p = 0$ for $y = 0, \pi$ ...
3
votes
2answers
56 views

Find solution of problem - Method of characteristics

I want to find the solution of the problem: $$(t+u(x,t))u_x(x,t)+tu_t(x,t)=x-t, x \in \mathbb{R}, t>1 \\ u(x,1)=1+x, x\in \mathbb{R}$$ I have tried the following: $$(x(0), t(0))=(x_0, 1)$$ ...
0
votes
0answers
7 views

nonhomogeneous equation with nonhomogeneous boundary equation

I tried the usual u = v+w stuff that gave me the following: $v_{tt} - c^2 u_{xx} + m^2 v = -m^2\pi x\cdot1/2$ $v(x,0) = g(x) + \pi x/2; v_t(x,0) = h(x)$ $v(0,t) = 0; u(\pi,t) = 0$ I ...
0
votes
0answers
20 views

What are the solutions of the Laplace equations for two (or more) eccentric cylinders?

I am looking for solutions to Laplace equation for two eccentric cylinders in 3D with arbitrary boundary conditions. The boundary condtions also depend on the axial variable. I tried to work with ...
1
vote
1answer
22 views

logarithmic sobolev inequalities

I'm reading a paper about PDE in fluid dynamics, where it used something called logarithmic sobolev inequalities: $$||\nabla u||_{\infty}\leq C||\omega||_{\infty} (1+\ln ||u||_m)$$ Where ...
2
votes
1answer
29 views

Integrals with complex functions: integration by parts and conjugate

I am working with integrals of complex functions. I assume all terms are well-defined. If $u=u(x,t):\mathbb{R}^n\times\mathbb{R}_+ \to \mathbb{R}$ (a real function), I have \begin{equation} ...
0
votes
1answer
26 views

Separating variables in the PDE $u_{tt}+2u_t+u=u_{xx}$

Separating variables in the PDE $u_{tt}+2u_t+u=u_{xx}$ In the ODE of $T(t)$ (Second last equation) shouldn't it be the one stay outside the bracket?
1
vote
1answer
33 views

Prove $\int_U |u|^2dx \le C(\int_U|Du|^2dx+\int_{\partial U}u^2dx)$ by divergence theorem?

How to prove $$\int_U |u|^2\,dx \le C\left(\int_U|Du|^2 \, dx+\int_{\partial U}u^2\,dx\right)?$$ The hint my teacher gives me is to apply the divergence theorem to $(0,\ldots,x_iu,0,\ldots,0)$, $1\le ...
2
votes
1answer
38 views

Solving 2D parabolic PDE

Please forgive if this is simple, but I was wondering if one may be able to derive a closed-form solution to \begin{align} \frac{\partial u}{\partial t} & = \frac{\partial^2 u}{\partial x^2} + ...
0
votes
2answers
44 views

Solving the problem with Fourier

I want to solve the following problem: $$u_{xx}(x,y)+u_{yy}(x,y)=0, 0<x<\pi, y>0 \\ u(0,y)=u(\pi, y)=0, y>0 \\ u(x,0)=\sin x +\sin^3 x, 0<x<\pi$$ $u$ bounded I have done the ...
0
votes
0answers
9 views

If solution of NLS is smooth and decaying at infinity; how to justify it satisfy the conservation law?

We consider the cubic nonlinear Shr\"odinger equation(NLS) $iu_{t}+\frac{1}{2} \Delta u = |u|^{2}u, \ u(x, 0)= u_0(x), (x\in \mathbb R^{d}, t\in \mathbb R)$ I have been trying to understand the ...
0
votes
0answers
19 views

For which $(x_1,x_2)$ is this a solution to the minimal surface equation?

Let $u(x_1,x_2):=arcosh(\sqrt{x_1^2+x^2})$ then I want to find out for which $(x_1,x_2)$ this is a solution to the minimal surface equation in two dimensions that you can find for example here. ...
0
votes
0answers
26 views

Compute the solutions of the following equation in Fourier space:

$$\frac{d^3u}{dx^3} − αxu = 0, x ∈ R, $$ where $ α > 0$ is some constant and $u(x)$ is assumed to satisfy $\int_R u(x) dx = π.$ I know this is a ODE so this is what I came up with so far: ...
0
votes
1answer
27 views

Existence of Solutions to PDEs - How do I know I've got them all?

I'm taking a very computational course in partial differential equations. Because of this emphasis, I'm feeling very underwhelmed by the course, and have a lot of questions that really aren't ...