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

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3
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1answer
32 views

Finite differences and conservation law

I am using a Finite Difference scheme to solve a simple PDE in conserved form: $$\partial_t u = \partial_x (\partial_x u +au\partial_x u) = (1+a)\partial_x^2u +a(\partial_x u)^2 $$ $$\frac{u_{n+1,j} ...
4
votes
1answer
104 views

Solving a PDE with Feynman-Kac Formula

I'm trying to solve this PDE using Feynman-Kac formula Now i follow the regular steps Here is where I don't know how to proceed. How do I calculate this expectation?
0
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0answers
19 views

Problem 5.10.21 in Evans' PDE.

Show that if $u,v \in H^s(\mathbb{R}^n)$ for $s > n/2$, then $uv \in H^s(\mathbb{R}^n)$, and: $$||uv||_{H^s(\mathbb{R}^n)} \leq C ||u||_{H^s(\mathbb{R}^n)}||V||_{H^s(\mathbb{R}^n)}$$ Does someone ...
0
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0answers
14 views

How to construct a integral solution to poisson equation with green functions

How to construct a integral solution to poisson equation with green functions
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0answers
21 views

Comparison principle for heat equation with smooth nonlinearity

Let $f : \mathbb{R} \to \mathbb{R}$ be $C^\infty$ and satisfy $f(0)=0$. Suppose $u_1,u_2 : \mathbb{R}^d \to \mathbb{R}$ are $C^2$ and satisfy $$\frac{\partial u}{\partial t} - \Delta u = f(u)$$ for ...
1
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0answers
9 views

method of characteristics in a nutshell

Good morning everybody, I need a quick reference for the following inhomogeneous first-order pde... namely $$f(x,y,z)=A\partial_x\varphi+B\partial_y\varphi+C\partial_z\varphi,$$ where $\varphi\in ...
3
votes
1answer
55 views

solution of multidimensional PDE

I'm looking for a way to find a solution 'f' to the following PDE. $$ y \frac{\partial f}{\partial r} + g_1(r)\left(z\frac{\partial f}{\partial y} - y\frac{\partial f}{\partial z}\right) + ...
0
votes
0answers
18 views

Weak time derivative for functions $u \in L^2(0,T;L^2(\Omega))$

The weak time derivative of a function $u \in L^2(0,T;H^1)$ is defined to be $u' \in L^2(0,T;H^{-1})$ satisfying $$\int_0^T \int_\Omega u(t) \varphi'(t) = -\int_0^T \langle u'(t), \varphi(t) \rangle$$ ...
1
vote
1answer
18 views

Solving characteristic base curves initial value PDE

I'm trying to solve the characteristic base curves of an initial value problem. $$ \left\{ \begin{matrix}\ xy\frac{\partial u}{\partial x} + (2y^2 - x^6)\frac{\partial u}{\partial y} = 0 ; ...
0
votes
1answer
26 views

Diffusion-Reaction PDE - radial coordinate

I am trying to obtain an expression for the concentration $C$ based on this stationary equation : $\frac{\partial C}{\partial t} = \frac{1}{r} \frac{d}{dr} \left(r \frac{\partial C}{\partial ...
0
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1answer
23 views

Solve laplace equation inside a rectangular

My answer is $U = Acos(nπx/L)e^-nπy/L$ I really have no idea how to solve the particular solution. Please advise me.
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0answers
14 views

Solution to parabolic PDE with periodic boundary

I have a question considering parabolic PDEs, which I thought to be trivial. I want to show existence and uniqueness of a weak solution to the Fokker-Planck equation on a manifold without boundary: ...
1
vote
0answers
31 views

Transforming 1D Burger's Equation into infinitely many coupled ODE's

I've been working on the following problem but I can't justify my steps, would a savvy mathematician kindly tell me what, if any, violations I've made. Problem: Show Burger's equation can be written ...
3
votes
0answers
23 views

when does a partial differential equation have unique solution?

The differential equation $ xu_x + yu_y = 2u$ satisfying the initial conditions $y = xg(x), u=f(x)$ with $f(x) = 2x, g(x) = 1$, has no solution $f(x) = 2x^2, g(x) =1$, has infinite number of ...
3
votes
1answer
29 views

Is fractional order Sobolev spaces reflexive?

Let $0<s<1$, we define $$ W^{s,p}(\Omega):=\left\{u\in L^p(\Omega),\,\frac{|u(x)-u(y)|}{|x-y|^{\frac{N}{p}+s}}\in L^p(\Omega\times\Omega)\right\} $$ with norm $$ \|u\|:=\left(\int_{\Omega} ...
0
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0answers
14 views

Solution of a differential equation with problem of Cauchy

The question is the next: What can I say from the existence, uniqueness and continuos dependence of the solution? Is this a strongly continuos one-parameter group or a semigroup. $ \left\{ ...
0
votes
1answer
36 views

2D linear inhomogeneous wave equation with inhomogeneous time-independent initial conditions

I'm looking for any insight into solving the following PDE: $$u_{tt}=c^2 (u_{xx}+u_{yy})-\sin(y)$$ $$u=0, y\in {0,\pi} $$ $$u_x=0, x\in {0,1}$$ $$u(x,y,0)=\cos(\pi x)\sin(3y) $$ $$u_t(x,y,0)=0$$ ...
4
votes
2answers
49 views

Convolution integral problem

In the process of solving a certain PDE, I've arrived at a convolution integral: $$\int_{\mathbb{R}^3} G(x-y) \nabla p(y) dy$$ where $x \in \mathbb{R}^3$, $G(z)=\frac{1}{\| z \|}$ and $p(z) = ...
0
votes
0answers
14 views

Bump function construction with positive Fourier transform [duplicate]

Fellow math people, I am looking to construct a bump function with a positive and rapidly decaying Fourier transform. In particular, the function f should satisfy: (1) f non-negative and smooth and ...
0
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0answers
3 views

the notion of Wazewski set in shooting method

I come across the notion of Wazewski set when studying shooting method for proving existence of a boundary value problem. Some authors who use shooting method to prove existence start with ...
2
votes
0answers
33 views

Solving a system of two linear PDE: $u_x+v_x +u_y=0$ and $v_x+u_y-{1\over 2} v_y=0$

trying to solve the following cauchy problem: $$u_x+v_x +u_y=0\\v_x+u_y-{1\over 2} v_y=0\\u(x,0)=1-x,v(x,0)=x$$ my solution is: 1. multiply each equation by $t_1,t_2$ and sum the two equations like ...
-1
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0answers
10 views

Laplace Equation- Dirichlet Problem- superposition approach?

I have tried to treat this as a seperable partial differential equation but I can't seem to get an equation for the product functions.
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0answers
50 views

If the limit matrix of a linear system has eigenvalues of negative real part, then the system is asymptotically stable

I appreciate if anyone can help me on this question: You are given the following linear system: $x'(t)=A(t)x(t)$. Suppose that $\lim_{t\to \infty}A(t)=A_{\infty}$ and that all eigenvalues of ...
2
votes
1answer
46 views

Solving a first order nonlinear PDE

I have to solve this equation : $$ \frac{\partial u}{\partial x} \cdot \frac{\partial u}{\partial y} = xy $$ with initial condition $u(x,0) = x$. I know it is easy with separation of variables, but ...
2
votes
1answer
36 views

Construction bump function with positive Fourier transform

I am looking for the construction of a smooth bump function, $f$, mapping the real line to itself which has two special properties: (1) $f$ is constant on some interval in its support (for instance ...
0
votes
1answer
31 views

Can you help with the Method Of Eigenfunction Expansion of a Non-Homogeneous PDE problem?

Here is the Problem: Solve $\frac{\partial T(x,t)}{\partial t} = \frac{\partial^{2} T(x,t)}{\partial x^{2}} +2xe^{-t} $ with the following boundary conditions $T(0,t)=10, and \frac{\partial ...
1
vote
0answers
13 views

Inhomogeneous Wave Equation: can a force acting along a line segment drive solution to zero?

Suppose $u$ solves $$ \begin{cases} \partial^2_{tt}u - \nabla\cdot\left(c^2(x)\nabla u\right) = F(t,x) & (t,x) \in \mathbb R \times \mathbb R^n \\ u(0,x) = u_t(0,x) = 0 & x \in \mathbb ...
2
votes
0answers
40 views

Is the following function absolutely continuous?.

For a fixed $x\in\Omega$ is the function $F(x,y)=min\{1,\frac{\delta_{\Omega}(x)\delta_{\Omega}(y)}{|x-y|^2}\}$ - where $\delta_{\Omega}(x)$, $\delta_{\Omega}(y)$ are the distances from $x$, $y$ to ...
2
votes
0answers
28 views

Calculating gradient from finite difference results

I am solving the steady-state heat equation in two dimensions: $$\frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right) + \frac{\partial}{\partial y}\left(k\frac{\partial T}{\partial ...
2
votes
1answer
32 views

Inverse of a nonlinear heat equation

If u solves the equation $$ u_{t} = \frac{u_{xx}}{u_{x}^2}$$ in $\mathbb{R} \times (0,\infty)$and $v$ is the inverse of $u$ in $x$, as in $y=u(x,t)$ iff $x = v(y,t)$. I need to be able to show that ...
1
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0answers
15 views

The Courant Min-Max theorem of elliptic pdes.

This is an exercise function Evans PDE book, Chapter 6. The theorem states that for $Lu:=-\text{div}(A\cdot\nabla u)+cu$ where $c\geq 0$, we have the eigenvalue of $L$ can be written in the following ...
1
vote
1answer
35 views

A question about Green function in Laplace equations

Let $\Omega$ open bounded be given and $G(x,y)$ denote the Green function in Evans setting. That is, we have $$ \Delta_yG(x,y)=\delta_{x-y} $$ and $$ G(x,y)=0 $$ if $y\in\partial\Omega$. Now for ...
0
votes
1answer
25 views

Equilibrium solution for a heat equation

Find the equilibrium solution of $$ u_t(t,x) = u_{xx} (t,x) + x^2, \ 0<x<L\\ u(t,0) = 0,\ u(t,L) = 0 $$ I know that the equilibrium solution must satisfy: $[u_e(x)]' = 0 \ \forall t$. It ...
0
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0answers
34 views

Radial function

Suppose $u(x)\in C_{loc}^2(\Re ^3)$ satisfies $-\Delta u+(x+b(x))\cdot\nabla u\le 0, |u(x)|\le C(1+|x|)^{2014}$ where $b(x)$ is a smooth vector field in $\Re ^3$ with $|b(x)|\le (1/2)|x|$ for ...
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0answers
21 views

Why is a set of functions $v(t)$ dense in $L^2$

I was going through the following paper: http://www.emis.de/journals/HOA/AAA/Volume2011/142128.pdf In page 6, immediately after equation (3.15), its written that "functions of the form $v(t)$ are ...
1
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1answer
24 views

Entropy Solution of $u_t+(u^2/2)_x=0$

Given the initial data $$ g(x)= \cases{ 1 & x< -1 \\ 0 & -1 < x< 0 \\ 2 & 0 < x< 1 \\ 0 & 1 < x \\ } $$ What is the entropy solution of $u_t+(u^2/2)_x=0$?
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104 views

Separation of variables and quantum mechanics

In the book Quantum mechanics by Eugen Merzbacher, third edition, at page 462 he claims that this differential equation (for the unknown operator $F_0=F_0(x,y,z)$) can be solved by separation of ...
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0answers
20 views

Green function for a second-order elliptic PDE

Let $L = L^*$ be a second-order elliptic PDE with smooth and bounded coefficients in some bounded domain $\Omega \subset \mathbb R^d$, $d \geq 3$, with smooth boundary. Let $G(x,y)$ be a Green ...
1
vote
1answer
30 views

Problem of Partial Differential Equations

For this question, I get stuck when I apply the second initial equation. My answer is $θ= Ae^-(kλ^2 t)\cos λx$, where $A$ is a constant. Would anyone mind telling me how to solve it?
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0answers
30 views

What is a reasonable manufactured solution to test finite difference method?

What is a reasonable manufactured solution to test the following equation against its finite difference approximation? I want it to look like a Cosine function about $0$, rotated about $Z$ axis ...
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0answers
14 views

Conformal mapping

I know nothing about conformal mapping. I want to find analytical solution of Laplace equation in a hexagon with Dirichlet boundary condition at each wall. I already know the analytical solution of a ...
2
votes
1answer
42 views

Rayleigh-Bénard convection

I have a nondimensionalized linear perturbation system relevant to the appropriate pure conduction solution for Rayleigh-Bénard convection in upper planetary atmospheres under the compressible gas ...
1
vote
2answers
44 views

PDE Initial-Boundary value problem question

The problem: The ends of a stretched string are fixed at the origin and at the point, $ x=\pi$ on the horizontal x-axis. The string is initially at rest along the x-axis, and then drops under it's ...
1
vote
1answer
32 views

Mixed boundary condition for the wave equation, using reflection method

Solve for $u(.5,13)$: $$u_{tt}=4u_{xx}\quad 0<x<1$$ $$u(x,0)=x, \ u_t(x,0)=1$$ $$u(0,t)=0, \ u_x(1,t)=0. $$ Using D'Alembert formula I got that I must take the odd extension and then apply ...
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0answers
19 views

Two dimensional non homogenous pde

Would you please help me with some examples like this: I need 3 examples of 2D nonhomogenous PDE with complete solution (Sturm liouville) thank a lot.
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0answers
18 views

Estimates of partial derivatives

Suppose $f\in C^{\infty}(\mathbb{R}^n)$ is real analytic and $\Delta f(o)\not=0$. Are there pointwise estimates for $\frac{\partial^{\alpha}f}{\partial x^{\alpha}}$ in terms of ...
1
vote
1answer
35 views

Verifying Distribution Equivalence for Fourier Series Expansion

In my lecture notes, given a periodic distribution $T \in (C_{per}^\infty([-\pi,\pi]^n))'$, the Fourier coefficients are defined by $$\hat T(m) = T({1 \over (2\pi)^n}e^{-i m \cdot x}),$$ for $m \in ...
1
vote
0answers
32 views

Extend concept of weak derivatives?

Let $f \in H^m(\mathbb{R}^n)$, then we have for $|\alpha| \le m$ that $$\langle \partial^{\alpha}f, \phi \rangle = (-1)^{|\alpha|} \langle f , \partial^{\alpha} \phi \rangle$$ for all test ...
0
votes
1answer
39 views

Inverse fourier transform for the heat equation

I'm trying to find the Inverse Fourier Transform for the following heat equation. Question: Solve the problem $$u_t=ku_{xx}+\frac{1}{\sqrt{2kt}}e^{\frac{-x^2}{4kt}}, -\infty < x< \infty, t ...
1
vote
1answer
41 views

What concepts do I need to solve this nonlinear problem?

Problem Details: Let $L$ be a linear operator and $N(u)$ be a nonlinear function of $u$. Consider the IVP for the following nonlinear PDE: $$\partial_{t}u + L(u) = N(u)$$ defined on $(x,t) \in ...