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

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Two problems from Avner Friedman's PDE book.

The problems are as follow: Prove that if $Lu=0$ for any $u\in C^m(\Omega)$, then $L\equiv 0$ - that is all the coefficients of $L$ vanish identically. Prove that the assertion of the previous ...
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1answer
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Inhomogeneous Diffusion Equation - Maximum Principle

Let $u$ be a solution of the inhomogeneous diffusion equation $u_{t} = ku_{xx} + f$ on the rectangle {${0 < x < l, 0 < t < T}$} with initial data $\phi$ and boundary data $g$. Prove that ...
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9 views

Set of coupled partial differential equations

I've read that the Einstein equation is a set of 10 coupled partial differential equations. I know what a partial differential equation is, but I don't know what a set of coupled partial differential ...
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15 views

First Order Linear Inhomogeneous Partial Differential Equations

I've been trying to solve this one problem for days. Literally. Days. This is my method of last resort, so I'm praying someone can explain this to me. I understand the method of characteristics, ...
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1answer
17 views

Proving weak coercivity by young's and interpolation inequalities

Let be $$(P)\left\{\begin{array}{ll} &-\Delta u + V(x)u=f & \text{ in }\ \Omega\\ &u=0 & \text{ on } \ \Gamma \end{array}\right.$$ with $V \in L^r(\Omega)$, for some ...
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23 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 ...
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16 views

Maximum Principle of the Diffusion Equation

Consider a solution of the diffusion equation $u_{t} = u_{xx}$ in {$0\leq x\leq l,0\leq t<\infty$} a) Let $ M(t)$ = the maximum of $u(x,t)$ in the closed rectangle {$0\leq x\leq l,0\leq ...
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1answer
8 views

1D diffusion equation with boundary condition

Suppose we have the diffusion equation defined by $$u_t(x,t) = \Delta_x u(x,t) \ \ \ \ \text{in}\ \ (0,L)\times (0,T)$$ with the boundary condition $$u(0,t)=f(t)$$ $$u(L,t)=g(t)$$ and the initial ...
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14 views

Solving 1-d wave equation as conservation law

So the long story short is the following. I break the 1-d wave equation ($\frac{\partial^2u}{\partial x^2}=\frac{\partial^2u}{\partial t^2} $) into a system of first order hyperbolic equations. I ...
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Fundamental solutions for pde [on hold]

I am looking for fundamental solution for the following PDE problem. $$2xy(u_{xx} + u_{yy}) + yu_{x} + xu_{y} =0$$ Any help will be appreciated. Thnak you.
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27 views

Bounding an integral

I'm trying to show that the following integral ( a solution for the non-homogeneous transport equation ) has this bound: $$ \begin{equation*} \left\|{ \int_{0}^{t} f(x+b(w-t),w) dw ...
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19 views

Fourier transform method of solving the heat equation

We have a heat equation given by: $$\frac{\partial u}{\partial t} = \frac{\partial ^2 u}{\partial x^2}$$ We know that $\mathcal{F}\{ u'' \}=-\xi^2\hat{u}$. Defining ...
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15 views

Showing coercivity of the bilinear form associated with a robin boundary value problem

I'm trying to show the existence and uniqueness of weak solutions to the following boundary value problem: \begin{align} -\nabla \cdot ( k \nabla u) &= f \quad \text{in } \Omega \subset ...
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17 views

Let $f \in L^2(\mathbb{R})$ and let $u=u(x,t)$ solution of following problem

Let $f \in L^2(\mathbb{R})$ and let $u=u(x,t)$ solution of following problem: $$\left\{\begin{matrix} u_t=\frac{1}{2}u_{xx} & x \in \mathbb{R} &, t > 0\\ u(x,0)=f & ...
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1answer
20 views

Find the strong form of a PDE from the weak form.

I'm having a little difficulty understanding how to find the strong form of a PDE given the weak form. For example, I have the weak form as: $\displaystyle\int_\Omega [a(x)\nabla u\cdot\nabla ...
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14 views

Duhamel formula of semi linear heat equation

Can someone please help me to prove that the solution of semi linear heat equation with initial data in $H_m$ and which is given by Duhamel formula is in $C1((0,T],H_m)$ where $H_m$ is the Sobolev ...
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1answer
18 views

Finding First Integrals in the case $2xy u_x - (x^2+y^2) u_y =0$

Good day, As described in the title, I want to find two First Integrals (FI) to the PDE $$2xy u_x - (x^2+y^2) u_y =0$$ Of course, $u$ is a FI and the solution of the PDE ist $u(x,y)=u_0$. But I want ...
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2answers
27 views

Canonical form and fundamental solution of pdf

Can someone help with these two PDE problems? Thank you. Reduce to Canonical form and find the fundamental solution if possible. $$y^2u_{xx} + x^2u_{yy} = 0.$$ What type of transformation should I ...
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Lebesgue Spaces on the Boundary: Outward Normal Vector to ∂Ω [on hold]

So basically, there is an outward normal vector from the boundary of an open set $\Omega$. What i need to do is to prove that this is actually a unit vector. Can anyone please help? Other sources like ...
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10 views

How can I solve the conservation of traffic PDE?

I'm trying to solve the conservation equation for traffic flow so that I can use it for an example. It is stated as follows: $$\frac{\partial \rho }{\partial t} + \frac{\partial \rho v(\rho ...
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35 views

Fundamental Theorem of calculus for multivariable function

I'm reading a research paper on PDE (http://arxiv.org/pdf/1011.0949v5.pdf). On page 2, the authors said: "From $\partial_{t} T_{00} = \partial_{x} \, T_{01}$, and the fundamental theorem of calculus, ...
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1answer
30 views

To understand elliptic partial differential equations

I am a graduate university student of mathematics. I would like to study elliptic partial differential equations on my own. I have tried this lecture note though I cannot understand it all as I never ...
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1answer
21 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 ...
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1answer
31 views

Separable solution to a nonlinear parabolic PDE

I seek a separable solution to the nonlinear parabolic partial differential equation, $\frac{\partial u}{\partial t} = u \frac{\partial u}{\partial x^2} + u^2.$ The physics of the problem allow ...
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1answer
23 views

A linearly independent set that spans a space

So, in partial differential equations, we generate solutions for PDEs (kind of obviously). However, while the solutions we generate span the space of all solutions and are all linearly independent, ...
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1answer
25 views

Is this element of $H^1(\Omega)^*$ actually in $L^2(\Omega)$?

Let $\Omega$ be a smooth bounded domain. Let $v \in H^2(\Omega)$ satisfy $-\Delta v = 0$ on $\Omega$ with $\partial_\nu v = g$ where $g \in H^{1/2}(\partial\Omega)$ is normal derivative data. ...
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21 views

Galerkin method in Sobolev space

I've got this problem to solve: Using Galerkin method, prove that there exists a weak solution of this differential equation: $$-\Delta u = a(x) \circ \triangledown u - u_t +f(x)$$ on $\Omega$ $$u ...
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46 views

Several options using Black-Scholes equation(s)

Could someone provide me some information about the modelling of several options at the same time by using Black-Scholes (probably coupled) equations? Any reference to papers and/or books shall be ...
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Upper Bound of Fisher Equation

Could anyone please give me directions on how to establish a non trivial and as good as possible upper bound ($u(x,t) \le u_0$) of the Fisher equation? \begin{cases} u_t = u_{xx} + u(1-u) \\ u(x,0) = ...
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Canonical form transformation

​ ​ My subject about the canonical form of PDE. I had many exercises to do and they were fine, but I'm stuck with this one: ​ ​ $$U_{xx}-yU_{xy}+xU_x+yU_y+u=0$$​ ​ So first we have to calculate ...
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what is the role of adding numerical dissipation to solve partial differential equations

I usually solve the partial differential equations (PDE), but I have never used the numerical dissipation to have a optimal results in terms on accuracy and stability of PDE's solution in generals. ...
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36 views

Best Approximate Solution of Heat Equation (Diffusive Logistic Equation)

If $u(x,t) \ge 0$ in the domain $ (0 \times 1) \times (0,\infty)$, find a function that caps the value of $u(x,t)$ in the region $(0 \times 1) \times (0,T)$. $u(x,t)$ is a solution of the following ...
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1answer
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Showing that the map that takes $u_0$ to solution $u(t)$ is self-adjoint

Let $u$ and $v$ be the solution of the heat equation $$w'(t) - \Delta w(t) =0$$ with initial data $u_0$ and $v_0$ respectively, and with either homogeneous Dirichlet or Neumann BCs on a bounded domain ...
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1answer
35 views

Find the general solution to this PDE

I'm asked to find the general solution of $$au_{xx}+bu_{xy}=0$$ With $u=u(x,y)$ and $a$, $b$ real constants. I'm just starting with PDE's, haven't seen any resolution technique except for basic ...
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1answer
34 views

Eigenvalue equation and separable solutions

Recently, studying Quantum Mechanics I found a doubt regarding separable solutions and eigenvalue equations for differential operators. Suppose we are considering some space $\mathcal{H}$ of functions ...
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29 views

Does this inequality imply a uniform $L^\infty$ bound?

Suppose I have the estimate for $t > 0$ $$\lVert u(t) \rVert_{L^\infty(\Omega)} \leq Ct^{-1}\lVert u_0 \rVert_{L^1(\Omega)}$$ for the solution $u$ of a parabolic PDE with initial data $u_0$ on a ...
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2answers
58 views

Second order PDE with initial condition

How do I solve the equation $\frac{\partial^2}{\partial x\partial t} u(x,t)=\frac{\partial^2}{\partial x^2} u(x,t)$ with the initial condition $u(x,t=0)=\sqrt{\frac{\pi}{2}}\exp(-|x|)$ ? The solution ...
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0answers
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What does “loss of regularity” mean?

I have seen a lot the phrase "loss of regularity" in references regarding PDE. (For instance, there are questions like "do solutions of 3D Navier-Stokes equations lose regularity or not?") Could ...
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3answers
52 views

How to calculate $f(x, y, z)$ given $d f = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy +\frac{\partial f}{\partial z} dz$

On a manifold with local coordinates $(x_1, \ldots, x_n)$ I have a closed 1-form $\omega$ for which $d \omega = 0$ holds. This means There must be a function $f(x_1, \ldots x_n)$ for which $d f = ...
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1answer
37 views

An equivalent theorem for Sobolev spaces in infinite dimensions

There is a proposition which states: Let $f\in W^{1}(U)$ be real valued and $h\in C^{1}(\mathbb{R})$ with $h'\in C_{b}(\mathbb{R})$. We then have $h\circ f\in W^{1}(U)$ and $$\partial_{j}(h\circ ...
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Heat equation on a torus [on hold]

Can someone please help me to solve the following problem about the existance of a unique solution of the heat equation:here is the problem
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Nonlinear second order PDE

I need to solve the following PDE (which is a maximized Hamilton-Jacobi-Bellman equation) \begin{align} rV(\theta_1,\theta_2) = \frac{(\theta_1^\rho + ...
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Quasi-linear partial differential equations. Solving them.

This is what I have as a quasi-linear partial differential equation:$$u(x_1,...,x_n), \ \ \ \ \sum_{i=1}^{n}A_i(X,u) \frac{\partial u}{\partial x_i}=A_{n+1}(X,u) \ \ \ (1)$$ Then it says let ...
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Differentiability of PDE with respect to parameters

Consider a linear partial differential equation $$ (L u)(x)=f(x) \quad \forall x \text{ in } \Omega\\ u=g \quad\text{on }\partial\Omega. $$ Assuming that $f$ and $L$ depend on a parameter ...
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13 views

Study of heat equation on a torus [closed]

Good morning, can someone please help me to find a web site where there is the study of heat equation on a torus. thanks in advance
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23 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, ...
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25 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 ...
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A Poincare type inequality

How to prove: $$||u||_{L^1({\Omega})}\leq c(\Omega)(||u||_{L^1(\partial\Omega)}+||Du||_{L^2(\Omega)})$$ Suppose u is smooth enough and $\Omega$ is a bounded domain with smooth boundary.
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1answer
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PDE Proof that a linear combination of 2 solutions is also a solution [closed]

Can someone please help? I've been trying to figure this for a few days now. Consider the first order PDE: $au_t + bu_x$ = 0, where a and b are constants. Show that if $u_1$ and $u_2$ are solutions ...
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20 views

How to integrate to solve a PDE with mixed partials in the integrand

Problem Statement: Determine the equlibrium temperature distribution inside a circular annulus $r_1\leq r \leq r_2$. If the outer radius is at temperature $T_2$ and inner radius at temp $T_1$. So ...