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

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11 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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0answers
6 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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0answers
10 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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0answers
11 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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0answers
8 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
8 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
21 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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0answers
11 views

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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0answers
9 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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0answers
27 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
23 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 ...
2
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1answer
18 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
25 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 ...
2
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1answer
21 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
20 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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0answers
10 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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0answers
41 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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0answers
23 views

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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0answers
10 views

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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0answers
9 views

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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0answers
34 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 ...
3
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1answer
18 views

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
34 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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0answers
27 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
56 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
23 views

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
48 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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0answers
37 views

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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0answers
21 views

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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0answers
18 views

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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0answers
21 views

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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0answers
13 views

Study of heat equation on a torus [on hold]

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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0answers
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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0answers
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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0answers
13 views

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
29 views

PDE Proof that a linear combination of 2 solutions is also a solution [on hold]

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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0answers
19 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 ...
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1answer
35 views

Odd and Even Fourier Series Extension of $f(x)=x$ on $[0,\pi]$

I'm confused on finding the odd and even extensions of $f(x) = x$ on $[0,\pi]$. I know the general forms and how to find the co-efficients, but for the sin series, $f(0)$ =/= $f(\pi)$, so then I only ...
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1answer
22 views

Online course for numerical methods/analysis of PDEs

Could anybody recommend an online course for implementing numerical methods to solve PDEs which can supplement reading? This is with a view to writing an implementation to solve the Monge-Ampere ...
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1answer
13 views

Non-homogeneous First Order PDE Method

Just to be upfront, this is a homework question, but I'm just stuck on one particular part and I want to see what I'm doing wrong. The PDE in question is the following: ...
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0answers
19 views

How to find the general solution of this PDE

I've been assigned the following partial differential equation as an introductory exercise: $$u_{xy}+au_x+bu_y+abu=0$$ Where $a,b$ are constants and $u=u(x,y)$. Having seen barely anything except ...
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1answer
30 views

Partial differential equation 5 [on hold]

Which change of variable should I do to solve this PDE? $u_{xy}(x, y) + au_x(x, y) + bu_y(x, y) + abu(x, y) = 0$
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1answer
52 views

Compare analytic model with numerical, mass spring system.

So I'm trying to solve a problem here and I have been working on it all day, clearly i'm in need of some guidance. I have a rod of length $L$ and cross section area $A$, Young's modulus $E$ and ...
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0answers
29 views

The eigenfunction of $-\Delta$.

Let $u_k$ be eigenfunctions for operator $-\Delta$ over domain $\Omega\subset \mathbb R^2$, open bounded, smooth boundary. Then, we know that $u_k$ forms a basis for $L^2$. Let $u\in H_0^1(\Omega)$ be ...
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0answers
32 views

method of characteristics for non-linear PDE

I'm trying to solve the PDE $u_x^2-u_y^2=8u$ with initial conditions $u(x,x)=f(x)$. I have that $F(x,y,u,p,q)=p^2-q^2-8u$, with $p=u_x, q=u_y$, and then \begin{equation*} \begin{array}{ll} ...
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1answer
28 views

System of differential equation (Matrix form)

I'm trying to solve this system $$ M\ddot{X}(t) = KX(t) $$ where M is a known diagonal matrix and K is a symmetrical known matrix. I'm asked to do the ansatz $Y(t) = M^{1/2}X(t)$ where $M^{1/2} = ...
2
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0answers
19 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 ...
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0answers
16 views

Differentiation with integration region depending on $x$ to solve for decreasing energy of wave equation

I want to show that for the general wave equation $u_{tt} - \nabla \cdot (c^2\nabla u) + qu = 0, \quad u(x, 0) = \phi(x), \quad u_t(x, 0) = \phi(x)$ we have $$ E(t) = \int_{|x-x_0| < R_0 - c_2t} ...