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

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24 views

Nonlinear PDE $u_y=(u_x)^3$

I need to show that the only solutions of $u_y=(u_x)^3$ that are smooth on whole $\Bbb R^2$ are of the form $ax+by+c$, could anyone help me please?
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0answers
21 views

About composition of Holder functions.

Let $f,g$ be Holder continuous functions with respective exponents $\alpha, \beta \in (0,1)$. More precisely $f \in C^{\alpha}(\mathbb{R}^n;\mathbb{R}^n)$, $g\in C^{\beta}(\mathbb{R}^n,\mathbb{R})$. ...
3
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0answers
16 views

Solving the 2D Poisson equation with variable boundary location

I am trying to find $z(r,\phi)$ from the 2D Poisson equation in polar coordinates: $$\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial z}{\partial ...
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0answers
11 views

How the Jacobian is connected to the movement of particle from one domain to another? [on hold]

I am dealing with the proof of Reynold-Transport Theorem. There the Jacobian is used for the changing position of particles from one domain to another. Can anyone help me to understand what does ...
3
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1answer
29 views

first order linear PDE solving

$$\dfrac{\partial{\phi}}{\partial{i}}=0$$ $$\dfrac{\partial{\phi}}{\partial{v}}=E-v-i R_0$$ Where E,$R_0$ are constants. How do I solve these kind of PDE's.
2
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0answers
18 views

Second order wave equation

Consider the second order wave equation $u_{tt}=u_{rr}+1/ru_{r}+1/r^2u_{\theta\theta}$ with the boundary data $u(r, \theta, 0)=f(r)$ $u_t(r, \theta, 0)=0$ $u(1, \theta, t)=0.$ Assuming that $u(r, ...
2
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1answer
48 views

Confused over the solution of partial differential equation $xu_x+u_t=0$

Consider, $$ \displaystyle x\frac{\partial u}{\partial x}+\frac{\partial u}{\partial t} = 0 $$ with initial values $ t = 0 : \ u(x, 0) = f(x) $ and calculate the solution $ u(x,t) $ of the above ...
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1answer
13 views

discretization of mixed boundary conditions in the advection diffusion reaction equation, the Crank Nicolson method

Sorry, I have some doubts regarding the discretization of mixed boundary conditions in a PDE. I have discretized my equation, but I doubt about the boundary conditions. I dont know if you have to ...
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0answers
30 views

Dirichlet Eigenvalues of Laplace-Beltrami operator in Hyperbolic space

Consider the hyperbolic half-plane $\mathbb{H}=\lbrace (x,y)\in\mathbb{R}^2: y>0 \rbrace$ with standard Riemannian metric. The Laplace-Beltrami Operator can be written as ...
2
votes
1answer
22 views

Online resources to learn numerical methods for PDEs?

I would like to get into a career that uses alot of applied math. I took a numerical analysis course in undergrad and liked it, so I plan to self-learn numerical methods for PDEs. Other than the MIT ...
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0answers
24 views

System of First-Order Quasilinear PDEs: Burger's Equation

The Burger's equation is given by $u_{t}+uu_{x}=\nu u_{xx}$, where $u=u(x,t)$ and $\nu$ is the kinematic viscosity. How do I rewrite the equation (or any higher order PDE) as a system of first-order ...
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29 views

Careers in applied math with an MS other than in finance and data/machine learning?

Since I like math, I would like a career that uses alot of applied math. I'm about to complete my Master's and could do my thesis in numerical solutions of PDEs I'm already aware of careers such as ...
3
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1answer
34 views

Forced (-sin[x]) Burger's equation

While studying for an exam, I found this question from a previous exam: Consider the forced Burger's equation for $u(x,t)$ on the periodic domain $x \in [-\pi, \pi]$. $$u_t + uu_x = -\sin(x)$$ ...
1
vote
2answers
57 views

Eigenvalues of symmetric elliptic operators

As stated in one of my previous questions already, I have not had so much exposure to theoretical linear algebra. This time, I'm reading a theorem and proof from PDE Evans, 2nd edition, pages 335-356. ...
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1answer
26 views

Solution of eikonal equation is locally the distance from a hypersurface, up to a constant

Consider the Eikonal equation (with right handside 1) $$\sum_{i=1}^{n}(\frac{\partial u}{\partial x_i})^2=1$$ I want to see why any solution to this is locally the sum of a distance function from a ...
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1answer
20 views

Finding $k$ provided $F(x/y,z/x)=0$

If $F(x/y,z/x)=0$ we want to show that $$xz_x+yz_y=0.$$ I am not sure if I can write $z_x=F_x/F_z$ just to use the assumption? Thank you friends.
2
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2answers
28 views

Cauchy Problem and Region of Validity

I have the Cauchy Problem $$ 2xu_x+(x+y)u_y=2u $$ with data $$ u(x,-x)=\sqrt{x},x>0$$ Omitting details, my answer is $$u(x,y) =\sqrt{x}\left ( \frac{y}{2x}-\frac{1}{2} \right)^{-\frac{1}{3}} ...
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1answer
16 views

Finding $w_{x_k x_l}$, given $w := \sum_{i,j=1}^n a^{ij} v_{x_i} v_{x_j}$

I know I've asked for the umpteenth time on taking partial derivatives to solutions of elliptic PDE. But I have yet one more. Page 351 of PDE Evans: Proof. 1. We may assume $u > 0$ in $U$, for ...
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0answers
27 views

the geometry of level set of solution of elliptic PDE

Let me make my question more clear. Suppose I have a nonlinear elliptic PDE, say $$-\triangle u = u^2$$ and I am solving this problem on a nice domain, say the unit ball $B(0,1)$ and I have the zero ...
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0answers
28 views

Weakening Sobolev coefficient in an elliptic estimate.

One of the classical estimates for Sobolev norms and elliptic operators is the following: let $L$ be an elliptic operator of order $l$, then there is some $C > 0$ such that for all $u \in H^{s+l}$ ...
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1answer
38 views

Proof of Hopf's Lemma

This is a segment of the proof to "Hopf's Lemma," from page 348 of PDE Evans, 2nd edition. I have a question regarding this, at the bottom of this post. Proof. 1. Assume $c \ge 0$. We may as well ...
2
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0answers
34 views

Proof of Second Partials Test

How does one rigorously prove the second partials test without firstly assuming that $D(a,b)=AC-B^2$ that states the following: $ A=\frac {\partial^{2}f(a,b)}{\partial x^{2}},B=$$\frac ...
2
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1answer
24 views

Convergence in $L^p(0,T;L^q(\Omega))$

If $\Omega\subset\mathbb{R}^3$ is bounded, $$f_n\to f\mbox{ in }L^q(0,T;L^p(\Omega)),\,1\leq q<\infty,\,1\leq p<2 $$ and $$f_n\to g\mbox{ weak-star in } L^\infty(0,T;L^2(\Omega)),$$ then $f=g$ ...
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1answer
20 views

Products of distributions ill-defined

This question concerns distributions, as often encountered in PDE theory, which are defined as continuous linear functionals on the space $C_0^{\infty}(\Omega)$ of test functions. The product of two ...
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1answer
41 views

solving a simple inverse problem related to elliptic pde

Suppose that I have the elliptic PDE $\nabla(\nabla A(x)\cdot U(x)) = 0$ where $x \in [0,l_1]\times [0,l_2]$ with boundary conditions $U(0,x_2) = 0, U(l_1,x_2)=1$ and $U_{x_1}(x_1,0)=0, ...
1
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1answer
35 views

Linear algebra - as applied to the weak maximum principle

Currently I have little background in theory-based linear algebra (I never took an upper-division linear algebra course), so I am asking this question here. Although this is from page 345 of PDE by ...
3
votes
0answers
84 views

quasilinear partial differential equation

Given a PDE $ f e^2 \frac{\partial f}{\partial x}- e f^2 \frac{\partial f}{\partial y} + M_1 f^4 + M_2 f^2 + M_3=0 $ Note that $M_1$ , $M_2$ and $M_3$ are functions of $\cos (x-y)$ and $\sin (x-y)$ ...
2
votes
1answer
40 views

Solving the Laplace partial differential equation with particular boundary conditions [on hold]

How this Laplace partial differential equation $$ u_{xx}+u_{yy} =0 $$ with initial conditions on $y=0 $ as $$ u(x,0)=0 $$ $$ u_{y}(x,0)=n^{−1} \sin{nx} $$ has solution $$u(x,y)=n^{−2} \sin({hny}) ...
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0answers
10 views

Examples of quasilinear wave equations

Consider a quasilinear wave equation equation of the form $\sum g^{ij}(u, Du)\partial_i\partial_j u = F(u, Du)$ on $R \times R^n$ subject to initial data $u(0,x)=g, \; \partial_t u(0,x)=h.$ Given ...
1
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1answer
36 views

To show unique solution for the Laplace equation

The problem is in the top while its weak form at the end, source $\hskip 1in$ I know that the solution is unique because the boundary condition is Dirichlet. But I want to show this. How can you ...
4
votes
1answer
36 views

What is the purpose of studying Sturm-Louville eigenvalue problem?

After a cursory read on the SL eigenvalue problem, I did not immediately feel enlightened and failed find much usefulness except for knowing that SL generalizes a broader class of differential ...
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0answers
36 views

An exercise about Fourier transform and $H^s$ in Treves

In Treves, the Fourier transform is defined by $\hat{f}(\xi)=\int {e^{-i\langle\xi,\space x\rangle} f(x) dx}$. The following is the problem, where I have figured out almost all of the questions except ...
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0answers
28 views

PDE using Laplace transform

! Can anyone please explain how to solve this question?
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1answer
27 views

In what sense is the Ricci-Flow equation a “distant relative” of the Black-Scholes equation?

In the book "The Poincare Conjecture: In Search of the Shape of the Universe" by Donal O'Shea, the author states that, "The Ricci-flow equation Perelman wrote, a type of heat equation, is a distant ...
3
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3answers
130 views

Initial-value problem for non-linear partial differential equation $y_x^2=k/y_t^2-1$

For this problem, $y$ is a function of two variables: one space variable $x$ and one time variable $t$. $k > 0$ is some constant. And $x$ takes is value in the interval $[0, 1]$ and $t \ge 0$. At ...
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1answer
16 views

Reference on Riesz representation theorem for $L^p(0,T,X)$ spaces.

Brezis Functional Analysis book proves the following Riesz representation theorems for usual $L^p(\Omega)$ spaces: In what book can we find an analogous of these theorems for $L^p(0,T,X)$ spaces? ...
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1answer
12 views

Definition of $H^{-1}$ space in Evans' PDE book

Let $U$ be an open, bounded subset of $R^n.$ Evans' well known PDE book defines the spaces: -$H_0^1(U)$:= $\{f\in H^1(U): \text{there exists a sequence} \; \phi_n \to f \; \text{in the} \; H^1(U) ...
4
votes
3answers
113 views

Solution of $\frac{d^2y}{dx^2} - \frac{H(x) y}{b} = H(-x)$

Does the equation $$\frac{d^2y}{dx^2} - \frac{H(x)}{b} y = c H(x)$$ have a solution where $H(x)$ is the Heaviside step function and $b$ and $c$ are constant? Update: What about the second step ...
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0answers
31 views

Parabolic PDEs and Gradient Systems

Apologize in advance for the length of this question, I need some help in clearing some things up that I haven't quite got my head around yet. It seems to be easy to find things out about finite ...
1
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1answer
24 views

Skew-adjoint differential operator $B$ with spectrum $\sigma(B)=i(-\infty,-1]$

Consider the Hilbert space $X=L^{2}\left(\mathbb{R}^n\right)$ and the Schrödinger operator $A=i\Delta$ defined on the domain $D(A)=H^2(\mathbb{R}^n)$. It is known that the spectrum of $A$ is ...
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0answers
19 views

harmonic functions: comparison of gradients

Consider $\Omega$ a open, bounded, convex domain in $R^n.$ I am trying to justify this: Let $u, v$ non negative harmonic functions in $\Omega$ (in the Sobolev sense). Suppose that $ u = v =0$ in ...
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0answers
15 views

Heat semigroup on Morrey spaces

I have a few questions concerning the heat semigroup $e^{\Delta t}$ on Morrey spaces. (1) I read that the heat semigroup on $ M^{p}_{q}(\mathbb{R}^n):= \left\{ f \in L^{q}_{loc}(\mathbb{R}^n): ...
2
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1answer
24 views

Does separation of variables in PDEs give a general solution?

When a partial differential equation is solved using the separation of variables method, is the produced solution the most general one that satisfies the equation or have we lost some forms of the ...
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0answers
12 views

Method of Characteristics (Change of Co-ordinates)

Here below is the notes about the change of co-ordinates from $xy$-plane to $\xi\eta$-plane. I wanna ask for why dot product works for the change, i.e. $\xi=(x,y) \cdot (a,b)$ and $\eta=(x,y) \cdot ...
2
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0answers
29 views

Method of characteristics for systems of PDE (vs. Lewy's example)

Main question: How does the method of characteristics generalize for systems of first order PDE, as opposed to scalar PDE? Namely, is there such a generalization at all, and if so what information ...
2
votes
1answer
66 views

ODE $d^2y/dx^2 + y/a^2 = u(x)$

Does the following ODE: $$d^2y/dx^2 + y/a^2 = u(x)$$ have a solution? where $u(x)$ is the step function and a is constant.
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23 views

Is my function singular at these two points?

My function $S(x,y,t)$ satisfies the following PDE $$\frac{\partial S(x,y,t)}{\partial t}=-H(x,y)$$ where the known function $H=x^2+xy+y^2+\frac{1}{x-\alpha}+\frac{1}{\beta-x}$. It is clear that $H$ ...
2
votes
1answer
87 views

Solution to wave equations with predefined phase

I want to construct special solutions to wave equations, specifically to Helmholtz-equation and paraxial wave equation (PWE). Let us consider the PWE $$(\partial_x^2 + \partial_y^2 - ...
3
votes
2answers
74 views

On the solution of constant coefficients PDEs (exponential method)

Having a look to my old PDE notes, I have come across with the following problem: Consider the 2nd order PDE: $$ \varphi_{xx} - \varphi_{xy} = 0, \quad (x,y)\in \mathbb{R}^2, \quad \varphi = ...
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2answers
14 views

Prove $F(r)=Cr^2\log(r)$ is a fundamental solution for $\Delta^2$ in $\mathbb R^2$

Prove that for some $C$, $F(r)=Cr^2\log(r)$ is a fundamental solution for $\Delta^2$ in $\mathbb R^2$. Recall that $$ \Delta^2u=u_{rr}+r^{-1}u_r+r^{-2}u_{\theta\theta}$$ My answer is below.