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

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Integral form of Bessel's (n = 0)

While studying for a comprehensive exam, I have come across this old problem: Consider the Helmholtz equation in the $\mathbb{R}^2$ plane $$u_{xx} + u_{yy} + \omega^2u = 0$$ Derive an ...
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Form-invariant solution to PDEs

I'm trying to understand how to create form-invariant solutions to PDEs: $$\hat{L}u(x,t)=0$$ with the constraint $|u\big(x,t\big)|^2=|u\big(f(t)\cdot x + g(t),0\big)|^2$. During evolution, the ...
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Derivation of Lagrange-Charpit Equations

I am working through the derivation of the Lagrange-Charpit equations presented in this Wikipedia article: http://en.wikipedia.org/wiki/Method_of_characteristics#Fully_nonlinear_case I am interested ...
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1answer
28 views

About composition of Holder functions.

Let $f,g$ be Holder continuous functions with respective exponents $\alpha, \beta \in (0,1)$. More precisely $f \in C^{\alpha}(\mathbb{R}^n;\mathbb{R}^n)$, $g\in C^{\beta}(\mathbb{R}^n,\mathbb{R})$. ...
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1answer
23 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, ...
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1answer
126 views

How to construct pseudospherical surfaces from sine-Gordon solutions?

Due to my not being very skilled in differential geometry, I want to ask if there is a reference (book, paper, etc.) that explicitly works out how one constructs the parametric equations of a ...
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27 views

System of First-Order Quasilinear PDEs: Burgers' Equation

The Burgers' 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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24 views

How to interpret tensor form PDE in terms of matrix algebra

From this mathwork page "c for system", the usual second order PDE is written in tensor form: $$ -\nabla\cdot(\mathbf{c} \otimes \nabla \mathbf{u})+\mathbf{a}\mathbf{u}=\mathbf{f} $$ and ...
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1answer
52 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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2answers
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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.
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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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1answer
218 views

PDEs with non-local terms

Not sure if I've used the correct terminology here (`non-local'). I think the lack of knowing the correct terminology is why I haven't been able to find any information about my query thus far. I'm ...
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1answer
23 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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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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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 ...
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1answer
14 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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1answer
43 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, ...
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1answer
89 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 - ...
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3answers
131 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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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 ...
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2answers
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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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PDE using Laplace transform

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

Show that $L[v^2] := \Delta(v^2) + \frac{2}{w} \sum_{i=1}^n w_{x_i} (v^2)_{x_i} \ge 0$

Let $u$ be a function such that $$ \Delta u + \lambda u = 0 $$ for some $\lambda \in \mathbb R$, also let $w$ be a function such that $$ \Delta w + \beta w < 0. $$ for some $\beta \in \mathbb R$. ...
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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 $x \in [-\pi, \pi]$. $$u_t + uu_x = -\sin(x)$$ ...
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2answers
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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
21 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
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.
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Higher interior regularity

From PDE by Evans, 2nd edition, pages 332-333. My question and work shown are at the bottom of this post. THEOREM 2 (Higher interior regularity). Let $m$ be a nonnegative integer, and assume ...
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1answer
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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
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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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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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1answer
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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
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 ...
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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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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 ...
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Prove the maximum principle for a solution of the PDE $(-\Delta + \lambda)u=0$

Show that if $u$ satisfies $(-\Delta + \lambda)u=0$ in a smooth region $\Omega$ where $\lambda$ is a nonnegative real number, then u satisfies the maximum principle. We can just assume u is smooth, ...
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1answer
37 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 ...
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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 ...
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1answer
40 views

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

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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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 $\sin (x-y)$ ...
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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 ...
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Heat Equation & Fundamental Theorem of Calculus

While studying the heat equation, I ran into the following exercise: Consider conservation of thermal energy $(2)$ for any segment of a one-dimensional rod $a\leq x\leq b$. By using the ...
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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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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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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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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 ...
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How to solve a time-dependent Schrodinger equation in periodic Dirac delta potential

I'm trying to solve a 1D time-dependent Schrodinger equation: $$ i\frac{\partial \psi(x,t)}{\partial t}=\left[-\frac{1}{2} \frac{\partial^2}{\partial x^2}+V(x)+F(t)*x\right]\psi(x,t) $$ where $V(x)$ ...
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91 views

Checking Boundary Conditions for Candidate Solutions to PDE

Consider the one-dimensional heat equation $ u_{xx}=u_t$ with the boundary data $$u(x, 0)=f(x), \quad u(0, t)=u(L, t)=0.$$ The standard method of solving this equation is by finding a candidate ...
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1answer
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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? ...