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

learn more… | top users | synonyms (2)

0
votes
1answer
11 views

Meaning of $ \Delta u \in L_{loc}^1(\Omega)$ in the sense of distribution

I have seen a sentence that says $ \Delta u \in L_{loc}^1(\Omega)$ in the sense of distributions. What does it mean?
0
votes
0answers
12 views

Is it always possible to build a fundamental solution from a parametrix?

A parametrix $F$ of an operator $P$ is almost a fundamental solution in the sense that it satisfies: $$PF=\delta+K$$ where $K$ is a correction term. What are the conditions needed on $K$ such that ...
0
votes
0answers
12 views

Exercise Problem #$3$ of Section $5.2$ of McOwen

The problem is stated via several steps: a) Given the relations: $\partial_{x} K(x-y,t-s) = -\partial_{y}K(x-y,t-s)$ & $\partial_{t} K(x-y,t-s) = -\partial_{s}K(x-y,t-s)$ PROVE THAT: If $f$, ...
3
votes
1answer
22 views

Mathieu-like equation and its analysis

I have an equation $$ \tag 1 \frac{\partial^{2} y}{\partial t^{2}} - \frac{\partial^{2}y}{\partial z^{2}} + i\frac{\partial a(t)}{\partial t}\frac{\partial y}{\partial z} = 0 $$ Here $$ a = ...
0
votes
0answers
23 views

How can I modify this simple code to include the pressure term? (1-D Navier Stokes)

I have a mathematical model that involves a cylindrical container that is being modeled with a one dimensional simplification as the system is isotropic with respect to the z-axis. As part of the ...
0
votes
0answers
28 views

step by step solution of $y''+(x-1)y=0$ by Frobenius method

I've tried solving this equation, $y''+(x-1)y=0$, but honestly I'm not sure if I've done it right. Using Frobenius Method, I got $r_1=1$ and $r_2=0$ as the indicial roots which I think under case 3 as ...
0
votes
1answer
22 views

Calculating Rate of Change

At the point $(0, 1, 2)$ in which direction does the function $f(x,y,z) =xy^2z$ increase most rapidly? What is the rate of change of $f$ in this direction? At the point $(1, 1, 0)$, what is the ...
0
votes
0answers
19 views

About the dual of Sobolev spaces

I'm reading a paper about PDE, in which the author mentioned the space $W^{-1,1}(\Omega)$, does this mean the dual space $W^{1,\infty}(\Omega)^{*}$ ? I hope not. I only know the Sobolev dual space ...
4
votes
0answers
62 views

Assumption in PDE theory

I have an exercise in PDE theory. Let $w \in C^2(U)\cap C(\overline{U})$ where $U$ is open, bounded and connected and $c \in C(\overline{U},\mathbb{R})$ with $c(x) \le 0$ everywhere. Moreover, ...
2
votes
0answers
34 views

Solving Bessel's ODE problem with Green's Function

If we have an inhomogeneous boundary value problem $x^2 y'' + xy' + (x^2 -1)y = x,$ $y(0) = y(b) = 0,$ where $b>0$ How to use Green's Funtion to Solve this problem. I am facing issues with ...
1
vote
1answer
22 views

Godunov scheme for advection equation

I'm trying to solve the advection equation $$m_t+(\alpha m)_x=0$$ with $m(0,\cdot)=m_0$ numerically using the first order Godunov scheme. Hence I write ...
3
votes
0answers
46 views

Decomposition of measures acting on sobolev spaces

This is a follow-up question to Decomposition of functionals on sobolev spaces. Let $\Omega \subset \mathbb{R}^n$ be a bounded, open set and $\mu \in H^{-1}(\Omega) = H_0^1(\Omega)^*$. Moreover, let ...
3
votes
1answer
58 views

Finite difference method works for $\frac{\partial u}{\partial t} = \frac{du}{dz}$ but not for $\frac{\partial u}{\partial t} = - \frac{du}{dz}$?

I am using the method of lines with forward differences to solve the transport equation $$\frac{\partial u}{\partial t} = \frac{du}{dz}$$ with initial condition $u(z, 0) = z$ and boundary condition ...
0
votes
0answers
31 views

Singularity in Ricci flow vs Ricci soliton

In the paper "The formation of singularity in Ricci flow" Hamilton studied systematically the possible singularities of the flow.My question is why it is important to classify Ricci solitons in order ...
1
vote
1answer
45 views

Mass conservation for heat equation with Neumann conditions

I'm trying to implement Neumann boundary conditions to solve the heat equation with an explicit scheme. I checked mass conservation, but it doesn't seem to hold. Could someone check if my boundary ...
5
votes
0answers
33 views

Solving a 2nd-order elliptic PDE with non-constant coefficients

I wonder how I can solve the following 2nd-order PDE on the positive semiplane $\{x>0\}$: $$(\partial_x^2+\frac{1}{x}\partial_y^2)\phi=\delta(x-x_0)\delta(y).$$ I notice that the l.h.s. is the ...
0
votes
1answer
37 views

$\Delta f=0$ in $\{x\in U:f(x)>0\}$ $\Rightarrow$ $\Delta f=0$ in $U$?

Let $f\geq0$ be a continuous function satisfying $\Delta f=0$ in $\{x\in U:f(x)>0\}$. I was wondering if one could follow $\Delta f=0$ in $U$, especially in the cases $f\in C^2$ or $\Delta f=0$ in ...
1
vote
0answers
23 views

Under what hypothesis on the domain is the X-ray transform/John transform operator bounded?

Let $\mathcal{L}$=space of all lines $L$. The X-ray transform is defined here: https://en.wikipedia.org/wiki/X-ray_transform $Xf:\mathcal{L}\to \mathbb{R}$ is defined by: $Xf(L)=\int_{t \in ...
1
vote
0answers
25 views

Question about equivalent norms on $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$.

Assume $\mathbb{‎‎H}=W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ with the norm induced from inner product $$\langle u,v\rangle_{‎\mathbb{‎‎H}}=\int_\Omega \Delta u \,\Delta v\, dx$$ for any $u \in ...
0
votes
0answers
9 views

book for parabolic partial differential equation

I need the book "Linear and quasilinear equations of parabolic type" By Olʹga Aleksandrovna Ladyzhenskai͡a, Vsevolod Alekseevich Solonnikov, Nina N. Ural'tseva.The price range of the hard copy is ...
0
votes
0answers
15 views

PDE in $2$ dimension involving the Laplace operator and mixed boundary conditions

I have a problem in resolving the problem of $$\begin{cases}-\operatorname{div} (\nabla u)=1 & \text{ in }\quad\Omega =[0,1]\times[0,1] \\ u=0 &\text{ on }\quad \partial \Omega (0,y) \\ ...
4
votes
2answers
43 views

How does the PDE $\,\dfrac{d^2u}{dx^2} = 0\,$ become $\,u=x\,f(y)+g(y)\,$ when integrated?

Given that $u(x,y)$ can someone please explain to me how the result as asked in the question is achieved? Steps would be really appreciated, thanks.
5
votes
0answers
151 views

Doubts relating to Spaces of type $\mathcal{S}$

I have doubts in the following two questions : 1) What is the Laplace transform of $[x^k\varphi(x)]^{(q)}$, where $\varphi\in \mathcal{S}_\alpha^\beta$ and $-\infty<x<\infty$ , ...
2
votes
0answers
28 views

Mixed Dirichlet-Neumann eigenvalue problem

Let $\Omega\subset\Bbb R^2$ be a bounded $C^2$ domain. Let $\partial\Omega=\partial\Omega_1\cup\partial\Omega_2$. Does anyone know about the existence of eigenvalues and eigenfunctions for the ...
2
votes
1answer
21 views

If the gradient of $f$ at $x$ has the same direction with $x$ for all $x$, is $f$ radial?

I would like to ask the following question: If $f:% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{n}\rightarrow% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ is ...
0
votes
0answers
6 views

General theory of Galerkin approximations for evolution equations

I'm studying parabolic evolution equations from Lawrence Evans's book and I encounter the Galerkin method for finding weak solutions. I wonder if there is a general theory (for abstract equations on ...
2
votes
0answers
39 views

$u=xf(xy)$, show that $xu_{xx}-yu_{xy} = 0$

I need to show that: $$xu_{xx}-yu_{xy} = 0$$ when $$u=xf(xy)$$ So, I did: $$u_x = xyf_x(xy)+f(xy) \implies $$ $$u_{xx} = xy^2f_{xx}(xy)+2yf_x(xy)$$ and $$u_{xy} = ...
2
votes
0answers
20 views

$1$st Order PDE

I was solving 1st order PDE which was \begin{equation} (x^2-y^2-z^2)p+2xyq=2xz. \tag{1} \end{equation} I had tried to solve this. Please tell me whether it is correct or not. ...
0
votes
0answers
15 views

Fourier Series Representation for Solid Vibrations in a Ball

I am trying to solve the following PDE eigenvalue problem for solid vibrations in a ball. \begin{cases} u_{tt}=c^2\Delta u&\text{in }D\\ u=0 &\text{on }\partial D\\ u=\phi (r,\theta) \text{ ...
1
vote
0answers
18 views

Consider a first order quasi linear equation under various Cauchy Data

Consider the first order Quasi linear PDE given by $$uu_x+u_y=1$$ As usual $$J=\begin{vmatrix} f'(s_o) & a(f(s_0),g(s_0),h(s_0))\\ g'(s_o) & b(f(s_0),g(s_0),h(s_0)) \end{vmatrix}$$ The ...
-1
votes
0answers
10 views

Inverse Parameter Estimation

I want to solve for the unknown parameters $K$, $S$ and $R$ for the flow equation $$ \nabla \cdot [K(x) \nabla w(x,t)] = S(x).\frac{\partial w}{\partial t} + R(x,t) $$ with Dirichlet boundary ...
0
votes
1answer
25 views

Problem with initial conditions and partial differential equations in Maple

I am new to Maple, trying to see how do I add initial conditions to a system of partial differential equations. Here is an example problem, which has some error in it. What is wrong? This appears on ...
0
votes
2answers
57 views

solution of variable coefficient equation

Consider the equation $$u_x + yu_y = 0$$ and I know that this PDE has solution $u(x,y) = f(e^{-x}y)$ Can someone help me to derive this PDE to get the solution? Thank you
1
vote
2answers
21 views

Quasilinear PDE:$(x^2+1)u_x +2xu_y = 0, u(0,y)=\phi(y)$

I am trying to solve the PDE $$(x^2+1)u_x +2xu_y = 0, u(0,y)=\phi(y)$$ The solution is given here at the end of the third page and begining of the fourth and the author concluded at a step that ...
3
votes
1answer
31 views

Shock formation in an inviscid Burger's like equation

Consider the equation $$\frac{\partial u}{\partial t} +\frac{\partial}{\partial x}(u^2+uf(x,t)) =0$$ where $f(x,t)$ is a suitably well behaved function. Given the initial condition $u(x,t=0)$, for ...
3
votes
1answer
39 views

Extending Sobolev functions by zero

I believe that if you have a sufficiently regular (say Lipschitz) bounded domain $\Omega\subset\Bbb R^n$, then you can extend a function $u\in H^1_0(\Omega)$ by zero, and the extension lies in ...
5
votes
0answers
94 views

Functions Satisfying $u,\Delta u\in L^{1}(\mathbb{R}^{n})$

In this paper, the authors assert that ...the domain of realization of the Laplacian in $L^{1}(\mathbb{R}^{n})$ is not contained in $W^{2,1}(\mathbb{R}^{n})$ if $n>1$. However, it is ...
2
votes
0answers
17 views

How does one find the coefficients in the solution to the Laplace equation?

I'm reading this. Equation (522) gives the general solution to the Laplace equation. What I'm stuck about, is how to determine the coefficients $a_m$, $\beta_m$, and $\theta_m$ for non-trivial ...
1
vote
1answer
31 views

Solution for PDE $f f_x$ = a $f_t$ + b

Does anyone know the solution to this PDE, with $f = f(x,t)$ $$f f_x = a f_t + b$$ the boundary conditions are: $f(0,t) = 0$, $f(L,t) = \text{const}_1$, $f(x,0) = \text{const}_2$
0
votes
1answer
62 views

About PDE problem

Let $ \, u(x,t) : \Bbb{R}\times(0,\infty) \rightarrow\Bbb{R}\, $ be $ \,C^2 \,$ such that $$ {\partial u\over \partial t} (x,t)-{\partial^2 u \over \partial x^2}(x,t)+x^2 ...
0
votes
0answers
11 views

recovering pressure term via Helmholtz-Hodge decomposition

Looking at the classical Navier-Stokes equations here. I wish to know how the pressure gradient can be recovered. In particular, I have trouble understanding how eq (1.7) can be stated as it is. ...
0
votes
1answer
36 views

When does a weak solution give rise to a strong solution?

When does a weak solution of a PDE give rise to an actual solution? I know that this is true for the Laplace equation thanks to Weyl's lemma. Does this happen in other cases? More generally what can ...
0
votes
1answer
43 views

Splitting a 2nd order PDE into a system of first order PDEs/ODEs

Consider a standard wave equation: $ \frac{\partial^2 p}{\partial t^2} = c^2 \frac{\partial^2 p}{\partial x^2} $ The question is how to formulate this as a first order system: $ \frac{\partial ...
0
votes
1answer
27 views

Inequality in Sobolev Space Involving Time

In Evans PDE book, I have the next Theorem: If $u \in W^{1,p}([0,T],X)$ then: i) $u(t) = u(s) + \int_{s}^tu'(\tau) d\tau $ for $0\leq s\leq t \leq T$ ii) $\max_{0\leq t \leq T} \| u(t)\|_X \leq C ...
1
vote
1answer
25 views

Erroneous reasoning over equivalence of hankel function with logarithm

We have the following equation/solution pairs: $$(\nabla^2+k^2)G(\mathbf{x},\mathbf{x'}) = \delta(\mathbf{x}-\mathbf{x'}) \to G(\mathbf{x},\mathbf{x'}) = ...
2
votes
0answers
23 views

Analytic version of Hilbert's XIX problem

The famous Hilbert's nineteenth problem, initially stated in the $C^\omega$ category, was reduced by Bernstein and Petrowsky to the analogous statement in the $C^\infty$ category (and, after ...
1
vote
1answer
26 views

a nontrivial inequality in the proof of weak solution of biharmonic equation

Hi I am looking at the post discussed about weak solution of biharmonic equation Proving unique weak solution. I am having trouble verifying statement 2: The bilinear operator is coercive, The claim ...
0
votes
0answers
22 views

derivatives with respect to outward pointing normal

I'm a bit curious about applying the divergence theorem (or a particular case of it, not sure). Let's say we have a twice continuously differentiable function $u$ in some open subset of ...
6
votes
4answers
178 views

$2^{nd}$ order PDE: Solution

I am trying to solve the following equation: $$\frac{\partial F}{\partial t} = \alpha^2 \, \frac{\partial^2 F}{\partial x^2}-h \, F$$ subject to these conditions: $$F(x,0) = 0, \hspace{5mm} F(0,t) = ...
3
votes
2answers
190 views

Power series expression for $\exp(-\Delta)$

I know it should be true, but for some reason I can't get the calculations to work out in order to show that if $f$ is smooth and compactly supported, the power series $\sum_{j=0}^\infty ...