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

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Use the Laplace Transform to solve the following PDE.

I need to use the Laplace Transform to solve the following PDE, but I don't think I'm doing it correctly. $u_{t}(y,t)=\nu\nabla^2 u(y,t)$ with $u(0,t)=u_{0}$ and $u(y,0)=0$. What I have so far: ...
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0answers
14 views

Show that boundary layers diffuse out from the plate with speed $\sqrt{\frac{\nu}{t}}$

I was wondering if somebody would be able to help me with this problem. I know how to solve it using dimension arguments but I'm unsure what is meant by 'transform techniques'. Any help would be ...
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2answers
17 views

Difference between two solution of inhomogeneous linear equation

Show that the difference between two solutions of an inhomogeneous linear equation $Lu =g$ with the same $g$ is the solution of the homogenous equation $Lu=0$ I know the definition of linearity, but ...
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1answer
35 views

Why does the limit $ \lim_{\varepsilon\to 0+}\int_{\varepsilon}^M \frac{\varphi(x)-\varphi(0)}{x}\ dx$ exist for smooth $\varphi$?

Let $D(\Bbb{R}):=C_c^\infty(\Bbb{R})$ and $$ p.v.(1/x)(\varphi):=\lim_{\varepsilon\to 0}\int_{|x|>\varepsilon}\frac{\varphi(x)}{x}\ dx. $$ for $\varphi\in D(\Bbb{R})$. I'm trying to understand ...
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1answer
34 views

1D Wave PDE with “strange” Boundary Conditions

I've just arrived home from an exam and cannot come to terms with the fact I couldn't solve the following question: Find a solution to $$ \left\{ \begin{array}{ll} u_{tt} - u_{xx}, & ...
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0answers
25 views

Solving Heat Equation with Laplace Transform

I am trying an alternative method to separation of variables to the following equation $$ \begin{cases} u_{xx} =4u_t , 0 < x < 2, t>0\\ u(0,t)=0, u(2,t)=0, t>0\\ u(x,0)=2\sin(\pi x), 0 ...
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1answer
30 views

Fundamental solution Laplace-Poisson equation

Let $\Phi:\mathbb{R}^n\setminus\{0\}\rightarrow\mathbb{R}$ be the fundamental solution of the Laplace equation (see e.g. in the book of Evans). For a function $f\in\mathcal{C}_c^2(\mathbb{R}^n)$ we ...
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1answer
27 views

Transport equations with constant coefficients

Let $X$ be the vector field given by $X = b \cdot \nabla_x + \partial_t$ where $b \in \mathbb{R}^n$ is fixed. Let $f \in C^1(\mathbb{R}^{n+1})$. Assume that $u \in C^1(\mathbb{R}^n \times ...
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1answer
16 views

Partial Differential Equation Formation (Arbitrary Functions) [on hold]

Form the partial differential equation by eliminating the arbitrary functions from: $$z=f(x^2+y^2)+x+y$$
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1answer
25 views

System with arbitrary function of an unknown

How can I solve the following system $$ (u_x)^2 - (u_t)^2 = 1 \\ u_{xx} - u_{tt} = f(u) $$ where $f$ is an arbitrary function of $u$, $u$ and $f$ to be determined. I don't know any approach, ...
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1answer
15 views

Bounded linear functionals in solving PDE

Many theorems in functional analysis, Rietz, Hahn-Banach, for example, are used to find linear functionals in certain spaces. But why are bounded linear functionals useful in solving PDE? This ...
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12 views

Steklov eigenvalue on unit ball [on hold]

Show that the eigenvalues of the Dirichlet-to-Neumann map of the unit ball $B^n$ of the n- dimensional euclidean space $R^n$ are 0, 1, 2, ... . Furthermore, the eigenspace of k is given by space of ...
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0answers
19 views

Existence of solution to a linear second order PDE

I want to show the existence and the uniqueness of a solution to the following partial differential equation: \begin{equation} ...
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0answers
27 views

Banach contraction mapping to find unique solution of $y(x) = \int_0^1 G(x,s) f(s,y(s)) ds$

Problem statement: Use the Banach contraction mapping theorem to show that $$y(x) = \int_0^1 G(x,s)\ f(s,y(s))\ ds$$ has a unique, continuous solution $y = y(x)$ if (1) $G(x,s) \ge 0$ for $0 \le x, ...
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1answer
28 views

Evans pde book: details on an bound for a Sobolev norm in the proof of the Meyers-Serrin theorem

Let $U$ be an open subset of $\mathbb{R}^n$ and $f\in W^{m,p}(U)$. Suppose that $$\|f\|_{W^{m,p}(V)}\leq\delta\tag{1}$$ for all $V\subset\subset U$ (that is, all $V$ such that $V\subset\overline{V} ...
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1answer
24 views

Prove why the shock equation is not linear.

I need a hint not an answer before answering this question: Two part question: The homogeneous shock equation is given by $u_x$ + $u$$u_y$ = 0 Part 1) Show why the shock equation is not linear. ...
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1answer
36 views

Locally flat manifold from Frobenius, differential forms approach

It is a known fact that a Manifold $M$ is locally flat if and only if the Riemann curvature tensor (Levi-Civita connection) vanishes. To show the if part is easy, but I am trying to follow the details ...
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4answers
56 views

Solving this 1st Order PDE [on hold]

I am trying to solve the following PDE with an initial condition: $$u_x + u_y = x + y$$ with $$u(x, 0) = 0$$ I am not sure which method to use to solve this. Thanks
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1answer
30 views

Solving this 2nd Order non-homogeneous PDE

I am trying to solve the following equation: $$3u_{xx} - 10u_{xt} - 3u_{tt} = \sin(x + t)$$ I know that the left hand side is a quadratic equation which I have to factorise. Then I let one of the ...
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1answer
36 views

Gradient flow of Dirichlet energy

I have heard that the gradient flow of the Dirichlet energy gives a solution of the heat equation, i.e. if $u(t,x) \in C^1( [0,\infty) \times \mathbb R^d)$ solves $$ u_t(t,x) = - dE(u(t,x)), $$ where ...
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0answers
26 views

Derivative of mollification

This is in response to a claim made in the second line of the question here, namely: Given the standard mollifier $\eta$ and a locally integrable function $f:U \rightarrow \mathbb{R}^n$, by defining ...
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0answers
19 views

Uniqueness of the Green's function

Given a linear operator $L$, a Green's function $G(x,s)$ is any solution of $$\tag{1} LG(x,s) = \delta(x-s)$$ where $\delta(x-s)$ is the Dirac Delta function. The Green's function can also be used in ...
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1answer
32 views

Finding general solution to Partial Differential Equations

I am asked to find the general solution $f(x, y)$ of the partial differential equation: $\frac{\partial ^2 f}{\partial x \partial y}=e ^ {x+2y}$ I know these are relatively easy to solve, I haven't ...
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0answers
7 views

What is the initial data of a fundamental solution?

Let $(M,g)$ be a globally hyperbolic Lorentzian manifold. Consider now the differential equation $\square_{g}\phi=0$. I am aware that a fundamental solution is a distribution that must satisfy: ...
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4answers
52 views

What is the essential difference between ordinary differential equations and partial differential equations?

Please forgive my stupidity. So many years after my undergraduate study and so many years after dealing with various concrete ODEs and PDEs, I still cannot tell the essential difference between ...
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1answer
20 views

An easy computation of an integral

I have a question about an integral. I don't know how to quick compute $$\int_{\mathbb{R}^n}\phi(x,1)|x|^2dx$$where$$\phi(x,1)=(4\pi)^{-\frac{n}{2}}e^{-|x|^2/4}$$ which is the fundamental solution of ...
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2answers
54 views

Which of these 1-D representations of the Navier-Stokes equations is correct?

The incompressible Navier Stokes equations can be written as A. $$\frac{\partial (\rho \mathbf{v})}{\partial t} + \nabla \cdot (\rho \mathbf{v} \mathbf{v}) = S$$ or B. $$\frac{\partial ...
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1answer
18 views

Heaviside function and weak solutions

If a solution to a PDE has a solution f(x) in which f(x) is a Heaviside function, independent on its argument, can I say the solution is unstable, therefore being weak?
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23 views

A question on vanishing viscosity in Evans PDE

In Evans book on PDE (2nd edition) he uses the vanishing viscosity method to prove existence of a weak solution to the following linear hyperbolic system ...
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1answer
45 views

Solve the PDE: $u_{xx} - 3u_{xt} - 4u_{tt} = 0$

It is asked to solve the PDE $$u_{xx} - 3u_{xt} - 4u_{tt} = 0$$ using a factorization, that consists in $$\left( \frac{\partial}{\partial x} - 4 \frac{\partial}{\partial t} \right) \left( ...
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1answer
57 views

What Does the Term “Regularity” Mean?

When I was an undergraduate, I took a course on regularity theory for nonlinear elliptic systems. This included topics such as the direct method of calculus of variations, mollifiers, integration over ...
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1answer
42 views

What can we conclude about the solution when integrating the equation of characteristics in this PDE problem?

I've come across the following PDE problem: $$\frac{\partial u}{\partial t} + 2tx^2\frac{\partial u}{\partial x} = 0, \\ u = u(x,t) \\ u(x,0) = x^3$$ This is a first order, linear, homogeneous ...
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21 views

Trying to model a substance settling in water using an advection equation?

I am trying to model a substance dispersed in a container of water gradually settling at the bottom. I am considering only one dimension. The top is at $z = 1$, and the bottom is at $z = 0$. So at $t ...
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14 views

Query about estimating an integral in Heat Equation

While studying the Heat Equation (P-309) from the book : 'Front Tracking From Conservation Laws' by Holden & Risebro; I have gone through the following calculation: " $\int_{\mathbb R} ...
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1answer
39 views

What does the symbol $H^1_0(\Omega)$ mean?

Here $\Omega \subset \mathbb{R}^n$ is a closed disc centred at $0$ with radius $r$.The book I am reading is assuming the Dirichlet boundary condition on $\Omega$ and claiming that the dual of ...
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The Role of the Courant-Friedrich-Levy Condition in Computational Partial Derivatives

The Role of the Courant-Friedrich-Levy Condition in Computational Partial Derivatives In Computational Partial Differential Equations (CPDEs) difference schemes are used to workout an approximate ...
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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?
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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 ...
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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 $f$, ...
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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 = ...
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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 ...
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ELITIC PDE - derivatives and Hessian estimates [closed]

I am dealing with minimal surface problem, from which I need to obtain gradient and Hessian informations $$A= \int \int (1+u^2_x + u^2_y)^{1/2} dx dy$$. Does anybody know, how to proceed? thanks!
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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 ...
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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 ...
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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 ...
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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, ...
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0answers
33 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 ...
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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 ...
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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
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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 ...