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

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PDE question: heat equation (third order??)

I am familiar with the usual heat equation, however, my lecturer gave me this problem and it does not look like anything I have ever seen (in my whole entire life and I am not just being dramatic). ...
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27 views

PDE Heat Equation with Variable Coefficient {Second ODE Variable Coefficient}

Another PDE question: If I have a non constant coefficients in my heat equation (PDE), how do I solve it? For example we have: $\frac {\partial T}{\partial t} =\frac {\partial ^2 T}{\partial r^2} + ...
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0answers
11 views

Can we use method of reflection to find Green's function in infinite strip?

I have learned how to use method of reflection to find Green's function of Laplacian equation for Dirichlet problem in half-space or quadrant in my undergraduate pde course. Now I am wondering how to ...
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1answer
401 views

Use the method of characteristics to solve nonlinear first order pde.

I find this problem challenging: Use the method of characteristics to solve $u_t+u_x^2=t$ with $u(x,0)=0$. I know I'm supposed to let $p=u_x$ and $q=u_t$. Then I get $F(x,t,u,p,q)=p^2+q-t=0$. But ...
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1answer
42 views

Wave Equation Partial Differential EEquation

Basically I got a simple wave equation with an extra twist. The PDE is $\frac {\partial^2 y}{\partial t^2} = c^2\frac {\partial^2 y}{\partial x^2} + L $ with homogeneous boundary condition As usual, ...
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2answers
84 views
+150

Why do we take the odd extension?

When we have the initial and boundary value problem $$u_{tt}(x,t)-c^2u_{xx}(x,t)=0, x>0, t>0 \\ u(0,t)=0 \\ u(x,0)=f(x), x \geq 0 \\ u_t(x,0)=g(x), x \geq 0$$ can we apply Green's theorem or ...
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1answer
37 views

How to deal with $u_{ttt}$ in derivatives estimates of $u_{tt}$ if $u_{ttt}$ is not defined?

Suppose we have proved the equation $$u_{tt}-u_{xx}=0\quad\text{in}\quad(0,T)\times(0,\ell)$$ with some boundary and initial conditions has an unique solution $u\in C^2(0,T,H^2(0,\ell))$ and we need a ...
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2answers
37 views

Analytic solution to Poisson equation

I need to find the analytic solution to this equation, in order to compare it with solution I get from using a numerical solution. However, I have not been able to find the solution. I think I can't ...
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1answer
18 views

Harmonic function satisfying given condition

In trying to solve a homework problem I end up having the equation $\Delta f=0$ knowing that $f(x)=f(\||x\||^2)$ where $x=(x_1,\dots,x_n)$. In 2-d it leads me to $f_{x_1} + f_{x_2} + x_1 f_{x_1x_1} + ...
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0answers
7 views

$\Gamma$-convergence (Gamma-convergence) and PDEs?

My question is about the applying calculus of variations to solving Partial Differential Equations. In particular, what is the idea behind using $\Gamma$-convergence to find weak solutions of PDEs? ...
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2answers
52 views

The mysterious $\dot{H}^{-1}$ notation.

I have encountered the $\dot{H}^{-1}$ notation in one of the SIAM Journal on Mathematical Analysis articles. It appears to be standard (or at least not uncommon) to use this one in the field, since ...
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1answer
67 views

analytical solution for linear 1st order PDE using laplace and seperation of variables

I am looking for the solution of the following pde: $\frac{\partial y(x,t)}{\partial t} = a* \frac{\partial y(z,t)}{\partial x} + b* y(x,t) + c$ and need help with the boundary and initial ...
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1answer
28 views

Solving laplace equation and decomposing

I want to solve $$ 9U_{xx}+U_{yy}=\sin (2\pi x) + \sin(2\pi y) \label{eq:1}\tag{1} $$ with $U=0$ on the boundary of the unit square. I know you would have to decompose the problem to satisfy each of ...
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0answers
8 views

Given a piecewise initial condition, how can the characteristic curve x be sketched when the solution x does not contain u terms?

The charac equation for x: $$\frac{\text{dx}}{\text{d$\tau $}}\text{=2t}$$ The solution x is $$x=t^2+x_0$$ Note that $$\tau=t$$ There is a problem. In order to sketch x, I require some ...
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1answer
22 views

Solving ODES in PDE

The PDE given as: $$t^2u_t-\text{yu}_x+\text{xu}_y\text{=0}$$ The characteristic equations are: $$\frac{\text{dt}}{\text{dt}}=t^2$$ $$\frac{\text{dx}}{\text{dt}}\text{=-y}$$ ...
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0answers
20 views

What method do we use to find the solution?

Find the solution of the initial and boundary value problem $$u_t(x,t)-u_{xx}(x,t)=0, x>0, t>0, \\ u(x,0)=f(x), x>0,\\ u(0,t)=0, t>0 $$ (The solution should be expressed as an integral ...
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63 views
+50

Energy inequalities with negative sobolev number.

Let $\phi\in H^{s}$ such that the following energy inequality is true: $$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\cdot)\|_s \, dt $$ where $P$ is a strictly hyperbolic linear operator. For ...
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0answers
25 views

heat equation pde Newton law of cooling boundary condition discrepancy

If $T(t)$ denotes the temperature of an object in an environment with temperature $T_0$, then Newton's law of cooling says $$ \frac{dT}{dt} = -k(T(t) - T_0).\quad (*) $$ This is an ODE. Consider now ...
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0answers
58 views

What type of equation is this?

Is this equation an ODE or PDE $$ \frac{d^3u}{dx^3}−αxu=0, x∈R $$ The only thing given is $\int_R u(x) =\pi $ and $α>0$ is some constant. I have to find the solution using fourier ...
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0answers
11 views

Proper writing of IBVP PDE & Finite Difference Implementation

I've seen a few examples (see slide 4 ) of the 2D heat equation described as $$ \begin{cases} u_t(t,x,y) = \nabla^2 u(t,x,y), \quad t > 0, \quad (x,y) \in (0,L) \times (0,H), \\ u(t,0,y) = ...
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1answer
10 views

Dirac Delta function to solve PDE in the sense of distribution

For each Borel set $A$ and $x\in\mathbb{R}^n$, denote Dirac measure centered at $x$ as \begin{equation} \delta_x(A)=\begin{cases} 1 & \mbox{ if } x\in A \\ 0 & \mbox{ otherwise}. \end{cases} ...
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1answer
14 views

About fractional Sobolev space

I'm reading a paper used Sobolev space $H^s(\Omega)$ , I only know the definition of these space when $\Omega=R^n$ which used Fourier Transform, what if $\Omega$ is a bounded open set? And I also ...
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0answers
18 views

Expansion wave in PDEs

The book I'm using is "Applied partial differential equations" written by Richard Haberman. On page 556, he gives the definition of expansion waves as "The distance between $$p(_{x_{1}},0)$$ and ...
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1answer
39 views

If the integrals of a harmonic function over horizontal lines are uniformly bounded, it is identically zero

Let $u\colon\mathbb{R}^2\rightarrow\mathbb{R}$ be a harmonic function, such that $$\int\limits_{-\infty}^{+\infty} \lvert u(x,y)\rvert dx < C,$$ where $C>0$ is a constant not depending on ...
2
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1answer
40 views

Find a solution that satisfies Laplace's equation in polar coordinates

How may I find a solution that solves Laplace's equation in polar coordinates, subject to the boundary conditions? In particular, I need to find one solution that satisfies $$\Delta u = 0,$$ subject ...
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0answers
26 views

Solving the Wave Equation using Fourier Transforms

The problem is: \begin{equation} u_{tt} -c^2u_{xx} - a^2 u = 0 \end{equation} with $\hspace{2mm}-\infty < x < \infty $, $ \hspace{2mm} u(x,t) \hspace{2mm}$ bounded as $ x \rightarrow \pm ...
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0answers
40 views

Heat Equation in spherical coordinates

Consider the problem of a sphere of material that starts at a non-uniform temperature, $T = r^{2}$ and is covered with insulation on the outer surface so that no heat gets out. We take the coordinate ...
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1answer
25 views

Is there any nonnegative $u\in C^2(\mathbb{R}^n)$ with $-\Delta u=1$ in $\mathbb{R}^n$?

Is there any nonnegative $u\in C^2(\mathbb{R}^n)$ with $\Delta u=-1$ in $\mathbb{R}^n$? I think not, but how can we prove it? Let's assume that such a solution exists. Let $R>0$ and $B_R:=B_R(0)$ ...
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15 views

Problem on Majda's Vorticity and Incompressible Flow

I'm reading Majda's book, on page 110, I cannot understand how to get $v\in C_W (0,T;H^{m})$ from the above. And he wrote "$[\phi,v^{\epsilon}]\to[\phi,v]$ uniformly on $[0,T]$ " two times, do the ...
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3answers
2k views

How to solve the non-homogeneous PDEs: $u_x + u_y=2u,\ u(x,0)=h(x)$?

I have this first-order non-homogeneous partial differential equation with initial condition: $u_x + u_y=2u,\ u(x,0)=h(x)$ The following was what I tried: By the method of characteristic curves, we ...
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0answers
7 views

How to define boundary conditions for a sphere to run reaction-diffusion equations on its surface?

I'm in a Biology lab, and we managed to simulate reaction-diffusion equations on a torus using periodic boundary conditions for a 2D matrix. We want to try doing the same on a sphere, but I'm a ...
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1answer
14 views

PDE Boundary conditions for characteristic solution for when x=0 and t is some constant

Solve $$\frac{dw}{dt}+4\frac{dw}{dx}=0$$ With initial condition $$w(0,t)=Sin(3t)$$ The characteristic equations are: $$ \frac{dt}{dt}=1, \frac{dx}{dt}=4, \frac{dw}{dt}=0$$ The characteristic ...
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16 views

Solutions of $u(x)=\int_{\mathbb R^n} |x-y|^p u(y)^{-q} dy$ are bounded away from zero

In one of research papers I am interested in, see this link or this if you cannot access, there is a lemma, Lemma 5.1, saying that if $n\geq 1$, $p,q>0$ and $u$ a non-negative Lebesgue measureable ...
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22 views

Question on: gradient & laplace operator

given the function $f(x,y,) = x^2-y^2$. the gradient should be given by $grad f = (2x, -2y)$. If I'm drawing single of these vectors, I only get the ones on the positive x-axis. Is this correct? ...
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1answer
35 views

Difficult integral $\frac{du}{u}=\left(\frac{x+y}{x}\right)dx$ in PDE

The linear problem is given as $$x\frac{\text{$\delta $u}\backslash }{\text{$\delta $x}}\text{+y}\frac{\text{$\delta $u}\backslash }{\text{$\delta $y}}\text{=(x+y)u}$$ with $u = 1$ on $x=1$ with ...
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2answers
62 views

Calculating $a_n$ in $\sum_{n=1}^\infty a_n \sin(\frac{n \pi}{2})=T_0$

I'm looking to solve the following when $T_0$ is a constant: $$\sum_{n=1}^\infty a_n \sin\left(\frac{n \pi}{2}\right)=T_0$$ If it matters this was reached from the following: ...
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0answers
24 views

a question about prove an exponential matrix function can be infinitely differentiable

If I have an exponential matrix exp(t(U+sH)), can someone tell me what is the dirivative with respect to s? I am really confused. (where U and H are matrices,and s,t are real numbers). Thus,if I let ...
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1answer
16 views

Determine the Fourier transform of $f(x) &

f(x)=1 if |x| < a or f(x) = 0 if |x| > a We use the formula $$ {1\over 2\pi} \int_{\infty}^\infty f(\bar x)e^{i\omega \bar x} $$ So is $f(\bar x)$ the same as $f(x)$ ?? In an answer they ...
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10 views

What it means for a Jacobian determinant to be zero in the context of PDEs and their solution?

The book mentioned that if the Jacobian determinant is zero then no solution exists in the neighbourhood of the boundary curves. What does this means in simplified terms? What are boundary curves? I ...
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1answer
22 views

What is vectors straddle a plane mean?

There is a condition in a paper, saying that two vectors straddle a plane. How can we transfer this condition to a equation? Because I have another 5 equations and need this one to solve 6 unknowns. ...
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1answer
25 views

Proof of uniqueness for the Poisson equation

Show that the following problem has at most one solution: Given a continuous function $\rho(x,y,z)$ which is zero for $x^2+y^2+z^2>a^2>0$, find $\phi$ such that $$\nabla^2\phi=\rho$$ ...
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0answers
22 views

Eliminate the arbitrary Function of PDE

I need to solve this problem; Eliminate the arbitrary function $f$ from the equation: $f(x^2+y^2+z^2,z^2-2xy)=0$ I try this solution $u= x^2+y^2+z^2, \quad $ $\quad v=z^2-2xy, \quad$ so ...
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2answers
63 views

Eliminate the arbitrary funcion - PDE first order

I'm heading the book Elements Of Partial Differential Equations -Sneddon 1957. At chapter two exists this exercise "Eliminate the arbitrary function $f$ fron the equation $$ z= ...
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3answers
42 views

solving a second order nonlinear pde

I would like to solve the following PDE, $$f_{y}^{2} = 2 f f_{yy}$$ where $f= f(x,y)$ is a real function of two variables $x,y$. My solution : derivative of $f_{y}^{2}$ with respect to $y$ is itself, ...
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1answer
19 views

Probability density function for a PDE with random inputs

I am looking for a general method or alternatively few textbook examples of deriving a probability density function for a solution of partial differential equation with random inputs in the equation ...
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0answers
28 views

Initial and Boundary value problem [on hold]

Solve the following initial-boundary value problem: $$\frac{\partial u}{\partial t} = a^{2}\frac{\partial ^{2}u}{\partial x^{2}}-b(u-u_{0}), t>0, 0 <x<L, \\ u(0,t) = u(L,t) = u_{0}, t > ...
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1answer
27 views

Definition of 'blow up' in the context of PDEs

What exactly is 'blow up'? Is there a proper well-defined definition for this term? What does it means mathematically? Does it implies 'infinity'?
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3answers
40 views

Solutions of the Laplace equation

How do I find solutions $u=f(r)$ of the two-dimensional Laplace equation $u_{xx}+u_{yy}=0$ that depend only on the radial coordinate $r= \sqrt{x^2+y^2}$
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1answer
15 views

Superlinearity in the definition of the Legendre transform

Suppose the Lagrangian $L:\Bbb{R}^n\to\Bbb{R}$ satisfies the following conditions: $L$ is convex $$ \lim_{|v|\to\infty}\frac{L(v)}{|v|}=+\infty $$ Define the Legendre transform of $L$ as $$ ...