Questions tagged [partial-differential-equations]

Questions on partial (as opposed to ordinary) differential equations - equations involving partial derivatives of one or more dependent variables with respect to more than one independent variables.

20,384 questions
Filter by
Sorted by
Tagged with
21 views

Prove the distribution of the point $\rho(x,t)$ at time $t$ satisfies an equality

Suppose we want to study the behavior of the gradient descent algorithm on function $F$ . But the discrete case may be a bit hard to analysis. We try to study a continuous version of it as follows. ...
• 1
6 views

• 2,433
12 views

Prove comparison principle for $u_t + f(u)_x = k u_{xx} + g(t,x)$ with $g \in L^\infty(0,T; L^2(\mathbb R))$

Let us consider $$u_t + f(u)_x = k u_{xx} + g(t,x),$$ with $k>0$, $g \in L^\infty(0,T; L^2(\mathbb R))$ and $f \in W^{1,\infty}(\mathbb R)$ and $f \not \equiv 0$ (possibly, we can also add the ...
• 7,756
16 views

PDE $\partial_t f(x,y,t)=cy (\partial_xf)+3x\partial_y(yf)+\frac{1}{2}c^2y^2(\partial_{xx}f) \\$

I have the following PDE $\partial_t f(x,y,t)=cy (\partial_xf)+3x\partial_y(yf)+\frac{1}{2}c^2y^2(\partial_{xx}f) \\$ where $c \in \mathbb{R}$ I would like to know what type of PDE is, any information ...
36 views

Is there any theorem for the solutions $u(t)$, $v(t)$ of the following differential equation? [closed]

Suppose I have the following equation (where $a(t)$, $b(t)$ and $c(t)$ are continuous functions) $a(t)(u'(t))^2 + b(t)u'(t)v'(t) + c(t)(v'(t))^2 = 0$ Is there any theorem that could possibly tell me ...
17 views

CFL condition on 1 dimensional non-homogeneous hyperbolic PDEs

I want to find the CFL condition on the upwind scheme for the following type of 1-dimensional hyperbolic PDE or a system of PDEs $$w_t+Aw_x = f(w)$$ Where $A$ is a ...
25 views

How can you derive the spacetime Fourier transform of the free Schrodinger evolution rigorously?

I'm trying to compute the spacetime Fourier transform of the free Schrodinger evolution. Consider $f\in L^2(\mathbb{R}^d)$ and $e^{it\Delta}f=:\mathcal{F}^{-1}(e^{-it|\xi|^2}\hat{f}(\xi))$ its free ...
• 43
25 views

• 41
36 views

• 4,202
57 views

Using PDE to represent a markov process

Consider a population of constant size $N + 1$ that is suffering from an infectious disease. We can model that spread of the disease as Markov process. Let $X(t)$ be the number of healthy individuals ...
• 71
23 views

Partial Integration in solving a PDE [duplicate]

Background: I've been self-studying a book on Partial Differential Equations by Walter Strauss and ran across a particularly challenging problem in a section on First-Order Linear Equations. The ...
55 views

2D Poisson equation using finite difference method in python

enter image description here# 2D Poisson equation example using finite difference method in python (version 3.10). I want to print the matrix form after solving the equation as given in the image ...
11 views

Reference for time dependent traces

Consider the spaces $H^{1/2}(0,T; L^2(\partial\Omega))$, or $L^2(0,T;H^{3/2}(\Omega))$ and what not. I'm interested in a reference book illustrating the meaning, properties of these spaces (so, ...
• 1,625
16 views

68 views

1 vote
58 views

Solving an IVBP for a metal rod dipped from cold to warm water

I have to solve the diffusion equation for a solid rod, which is in a bath of 100 degrees. At $t=0$ it is moved to a second bath, where the water temperature is 0 degrees. I prepare the diffusion ...
• 951
45 views

I have just finished 2nd year of my maths degree and I was wondering if anyone had any good recommendations for summer reading, I want to make sure I keep practising so that I don't feel underprepared ...
1 vote
33 views

The method of separation of variables in solving a PDE

Suppose we have a PDE and we can solve it with the separation of variables and let's say the variables are $x,y,z$. Is the factorized function $f(x,y,z)=g(x)h(y)l(z)$ the only possible solution? Are ...
• 31
1 vote
23 views

Weak convergence in $H^1(\mathbb{R}^N)$ implies weak convergente in $H^1(B)$

Let $\{u_n\}$ be a bounded sequence in the sobolev space $H^1(\mathbb{R}^N)$ converging weakly to some $u \in H^{1}(\mathbb{R}^N)$. How can I prove that some subsequence converges to the same limite ...
13 views

Does Homogenity implies first order PDE

I have a pde say $$f(x,y,z, a,b)=0$$ with $x,y$ being independent variable and $z$ being the dependent variable. If i can write $z=f(x,y,a,b)$ and if $f$ is a homogeneous function then can i say that ...
• 2,557
1 vote
35 views

Proof of the embedding of time dependent Sobolev spaces

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$. $H^{-1}(\Omega)$ is the dual of $H_0^1(\Omega)$. For shorthand I write $\mathcal{H} = H^1(0,T,H_0^1(\Omega),H^{-1}(\Omega))$. I want to ...
• 567
43 views

• 441
1 vote
22 views

Why $w(0,t)=w(L,t)=0\Longrightarrow w_t(0,t)=w_t(L,t)=0$

Let $w$ be a $C^2$ function in two variables, $x$ and $t$. The domain of $x$ is $[0,L]$ whilst the domain of $t$ is $t\geq 0$. Suppose that $w(0,t)=w(L,t)=0$. The apparently $w_t(0,t)=w_t(L,t)=0$. I ...
• 1,355
Consider the following nonlinear diffusion PDE $$\frac{\partial U}{\partial t} = \frac{\partial^2}{\partial x^2}[\Phi(U)],$$ where $\Phi$ is a smooth nonmonotone function of ...