# 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.

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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 \begin{equation} w_t+Aw_x = f(w) \end{equation} Where $A$ is a ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
Consider the following nonlinear diffusion PDE \begin{equation}\frac{\partial U}{\partial t} = \frac{\partial^2}{\partial x^2}[\Phi(U)], \end{equation} where $\Phi$ is a smooth nonmonotone function of ...