Timeline for Linear dynamics in a function space
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Apr 24, 2023 at 18:14 | comment | added | CWC | @ChristianRemling Ah I see, the computation is logically very clear. Thank you. I guess I will need to ask the author of the paper regarding the notation. | |
| Apr 24, 2023 at 17:58 | comment | added | Christian Remling | I'm not sure either what the exponential notation is supposed to mean for $x\not= x_i$, but the issue itself is straightforward: for such $x$ (and let's write $y(t)=f(x,t)$ again), the equation is $y'=g(t)$, with $g(t)=\sum K(x, x_j) f(x_j,t)$ (assuming we have solved the equations for $f(x_j,t)$ already, as discussed above), so $y(t)=y(0)+\int_0^t g(s)\, ds$. | |
| Apr 24, 2023 at 17:47 | comment | added | CWC | I think my issue is mainly with the notation. I am not sure what it means to take an exponential of linear operator $t\Pi$. If I treat it like a matrix exponential, then I get a series, but then I am not sure what it means by $\Pi^2$, $\Pi^3$, $\Pi^4$, etc. | |
| Apr 24, 2023 at 17:13 | comment | added | CWC | @ChristianRemling yes, I agree that $f(x_i,t)$ for $i=1,\ldots,P$ evolves as a linear homogeneous ODE and therefore agree with $y=e^{tC}y(0)$. However, I am not sure how this generalizes to $f(x,t)$ for $x$ that is not in ${x_i}$. In the first comment, you mentioned we can find it by integration, but I am not sure what that means.. More precisely, I am not sure how to arrive at the following expression that I have in my post: $f(x,t)=e^{t\Pi}(f_0)$. | |
| Apr 24, 2023 at 16:43 | comment | added | Christian Remling | But now my first comment applies: if we relabel $K(x_i,x_j)=c_{ij}$, then this is a linear homogeneous ODE with constant coefficients for the $P$ functions $y_i(t)=f(x_i,t)$, which we can indeed solve as $y=e^{tC}y(0)$ by standard ODE theory. | |
| Apr 24, 2023 at 15:34 | comment | added | CWC | Thank you @FabianWirth I edited the post to provide more details. Yes, $x_i$ are fixed. Yes, writing as $\sum_i K_i(x)f(x_i)$ works too, assuming $x_i$ are implied in $K_i$. | |
| Apr 24, 2023 at 15:31 | comment | added | CWC | Thank you @ChristianRemling I edited the post to provide more details. $x$ is independent of $t$, and $f$ is explicitly dependent on $t$. | |
| Apr 24, 2023 at 15:27 | history | edited | CWC | CC BY-SA 4.0 |
added 816 characters in body
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| Apr 23, 2023 at 21:18 | comment | added | Fabian Wirth | And while you edit can you please also clarify the following: (i) are the $x_i$, $i=1,\ldots,P$ fixed, could we not simply write $\sum K_i(x) f(x_i)$? what are your assumption on the functions in $\mathcal{F}$? What assumption on $K$ ensures that $\Pi$ maps from $\mathcal{F}$ to $\mathcal{F}$? | |
| Apr 23, 2023 at 20:23 | comment | added | Christian Remling | @CWC: Thank you for clarifying. I don't think your question can be meaningfully answered without those details about the $x_j(t)$ you just referred to, so maybe it would be a good idea to edit your question along these lines. | |
| Apr 23, 2023 at 18:50 | comment | added | CWC | @ChristianRemling In my case, $f$ is a function of both $x$ and $t$; I should have been clear. The quantities $f(x_i)$ change over time according to the ODE. So yes, I can solve matrix differential equation to solve for the value of $f(x_i)$ at time $t$. However, I am not sure how exactly to arrive at the “functional” exponential equation that I wrote in the post. | |
| Apr 23, 2023 at 18:40 | comment | added | Christian Remling | @PiyushGrover: This is not a PDE though. It's an ODE for the quantities $f(x_i)$, and once we have solved this, we find $f(x)$ for $x\not= x_i$ simply by integration. This seems a rather strange structure, so maybe the OP can clarify if this is really what was meant. | |
| Apr 23, 2023 at 18:08 | comment | added | Piyush Grover | This is standard result in linear PDE theory. E.g., the solution of heat equation $\partial_tf(x,t)=\Delta f(x,t)$ is $e^{\Delta t}f(x,0)$. Numerous proofs...you can start by reading up Green's functions | |
| Apr 23, 2023 at 17:42 | history | asked | CWC | CC BY-SA 4.0 |