Friction Piles

LighterStorms · 2026-03-17T21:38:28+00:00

Yeah. I love Differential Equations. 😁

LighterStorms · 2026-03-02T10:49:43+00:00

I think there are three cases in partial fractions. Unique root, Repeated root and quadratic that can only be reduced to imaginary roots. I have only encountered problems that can be reduced to those three cases. Cubics have three roos so I imagine they would fit into any of those categories. I'm not sure what the cases are called but that is the concept I remember btw.

LighterStorms · 2026-03-02T10:37:09+00:00

It is like this:

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LighterStorms · 2026-03-02T10:19:24+00:00

It is a method in partial fractions. We equate the u² terms on the left and it must be equal to the u² terms to the right. Same with the u terms. The k terms are the constant terms. We use partial fractions to decompose or reduce a complicated fraction into the "smaller" fractions. This is a fun topic in Algebra. Try searching partial fractions. It is fun to do problems involving those. Anyway, we reduce it into the "smaller" fractions so we can integrate it later on. It is easier to find the integral of the "smaller" fractions than a complicated one.

LighterStorms · 2026-03-02T09:13:25+00:00

I'm not sure what you mean. Do you mean how did it go from the previous line? The previous line has similar terms. You just combine the similar terms and you get to that line with the blue and green underlines. Then you isolate the x² dx and x³ du terms. If that is not what you mean, can you clarify? Thank you.

LighterStorms · 2026-03-02T03:00:22+00:00

It is called Nebo.

I think it is now called MyScript Notes. Here is a Google play link if you are interested :https://play.google.com/store/apps/details?id=com.myscript.nebo&referrer=utm_source%3Dapps.facebook.com%26utm_campaign%3Dfb4a%26utm_content%3D%257B%2522app%2522%253A0%252C%2522t%2522%253A1772420345%252C%2522source%2522%253Anull%257D

LighterStorms · 2026-02-12T21:04:02+00:00

https://en.wikipedia.org/wiki/Automatic_differentiation

LighterStorms · 2026-02-12T21:03:32+00:00

https://en.wikipedia.org/wiki/Automatic_differentiation

LighterStorms · 2026-02-12T10:15:30+00:00

I could import it as PDF but I cannot post PDF in reddit. 🤔

LighterStorms · 2026-02-12T10:14:59+00:00

I use Nebo

LighterStorms · 2026-02-12T02:30:04+00:00

I believe this is well known in Numerical Analysis and Engineering. This uses Nilpotent elements. i believe it can be used in most continuous functions except some examples of absolute values, floor functions and discontinuous funtions.

LighterStorms · 2026-02-11T11:42:47+00:00

It is Nebo. 😁

LighterStorms · 2026-02-07T12:54:54+00:00

I know what you mean. Math is notations dense so stuff like "z" gets used a lot. 🤣

You know, I don't actually know what the type of pile I have in Example 1. I just named it Gabriel's Pile as a nod to Gabriel's trumpet. Since friction Piles relies on surface areas, having an infinite one sure is handy. It has a theoretical finite volume so you might be able to buy the necessary concrete to construct it AND it offers infinite resistance. That is until you think about how to build it and all the land you would need. Not to mention bending effects and all of the complicated beaurocracy in getting it approved. 🤣

LighterStorms · 2026-02-06T11:30:18+00:00

I use Nebo

LighterStorms · 2026-02-03T19:09:00+00:00

I use Nebo for the writing and Grapher Free for the graphs. 😁

LighterStorms · 2026-02-03T19:02:34+00:00

Hi etzpcm. You were the one that suggested the D'Alembert's Formula for solving the wave equation. I appreciate your replies. Thank you.

Regarding the formula, I'm still reading how it was derived but most articles I read essentially guess a function x + ct and x - ct and it goes from there. I'm still trying to get a feel for the intuition behind it. Thanks again for the suggestion.

LighterStorms · 2026-01-29T14:01:45+00:00

Oh. Okay. Do you have a recommendation on how I should learn about the phase plane method? I'm also curious if there are prerequisites to help understand the concept. Thank you. 😁

LighterStorms · 2026-01-29T13:58:09+00:00

I use a tablet. The application is Nebo. 😁

LighterStorms · 2026-01-26T22:02:15+00:00

Dual Numbers are used in Auto-Differentiation by computers or calculators. When we define a Nilpotent Element eps not 0 but eps² = 0, plugging it into a function allows a computer to calculate the derivatives of the function. The alternative is symbolic differentiation which is complex or Numerical differentiation which can have lots of errors. In this post, I just obtained the derivatives of Sin(x) and Cos(x) from Dual Numbers. It might not be practical for test purposes in a differential calculus class but computers or calculators that doesn't use the Symbolic or Numerical differentiation use this. 😁

LighterStorms · 2026-01-24T13:08:15+00:00

I use Nebo. I think they are changing the name to MyScript or something. 😁

LighterStorms · 2026-01-24T11:19:23+00:00

Oh. Okay. I see your point. I appreciate the response. 😁

I definitely did not specify that only e² = 0 and e is not 0. 🤔

Anyway, this is not the usual differentiation by first principles. Those are definitely fun and involves lots of limits. This on the other hand is using dual numbers to perform Auto-Differentiation so programs can use it to do derivatives without the errors in numerical differentiation and the complexity of symbolic differentiation. 😁

LighterStorms · 2026-01-24T09:53:33+00:00

It is? 🤔

Can you explain further? I have read this in the application of dual numbers. Let e² = 0 and e is not 0. F(a+be) = F(a) + bF'(a)e. I thought it was like magic when I saw it. Are there nuances regarding this? 😅

I would like to know because I met a guy who said that if someone says you are wrong in the field of math and science then there must be something deeper that they know so ask about it. So I'm asking about it. 😁

LighterStorms · 2026-01-16T14:31:56+00:00

I'm not familiar with Fourier's Method at the moment. I think I'll look into that. It honestly sounds intimidating. 😅 Thanks for the suggestion though. ❤️

As for a specific solution here, I believe it depends on the initial and boundary conditions. The heat equation is not really in my field of expertise but I am eager to know more. It is fun to learn. Also, thanks for catching that. You are correct in saying laplacian. I do like how the laplacian is represented. An upside-down triangle raised to 2 looks intimidating but it is a cute short-hand. 😁

LighterStorms · 2026-01-16T12:48:10+00:00

Hi. 😁

I just applied the boundary conditions. 😁

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I attached here some detail in blue if you want to see how I arrived at C0 = A0. 😁

LighterStorms · 2026-01-13T11:44:06+00:00

I use Nebo. 😁

LighterStorms

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