Thanks, this was cool. I've seen all of the parts of this, but not put together. I have questions. I'm familiar with the term singular in linear algebra, but not in this context. It sounds like singular in this context is an improperintegral that diverges? In the second example, it looks like you took the double integral and turned it into a convolution integral? I've run into them in probability, but I don;t know a lot about them. I'll have to watch your video on the ADM, but it looks like a taylor series? Thanks?
@MathematicalToolbox
5 ай бұрын
As far as question one goes, not necessarily! The integral can converge. Most examples in books of singular Volterra integral equations are given in the form of Abel's Problem or Abel's Integral Equation. It can definitely have a closed form solution. I apologize for not being more thorough. For question two: indeed it is! Astute observation, I actually had not made that connection yet. Thank you for that. I actually had to look it up to verify what you're asking. For question three, yes, the ADM utilizes Taylor series and geometric series. In particularly nasty problems it can utilize both or even the Taylor series of two functions. Great stuff. Thanks again!
@walter274
5 ай бұрын
@@MathematicalToolbox I really appreciate your response! I will have to look up Abel's problem. I'm not familiar with it. Thanks for the videos and discussion!
@kowalguitar
5 ай бұрын
What is the application?
@MathematicalToolbox
5 ай бұрын
Another subscriber asked me this question so I will copy paste what I told him: I asked ChatGPT and I summarized what it told me down below: -Lippmann-Schwinger Integral Equation from Quantum Mechanics -Ornstein-Zernike Integral Equation in Statistical Mechanics -Integral equations are used in solid mechanics to model stress and strain distribution within solid materials. -Integral equations are also used in field theory to describe the behavior of electromagnetic or gravitational fields. I also referenced the applications in the integral equations book by Jerri: -Population dynamics -Torsion of a wire -Shape of an elastic thread/The hanging chain -Small deflection of a rotating shaft -Abel's Problem (The Tautochrone) -Radiation Transport -Electric potential on the rim of a unit disc hope that helps!
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