In my old ODE textbook (Modern Differential Equations: Theory, Applications, Technology, 1st Edition (1996), Abell and Braselton), Green's Function is covered for a 2nd order linear non- homogeneous ODE in standard form where the initial conditions are bot set equal to zero. Variation of parameters is the technique used to generate it. Depending on the ODE and conditions, the function will be different I suppose.
It's extremely helpful...I would be glad if you would use it to show us some examples of the application of green function in solving boundary value problems of electrostatics
At 4:46 you have (lambda-lambda(n)). lambda(n) id OK, its the eigenvalue associated with eigenfunction (n). But what about the first lambda, where does this nunber comes from? There is no general lambda, they are all associated with some eigenfunction. This lambda, it seems, is not. Please explain.
As mentioned in my answer down below, we don't really define lambda as a particular number; it's just a real number and given its position in the ODE, it is called an eigenvalue. As you said, it's not associated with a particular eigenfunction. On the other hand, lambda_n is the eigenvalue that is specific to the solution y_n to the corresponding homogeneous S-L (Sturm-Liouville) problem.
All the terms in the sum go away except when m = n (see 5:00 onwards) because the integral is zero whenever m and n aren't equal. That's why the sum disappears from the expression, since only one term remains after all the cancellation.
I've watched the video and there are two things that aren't clear to me: -When do we define lamda with no subscript, how is it separated from the one with subscript? We carry it all the way to the green's function without specifying anything about what it might be: Is it a variable? some constant? -why would there ever be f(z)=delta(z-x)? f(x) is originally the given non-homogeneous function, so it could be given as f(x)=delta(x-a), but then f(z) will be delta(z-a) switching from x to z should eliminate all the x variable, how can it leave one behind for the impulse response?
1. We don't really define lambda as a particular number; it's just a real number and given its position in the ODE, it is called an eigenvalue. lambda_n is the eigenvalue that is specific to the solution y_n to the corresponding homogeneous S-L (Sturm-Liouville) problem. 2. I've used f(z) = delta(z-x) for notational convenience. Ultimately, I want my solution y to be in terms of a variable x (i.e. as y(x)). This is why I don't want to use x, because when I do the integration at 7:00 to get my final answer, I want to keep the x and remove another independent variable like z. Ultimately though, I could have used f(x) = delta(x-a) but then when I do the integration, I would have *had* to integrate over x and remove it from my final answer, leaving a function of 'a' only. That's why I made it f(z) = delta(z-x), so I wouldn't go through that hassle. If you need more clarification, let me know. Hope that helps!
Question: how do we know that the two solutions to the Sturm-Liouville Problem (SLP) are basis functions for a generic function in the function space? Also, a dumber question: what do you mean by the function space of the SLP? Thank you, I appreciate your content.
For your first question: This is actually a property of SLPs that we can prove. You can look it up yourself but I found two sources (see bottom of page 5 and top of page 6 respectively: www.math.usm.edu/lambers/mat606/lecture20.pdf math.la.asu.edu/~kuiper/502files/SLP.pdf). For your second question: The function space is just a fancy term for a very large set of functions. The function space of the SLP is the set formed by taking the linear combination of the solutions of the SLP. If you have any more questions, feel free to ask. Hope that helps!
Can I ask you one more question? Do you know of any good book for studying special function like these in your videos? Can you recommended me a book for self studying special function?
Hi. If the equation I want to solve has f(x)=0, the solution for the C_n coefficients that define y(x) is cero. Is there a way to still solve the problem using the Green’s Functions approach?
Why would you want to use Green's functions to solve a homogeneous ODE? They're normally used for non-homogeneous ODEs; there's probably some other things you could do with a homogeneous ODE (e.g. regular series solution or Frobenius Method).
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Great explanation of Green's Theorem. I've never seen the derivation before and no one has ever explained its importance. Thanks! Great work!
Thank you so much!
that's not Green's theorem...
The clearest and most comprehensible explanation of Green's functions. Thank you a ton!
Outstanding presentation....Best explanation of Green's functions I've seen yet.
Brian T Davis Ph.D Theoretical Physics/Mathematics
thanks sir. First time I understood greens function and it's application.
In my old ODE textbook (Modern Differential Equations: Theory, Applications, Technology, 1st Edition (1996), Abell and Braselton), Green's Function is covered for a 2nd order linear non- homogeneous ODE in standard form where the initial conditions are bot set equal to zero. Variation of parameters is the technique used to generate it. Depending on the ODE and conditions, the function will be different I suppose.
Thanks for the Great presentation,, save my time for exam preparation .
From WB, India (MSc student
Your handwriting is godly
Excellent video series!!!!plz carry on!!
It's extremely helpful...I would be glad if you would use it to show us some examples of the application of green function in solving boundary value problems of electrostatics
Thank you, and I will add your request to my to-do list!
bro thanks god bless you man.
sir please upload some videos on solving differential equations using Green's function.
Noted! I'll get to it once I get the time!
@@FacultyofKhan it's past 2years, where are our greens videos!! :(
I love the explanation .I missed such explanation on the web.
Glad I could help!
Great.. Lecture
Nice work 👏👏👏
Excellent. Thank you.
At 4:46 you have (lambda-lambda(n)). lambda(n) id OK, its the eigenvalue associated with eigenfunction (n). But what about the first lambda, where does this nunber comes from? There is no general lambda, they are all associated with some eigenfunction. This lambda, it seems, is not. Please explain.
As mentioned in my answer down below, we don't really define lambda as a particular number; it's just a real number and given its position in the ODE, it is called an eigenvalue. As you said, it's not associated with a particular eigenfunction. On the other hand, lambda_n is the eigenvalue that is specific to the solution y_n to the corresponding homogeneous S-L (Sturm-Liouville) problem.
At 5:31, what happened to the summation in from of c_n?
All the terms in the sum go away except when m = n (see 5:00 onwards) because the integral is zero whenever m and n aren't equal. That's why the sum disappears from the expression, since only one term remains after all the cancellation.
Khan, do u have discord or something community like for Q/A? That will be great...
I've watched the video and there are two things that aren't clear to me:
-When do we define lamda with no subscript, how is it separated from the one with subscript? We carry it all the way to the green's function without specifying anything about what it might be: Is it a variable? some constant?
-why would there ever be f(z)=delta(z-x)? f(x) is originally the given non-homogeneous function, so it could be given as f(x)=delta(x-a), but then f(z) will be delta(z-a) switching from x to z should eliminate all the x variable, how can it leave one behind for the impulse response?
1. We don't really define lambda as a particular number; it's just a real number and given its position in the ODE, it is called an eigenvalue. lambda_n is the eigenvalue that is specific to the solution y_n to the corresponding homogeneous S-L (Sturm-Liouville) problem.
2. I've used f(z) = delta(z-x) for notational convenience. Ultimately, I want my solution y to be in terms of a variable x (i.e. as y(x)). This is why I don't want to use x, because when I do the integration at 7:00 to get my final answer, I want to keep the x and remove another independent variable like z. Ultimately though, I could have used f(x) = delta(x-a) but then when I do the integration, I would have *had* to integrate over x and remove it from my final answer, leaving a function of 'a' only. That's why I made it f(z) = delta(z-x), so I wouldn't go through that hassle.
If you need more clarification, let me know. Hope that helps!
Question: how do we know that the two solutions to the Sturm-Liouville Problem (SLP) are basis functions for a generic function in the function space? Also, a dumber question: what do you mean by the function space of the SLP? Thank you, I appreciate your content.
For your first question: This is actually a property of SLPs that we can prove. You can look it up yourself but I found two sources (see bottom of page 5 and top of page 6 respectively: www.math.usm.edu/lambers/mat606/lecture20.pdf math.la.asu.edu/~kuiper/502files/SLP.pdf).
For your second question: The function space is just a fancy term for a very large set of functions. The function space of the SLP is the set formed by taking the linear combination of the solutions of the SLP.
If you have any more questions, feel free to ask. Hope that helps!
Can I ask you one more question? Do you know of any good book for studying special function like these in your videos? Can you recommended me a book for self studying special function?
Any book on Mathematical Physics would work. I've used Boas's book in the past and it's been pretty useful. Hope that helps and good luck studying!
Can you please make a video on the analytic continuation of Green's functions?
how do you know that the spaces in the sturm liouville theorem and the non homogeneous problems are the same.
Hi. If the equation I want to solve has f(x)=0, the solution for the C_n coefficients that define y(x) is cero. Is there a way to still solve the problem using the Green’s Functions approach?
Why would you want to use Green's functions to solve a homogeneous ODE? They're normally used for non-homogeneous ODEs; there's probably some other things you could do with a homogeneous ODE (e.g. regular series solution or Frobenius Method).
What happens if \lambda = \lambda_n for some n? Then we cannot solve for c_n?
The secret of universe must be hidden in mathematics
Is dit chinees of wiskunde?
In the end it should've been f(z) = \delta (z - a) with "a" being a parameter, instead of \delta (z - x) which makes no sense.
Swaggersouls teaching math?
Sir green function ki application or property Mill sakti hai appki koi site hai Google pe jisse me unne pad saku
is he a robot
sir isski koi PPT hai Google pe