Lecture 33 part 1 (The Bessel function of the first kind)

okay so let's continue our discussion with the Bessel equation and the Bessel functions so I mentioned already in the previous clip down on maple that the Bessel equations of order n as with any homogeneous linear equation of order 2 that is the world of the Eau de is two has the solution in terms of the combination of two independent functions so let's write formally our definition first of all so the Bessel equation so the Bessel we wrote that actually last time but let's write it again so the Bessel OD of order n now I should point it out here that the order here refers to that constant as appearance in the Bessel equation because as a differential equation it's a second order so they do want open differential equation that's always true in this case but then the best of of order n refers to that constant that appears as a coefficient of Y so the standard form of the Bessel of the order n will be x squared Y double prime plus XY prime plus x squared minus N squared times y equals 0 so n can be really any number but in our case as I mentioned before we can limit ourselves to the case when N is a natural number 0 1 2 3 and so on so as I showed you before in the maple exercise the general solution of this Oh de is a linear combination of two special functions the so-called Bessel functions of the first kind JN of X plus c2 the Bessel function of the second kind yn of X so JN of X can be shown I'm going to show you the proof I'll show you actually how you can check that this is indeed one of the solutions of this or the so one can show that Jo can be written in terms of a power series with this rather complicated format so summation from 0 to infinity of minus 1 to the K of K factorial and plus K factorial and then times x over 2 to the power n plus 2 K so this is called the Bessel function of the first kind so I'm not going to take any time now to waste with discussing how you obtain the Bessel function of the second kind because as I mentioned in the previous clip we don't use it for our purposes because of the continuity assumptions of the solutions that will impose when we solve the boundary value problem so suffice is to say at this point that yn of X the second component of the general solution also known as the Bessel function of the second kind is always discontinuous at 0 which means less Y this as a remark because of physical assumptions so because of physical assumptions in solving boundary Dali problems will require typically that this constant c2 equals 0 so again to anticipate a little bit the Bessel or de will pop up as a turbulent problem in so doing there is bounded other problems in particular the heat equation for example in cylindrical coordinates so the domain of this function will be all in that case or whatever is the notation for the radio domaine only between zero and some none right depending on the radius of the cylinder and for the solution to make sense of our physical perspective obviously at the center of that cylinder you should have some temperature which should be a number but definitely not infinity or minus infinity as it happens here so you could think of the so let me write it down here actually so we have C to equal zero because let's finish this phrase here because yn of X is discontinuous at zero and infinite actually minus infinity if you remember bad graph so that can happen from a physical perspective right so you could think of this c2 equals zero is coming up from one boundary condition but the boundary condition essentially is stating that the solution should be continuous so you'll see later that boundary conditions don't necessarily have to be in terms of there's a number specific number for this boundary point it could be also a continuity assumption in itself which will basically give you an information about some of these coefficients which in this case means c2 equals zero so that is the reason why we're not going to bother too much with the Bessel function of the second kind because it's coefficient in the solution will always be zero because of these physical assumptions and as I mentioned before since we don't have a solution in the closed form in terms of elementary functions one can obtain a solution in a serious form and that happens to be in this format so the way to obtain the solution is to assume a power series format and then you plug in that format into the or de and with some rather tedious work you can come up with a four enough of these coefficients so I know this looks kind of ugly and scary maybe but this is essentially a power series and these coefficients happen to be the way they look like just because well that's life you know that's how they come up to be once you impose these coefficients to satisfy the equation yo D so it might be useful to write maybe two terms in the beginning so for example this summation the first term when K is equal to zero is minus one to the zero over zero factorial n plus zero factorial x over 2 to the power n plus 2 times zero and then if the second one would be minus 1 to the first one factorial n plus 1 factorial x over 2 to the n plus two times one and so on so for example here minus 1 to the 0 0 factorial is 1 K number by convention so this would be 1 over N factorial x over 2 to the power n the second term will be what will be 2 minus 1 1 for total is 1 n plus 1 factorial x over 2 to the n plus 2 and so on so the Bessel function of order n starts with X to the power n if it's so depending on the water of the the Bessel equation so on the next page I'm going to actually show you how you check that this is indeed the solution of the Bessel equation it's going to be a rather tedious process but the reason I want to show it to you it's not because we're gonna use it in the class too much you will have a similar homework question it's just easier than the one I the case I'll show you in the in the lecture but I want to go over this process to give you as strong as possible on motivation why if we deal with Bessel functions we really need to use some computation rules so that we don't really have to replace the Bessel function all the time with the power series because the process will be very very tedious if if we don't come up with some computation rules so before we move on to showing showing this checking essentially that J of X is the solution of the Bessel equation of God our n let me just state a couple of things which will be useful for the mechanics of doing this in the in the next part when you have a power series let's say you have the summation from zero to infinity X to the power K it is often necessary to change the first value of the index so in this case if you expand the summation this is X to the 0 which is 1 plus X plus x squared and so on but often if you want to combine this with a similar series that goes to the same powers but with a starting index that is different than this one you should be able to change the first index of K and still have the same power series in this inequality so in other words let's say I want to help avoid the same summation but with the starting value of k equal to 1 whenever you advance the starting value of the index with one unit you can achieve the same powers if you decrease the power of X by the same unit so notice if I point right instead of summation from zero to infinity X to the K summation from 1 to infinity X to the K minus 1 this basically gives me the same series because now you have X to the 1 minus 1 which is again 1 X to the 2 minus 1 which is going to be able to the first x squared and so on and you could do this trick whenever you want to I mean for any number let's say I want to start from 2 and I want to have the same summation all I have to do is to subtract 2 from K everywhere in the series and that will give me again the same the same series so all I do I did I here I just change the starting value of the index of the power K so the reason I'm showing you this is because if you want as you will you will need when you check this series to see if it's a solution of the best electric whenever you need to combine various summations together usually have essentially the same starting value of the index so that the value of K when you combine the inside the series is the same and also obviously the power of of X should be the same when you combine them so with that in mind let's move on with the next part when we're gonna deal with checking that the vessel function of the first kind indeed solves the Bessel equation of our hand so stay tuned

Channel: Daniel Maxin

Views: 2,696

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Id: 9O826dZVNmE

Length: 11min 45sec (705 seconds)

Published: Thu Mar 26 2020