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COSC 4364 Numerical Methods - Lecture14_B-Splines_ODEs
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01:34 I don't know. Yeah. So right. So I was thinking
01:49 was it's the curse of construction. starting with his his press constant.
01:58 Spiers mm hmm. They ended up getting is probably gonna smoke and shapes
02:05 higher the degree for the spine. spine because of come on then you're
02:14 combination depending on that's obviously the degree the spine increases by one wherever step
02:26 . So what was it? Uh then there may I stand up.
02:32 great. Eventually something approximation function within combination of spine solved whichever degree he
02:46 . Um So in this case G just the representation or the consequence of
02:54 in this, in your combination of of the cave. And then when
03:00 comes to the place and then we're to use this d to approximate some
03:07 functions has said some of the Mm hmm. Then there was this
03:16 you towards the end of last slide last lecture to basically or as they
03:26 made for making things convenient instead of to go through. So these steps
03:34 for funding the cd case or the order these points the best to try
03:41 reduce it to um things based on piecewise constant. Mm hmm.
03:51 So this was just one step and to show that to start with something
03:56 is a combination of 70 degrees There is kind of way of reducing
04:04 instead of linear combination of one order over these points. And there is
04:13 way of computing the corresponding coefficients the combination of lower order these spines and
04:21 can continue with this process over and This was an expression of this
04:29 So one of the advantages of having kind of form is that for any
04:34 X there's only one of these guys is not zero. So once you
04:41 the interval then there's only one of eyes that is A non zero and
04:48 for that new gaps. The value this finger combination as being this
04:56 Now the coefficients that started out and guess it's expression the idea is that
05:07 are constant, suitably chosen constant together , I have desirable properties so they
05:14 not functions of acts in this process we see they become dependent on
05:24 And I'll show an example later I have to do this. So that's
05:30 thing that I want this like Uh she said yes. So so
05:39 though this is an infinite sum. from just this picture and some other
05:50 and not right for anything degree. , if it's part of this
05:58 they have just abounded support which was interval of the in this case independent
06:06 X Where there are non zero. basically this intimate sometimes of being a
06:14 some that only was a few of points and how many involves depends on
06:20 breed the big spot. So I as far as they were kind of
06:27 a little bit better and I don't there is not so pretty that just
06:31 of see what's going on. So ended up with this some of the
06:38 here where This is K two These are the ones that are start
06:47 here that are not dependent on facts then the subsequent columns here are ces
06:57 from the others in the first column saying this form right here. So
07:01 kind of the curse is done in superscript phones. Okay down so that
07:09 camera as well all the way into depths terrible and depending upon where you
07:17 your ex evaluation to be invested in interval that determines what the lower subscript
07:25 . Insurance objects. Oh in the points for. They're not in terms
07:34 defining a variable you can see So it's texas and just interval there
07:43 a few of them that that's what Hello playback. I am the degree
07:51 the this plan is the support interval the piece points stretches kind of further
08:00 from something notes. So that's why get a bunch of these things that
08:05 need that is determined by the degree he's fine. So, so it's
08:15 of things for me, come at least I think about it as
08:20 they want some point here. They depending on what degrees at this point
08:25 have kind of started um earlier That's right. So you have to
08:40 um Yeah, so we're going to this. We can conclude for
08:49 So let's see what oh and that's point here is to show that the
08:54 of them Um as 1 um that relevant there are degrees span because of
09:04 . There is only one of them any given X. Is mm hmm
09:08 and all of the others are So the internet, someone picking up
09:12 of the Australia and I'm also sure that's right too simplistically for first degree
09:20 because and in that case there's an between two slides because the support is
09:25 too well. So in that case ramped up at the same rate as
09:30 other one ramp down. So there's also ended up being one ah the
09:37 fascinated by picture. And then now can also ensure what's done by induction
09:42 is being done here uh for any green spine. Well that's okay but
09:52 of them are always fun. So yeah, so in this case it's
10:01 us over that the coefficients into one any given function is okay. The
10:08 is that these some of these. some of the beast that's quickly instructed
10:16 , then we have the formula here so on. There's conduction something but
10:23 far as simple. So this is , Both of these places are
10:30 whether it's for all lines forgiven supreme then we can see that these two
10:38 as for the access cancels out. what the gap is that uh,
10:47 same expression um the top and the . So either. Mm hmm.
10:58 and that best at getting this some wants. So it's started preserved If
11:02 happens three times for the one for Hello. So in that case baseball
11:14 have been used once here to start the and some of the values issues
11:21 the first month this formula and shoulder person steps here ends up being
11:31 So the minister becomes to some of zeros that we already suffering.
11:43 What you set out to do So any questions on that Yes manipulation of
11:51 expressions. So hopefully they want to now. So they want basically this
12:02 of these funds the transportation for some if more precisely the way we do
12:11 at this point. And then of we have to do even approximations now
12:17 interpolation is that they want this expression this sort of the exact same values
12:24 the selected points for the interpolation. we want the exact match and we
12:31 this impression to fill out an estimate my tea between the no more points
12:39 the interpolation. So the question is do they get these? And in
12:47 cases the simple process and for the for this zero in the first
12:55 Calling on House. It's a bit competent for the second renewing these points
13:03 I will go through don't 40 persons not former drank it. So,
13:17 one thing that I think it's but keep in mind certain things are very
13:26 or the same as for the spine based on binomial. So we did
13:34 about something. So also different. all these things are computed um It's
13:42 or less the same for 1st and degree polynomial. Zero, first degree
13:49 . These points. As for the , it's quite different. I would
13:53 for second to me these parts. one thing that is permanent also that
14:02 there at all for 2nd and 3rd second and higher degree. These lines
14:10 lines. These points based on the meals. The there was degrees of
14:17 . So we have to make Ah I think the prosthetics clients based
14:26 binomial. I was used to refer the second derivative at one of the
14:32 points. And so the doctor responds was second related to the both imports
14:38 2° of freedom. So all right with me. These plants and they
14:48 the very responsible for these first So that is pretty simple. So
14:55 you want to be zero degree. He's fighting for this asbestos.
15:03 But it's constant. The payments are and dispensed. So, but it
15:15 . So remember that necessarily or from properties of the least concept. They're
15:21 and non-0 and then there are one a particular in the morning and everybody
15:28 . So in this case to do in a combination of the zero degree
15:32 plans, it's pretty straightforward just to that's the conditions, the values that
15:41 want to interpolate at that point because of his eyes there one at the
15:50 hand point. That's quite one. argument absolutely close the installation point
15:58 Yeah. Mm hmm. So you spend an answer in constant why I
16:07 the interval from whatever the notes are inspirational. So that's pretty step
16:21 Oh first. So first responders, not that much more complicated. So
16:32 this case when I first really wants kind of help function. So The
16:38 is students of all this within your of two. Ah these fines that
16:43 kind of shifted 40 degree. Mm . These plants are shifted by
16:50 So About the functional one laps after . Staff. Yeah. So in
16:58 case that's the property again. Ah way the these points are index,
17:07 ends up being this practical function always when there is a difference in one
17:14 Arguments that the er here and then zero. The argument that ci But
17:20 arguments plus one. Then it's a . So by the display, there's
17:27 one of these first degree. He's standing. That is, wow Any
17:34 of the interpolation points because of these to the left. zero. They
17:38 down until the right. It just up. So there's only one of
17:43 that there is and don't tell your . So it's not gonna they just
17:49 the wait in the linear combination of functions or first degree boost lines as
17:57 the integration our values. That's the . So, but it's pretty
18:08 The new interpolation or the zero for something. There are first degree in
18:17 parts in the sense that was also because when we look at the these
18:28 ah they uh and our kids get about the first example. Mm
18:39 So that's the first order no Then the coefficients in the first order
18:48 . The two were fully determined by interpolation conditions. So we didn't need
18:52 else. And we couldn't specify anything except continue waiting. Not great.
19:04 that's fine. Yeah, it sounds how this works. That was
19:09 There's only one that they want to that nonsense. one thing to
19:18 No, it's tough. Ah for they left most. And also they
19:28 there's the first thing we explained that starch to the left for this.
19:33 don't necessarily have. Oh, the and similar. The last one first
19:42 spine that extends one in the yard you want to do. Not.
19:49 interpolation. So and that will come sooner degrees. Find steps. There
19:57 for the interpolation or any spines. has to date. Conscientious. All
20:04 to the left to the right. you are contribute to the population.
20:14 . Okay. 0 1st. Please . Okay, I'm sorry, a
20:24 bit. Mm hmm. So the general form again. Oh,
20:34 . We need at this point. . A millionaire combination. Okay.
20:42 . And for whatever it is in book and it's also reflected on the
20:47 that changed there. Simple for the from C to a now will
20:54 Well, a bunch of the slides , but there's nothing different compared to
21:01 was used before. Yeah. So that's when I discovered Tyrone in this
21:15 , inc conditions do not get sufficient to the all of the coefficients and
21:23 accommodation. And there's kind of one when K is to the face of
21:32 . Oh okay, that's fine. kind of respect what goes on
21:37 which they have. This problem And in order to account for the
21:47 interval, uh and the indexing that used for the playing the starting point
21:57 response. That invested support is three from the second baseline. There was
22:04 for them first through the spine. why it's convenient to use this form
22:12 there is a difference in the index that kind of reflects more or less
22:16 far to the left. You need account for the second degree spots.
22:23 they're further back then shifted by two there are zero. So you don't
22:28 to worry about them and there's so and that's what I think it's a
22:35 of details slides but I thought it be good to go through it against
22:42 for me first time I thought it fast and try to figure out how
22:46 actually works. So um so this again just from the definition that it's
22:56 to die. And then As in of three in the malls then you
23:05 the properties on the study states. um this is a claim right?
23:15 we started to plug in TJ is . Ah this point has to then
23:24 obviously yes cyril. And but it's then yeah so so they find that
23:42 not zero. So I did the that was zero and then it tells
23:48 of for any X. So they this pointy ear there This one is
23:56 zero And the other one. Yeah think they will get hurt daemon.
24:09 also is zero. So when you're obviously this is non zero and this
24:16 non zero and this guy is So so trump Davis directed this picture
24:23 any one of them here at any of the nose, two of them
24:28 non zero. The crm between So so then if I look at
24:41 infinite sum here and that particular Well known there's sort of a day
24:49 this one. Yeah, it just to worrying about to all of
24:57 2nd The Baby's Point. So this okay. So in order not to
25:03 to figure out what the A's Ah the first step we're going to
25:09 through them was I tried to do . The Persian steps. Mm
25:17 Mr using 2nd degrees. These points use when filling out too.
25:25 using there are thinking this one. um so this is the version but
25:38 uh with them, you know the . So we have this system specific
25:45 this diversion form of what we do just plugging in the proper governance arguments
25:54 instead of X. Then if I this and I'm going to have this
25:59 first women say it's too. So the first week this one is in
26:05 same one as food and it's the thing. Subscript genuine. Still.
26:09 the starting point. I'm just going the one shifted by one in the
26:14 . So that's why I shifted up part and otherwise this factors and TJ
26:23 statistics and that's changing. Mhm This gives us Uh huh. This
26:36 down there. Yeah. Technologies. started after looking at one of these
26:46 we needed. He's fine. And we have another one that we also
26:52 to do. But this this expression Doesn't need to win this one.
26:58 then we also from this movie too up what these please sponsor. So
27:09 was from the previous line so Somehow they don't cut. This is
27:20 first degree. So this is this here um that this has this support
27:28 so that is C. J. terrible so long this guy um from
27:42 first civilian back here um signed up by the by construction capable tomorrow.
27:58 so then again this is the help and so that's easy. Right?
28:03 have function as um So let's see plugging these two things that they stepped
28:13 up this one and this one now simplifies center to this special in
28:21 And the other one that was in expression that is you can do the
28:30 thing. And then This is important also that's another one that's zero sell
28:42 selling okay The texture on this one is young have function And one at
28:56 point. So now we have the term. Yeah. This expression
29:02 there's also some very racial between think all night. So hopefully now what
29:12 got is plugging things into this expression we Right, that's enough this to
29:19 any question here. Um Yeah. boats as unknowns please. Any values
29:31 coefficients are the weights of the difference part and eventually these needs to be
29:43 . So we got interpolation suffering. . So so you can see from
29:57 equation. Right? In some ways you know one of them said they
30:04 compute data plus one version. This not the incorporation condition of all the
30:11 . I know. So we have here. So basically if we have
30:15 zero then we could computer the other and this um the formula for this
30:25 please. Bye. So what they're is they want to get in the
30:34 or something support of the the second with these points, we ended up
30:44 one more um weighting coefficient a in of giving work these points than the
30:55 of interpolation forms. We have Interpolation . So one of the ice storm
31:03 next was and then still name any swan. So these are the last
31:09 values as opposed. Yeah. So the one thing we have freedom also
31:17 secondary at this time and that they want to be a freedom that with
31:21 being polynomial interpolation. So that's no in terms of degrees of freedom
31:29 And they're different ah for the These clients are only no more
31:37 But they expressed in this because she Well, this was economy and this
31:49 in common than you think. So we get so many questions. So
31:56 have one to do your freedom and question of producer and in the public
32:01 have a machinery that yes to something sensible. It's always that's I don't
32:14 what to do themselves and minimize um square of the magnitude of the coefficients
32:20 be more positive and negative. So kind of natural to square them in
32:25 to get rid of no science and look at magnitudes. Oh,
32:34 So yes, they set out to the square sum of all the coefficients
32:42 order to do that. Rewrite basically and reorganize this equation to this starts
32:52 one of the Okay, so you get the next one to this question
32:58 from the structure of this equation. figure out what the outputs and datas
33:04 . And that's the question. And was fairly easy to see the
33:10 The right hands are divided by the from the passport similar. It's not
33:16 hard to see that position for the is simply this no, this expression
33:27 . So it's fairly straightforward together this I guess a little bit more messy
33:35 drama. So what we want to , so that essentially is okay,
33:41 we have this kind of diversion formula carry out the substitution and his formula
33:47 the program. And then we actually something that is related to one of
33:53 coefficients and choose a zero Starting So watch your hand up if you
33:58 get all the other ones through this . And the second part of saying
34:04 how do you yet the gammas and and this equation from iterating their present
34:14 on this question. So then we an expression here for have to prosecute
34:20 gammas and deltas and now that's You get all the days I know
34:28 to a zero. So in order get an expression, well that's fine
34:34 . But now these professions that are independent, the ace just dependent on
34:44 intervals for the population and inflation So now until you can get an
34:51 for fine only being, you the function of a zero with the
34:59 for christians. And so this is it is being. So there's some
35:04 were not Now with the performance will for how to compute a one A
35:09 , et cetera. We got something this. So this is basically that's
35:16 for me living space. And then part of the usual thing you want
35:22 minimize something and context is there evidence the derivative to zero and protected the
35:30 of this question. You got something this and then they can just directed
35:35 herds another question here, a bunch petitions for the zero. There's no
35:43 of the renovation, there's no It's just that in your function in
35:48 zero. So you know, that's proportionality constant respect in front of eight
35:53 some costs and putting derivative is zero you find in zero sports basically.
36:05 this formula to them it's um contents the coefficients in front of the zero
36:14 to the delta squares of the in summer. That's where it's from the
36:22 postponing one time and you know what are. And then another procedure for
36:31 okay cereal. And once you have zero then you can use this is
36:36 question you get all the other So, so that was the question
36:48 difficult. Some more on the show next cervical or examples that they have
37:00 many questions on the legacy contact. relevant sounds simple. Some of them
37:13 ? So so the point in this we used to find the coefficients and
37:23 is used in the second degree spines the outputs of the balance. That
37:28 the interpolation values and they're not start . And of course you did with
37:34 the points of us began whatever the along the population of sports. But
37:39 that's really in the input this procedure was described and figures out for a
37:49 from doing minimization of the square some these competitions and then the substitution for
37:55 to get a place based on What? So so this is then
38:08 the obvious what goes on in this . I tried to comment on
38:13 Um, this is kind of obvious just use an age that's before for
38:19 interval life they can have variable with world lines index according to ah where
38:29 one remember that. So in that means that interval is between in this
38:36 zero and the -1 and again have -1. So that kind of arbitrary
38:43 2s to have the interval to the , 00. And similarly and the
38:58 one endpoint, right? But then extend and say well the interval to
39:06 point we don't have on the right able to the last thing we have
39:12 the integration points and then when they where this come from right. So
39:23 impression there was correspondent zero was That zero and then zero. Yeah.
39:34 intervals here come, they were the . So not it becomes a I
39:45 want the first one is higher. and then the post form On this
39:56 one. So that's why it's So that's probably get a negative
40:03 And then the other one was come . The first timer 0, 0
40:11 20 employees zero. This one and same thermal blankets too. And this
40:17 an acceptable one x 1. So that case we get the number two
40:21 front of West europe this expression to started. And then because this
40:29 so the first one is simply historic products. Mhm. At this
40:36 Now the first one before we started minus one squared is one. So
40:43 to get this to be. So to get things started. Well,
40:50 was located, I'm trying to, question was the hospital of the
41:01 that was Severa and Severa was the proper science between the festival of
41:10 Yeah. Things are we have to the relation between Yeah, the other
41:17 and 2 grandma was a little bit about formula was this guy. Now
41:25 can see that's supposed to be wrote text and the gold corners and stuff
41:33 the technical term is better consumers but the first time and then our clients
41:41 the inspirations conditions in it and Mm hmm. But you can trace
41:51 what they called. Lines are called to This formula is almost directly from
41:59 one and 2 and two is Um, this follows from this
42:09 So we can define what the United and other correspondents. Oh, what's
42:22 ? So that's true. Mm And I guess the last one was
42:33 to recur version actually computes a so gave us a zero and then,
42:38 know, it's actually rewriting this equation that his former The previous day one
42:45 then we started and then the next . So this routine is said and
42:55 that took them the inspiration values and and how many intervals we have and
43:03 just returns the coefficients and generally But but actually for me where,
43:15 , but it's essentially computes he's No. Such that within one degree
43:24 freedom that uses to realize where someone . Oh. Mhm. Okay.
43:35 the next thing is then, so , they have a machinery. He
43:44 the A's for each other such that also satisfies things. Operation competition.
43:54 . And I asked the question To this one and or to evaluate this
44:02 we developed this machinery to try to within their combination of 0° plants instead
44:09 directly within the second degrees plants. that's what we're using. And this
44:21 here and I looked at let's see . So I try to relate it
44:27 how things were done in this general of the way that statement Instead of
44:34 direct evaluation with respect to the 2nd is fine. Mm hmm developing in
44:40 recursive formula for coefficient. So I have to worry about one of the
44:46 you don't think we spot. The incursion former public coefficients now
44:54 But the second one against final A So A is here and this
45:01 formula that they have there it So then it was just the starting
45:09 in this case is our age that out of the proficient procedure and then
45:16 them into this formula and eventually ended The coefficient from zero linguine spine correspondent
45:25 the Evaluation .22s. So in this we have three for questions. Um
45:38 , yeah. Using the 2nd degrees famous too. and then the final
45:46 and it's also interesting -1 and So that there are three elements first
45:54 . And I noticed that student the . So in this case, so
46:05 first equation is in play computing the thing Now The 1st element in the
46:14 column. And the second line convinced new Senegal this formula substitution, therefore
46:25 . And then once we have those can be used again the same and
46:32 the same formula here again. The coefficient for zero, everything why?
46:43 then is one in the intervals for values that we want to be
46:48 Yes. Mm hmm, mm mm hmm. Any questions. So
47:05 wasn't an entirely Samuel machinery either for wow, 2nd degree and 3rd degree
47:14 versions. That's fine as you may , was quite a few decorations and
47:19 they get older values and then we to try the system in the
47:25 Um The gaps on professionals. ah, okay. Sorry. There
47:36 a picture and yeah, they don't all the difference a little bit
47:42 But the points of this comparison, the second degree. This point sources
47:51 degree Flatow 765. So this is polynomial, one higher order. That
48:00 isn't well. And in particular That's your mind and your friends is
48:09 yet right now to respond. But is not much different from this
48:14 but one is on the left of installation point. Come on mr
48:18 But otherwise both are very well behaved It's sort of 1° in some
48:26 Just kind of showing a little so there better behavior. It's still
48:31 better work or these points because it's of small the life I'm supposed to
48:38 sarcomas and that behavior as states formed basis increases. Mm hmm. I'm
48:53 comfortable doing that but it's just more remind you. But there's no need
49:03 inspiration points and and not to Someone can also define, okay,
49:14 you can find a dispatch from a thing. No lo forms from their
49:20 was just these points that I should like to be. So that's what
49:37 was going to stop mentioned in terms interpretation but this is fun.
49:48 so a couple of more points about No one is going through and ah
49:57 derivatives. So what we talked about there, it appears um get approximations
50:05 on derivatives are you said? Except functional values or you can also I
50:13 took the view that and I think the functionality is to use and just
50:20 approach as being points for which that used for doing interpolation and generate the
50:27 and the security of the polynomial. the same thing can be done in
50:32 of the approximate. So they will accept the functions for these points.
50:41 , they may also be useful to out how do they get derivatives of
50:47 . So that's what happens in the year, you know. So it's
50:52 having to start with the international formula that. These finds that this is
50:58 case that we can find something taking the minutes on it. So
51:04 have the definite movement directly. In , derivatives work it all out.
51:10 a performing arts. If they work out, take the derivative with respect
51:13 X products and activities. First Thanks the other one and then this
51:22 thanks to the attitude of this Which one of the best introduction to
51:30 figure it out. And then when done that to basically show what this
51:36 the form, the derivative of Chester that's so first, um, it
51:45 sense that there's one degree lower Because every time it takes the derivative
51:48 respect to the independent variable X. degree. So what's going on?
51:55 that's what you get. That's another . Then these finds have been used
51:59 yes, but you can use them represent the serenity for the case signals
52:05 wife and they said it was the working into what you get. And
52:15 more that's the question is of this came from this. So This is
52:24 of the stuff to show investors that you can express the derivative of
52:31 Again, I'll you have some approximation some function you have that was expressed
52:38 the linear combination of thinking despite, then each one of them ends up
52:46 so, okay, what is one in the combination of claimant is one
52:50 finds and it's the best for them it in because they've taken the derivative
52:57 doing a summation commute. So derivative that was going down here and then
53:05 plug in the expression take the derivative this that we have worked enough up
53:11 and then it gets this kind of and changed the summation index. You
53:21 the same door. Cool. Sounds on both terms here. But then
53:27 have to account for it when you at the different factors scenario, that's
53:33 else in terms of the derivative of approximation that case finds. It becomes
53:41 linear combination. Oh chain one is for one because you know, are
53:46 spots for christians here can be arrived the original coefficients and the order of
53:59 So it's best if you have essential desserts or derivative, these fines,
54:11 derivatives cylinder combination of the one that lower these spines. And since we've
54:19 of worked out how to evaluate such , then they can use it before
54:25 figure out what the yes they justify went through ah to this novel
54:34 coefficients are different than the original. , it's still there studying the combination
54:40 spots. The next one was the . So the other one that was
54:54 useful in the past integration. And we can also show that ah and
55:05 case is kind of an indeterminate what to do um, podcast and
55:12 respect one prediction. The spy is vineyard combination of these blinds that extends
55:22 I mean that's mm hmm. Finally are integrating Hello? Possibly to
55:31 So the rest of it depends on . Thanks. Some of us just
55:39 the younger. I don't want to just I think we're integrating mhm.
55:48 clients, they figured these kind of combination of them in order to approximate
55:52 function do than gaps. Another expression has different coefficients. But since we're
56:04 That means the degree of at this has to be 1° harder because if
56:09 want to go back to the instagram take the derivative on the right hand
56:15 , right hand down the degree and enormous forgot and then there's some questions
56:21 figure out what these coefficients are. integration. Yeah, it takes responding
56:31 to take Thanks. So yeah, is basically both point and you have
56:41 you can take the rivet in summit hi fingers but I'm not going to
56:50 you to do that and many times what's wrong principles. Okay. So
57:08 , making the terminal you want inspiration . You get unlimited some again because
57:17 only for any given excellent value over argument and the response that is on
57:24 final number of them. There are zero sum in the end finances of
57:35 the spines. Um as you can from this expression here when you get
57:41 higher or at this point, Interval support is longer one more interval and
57:49 why he said this. And they justice performance uh population. That was
57:55 escape. Also one step further to left depends on small the response I
58:06 Yeah, accounted for or included in submission the current. Well let's go
58:14 the point of interest. Les I was just taking a half far
58:20 to you need to go for the of the response to actually reach the
58:24 are interested. So that was pretty just size of business. Does he
58:36 to find space clients software from And I think most of the,
58:47 that's another one that mentioned in the is yeah. And then double
58:54 But as far as I remember, also America and most of them these
59:02 in particular house how kind of like news because so when they're small of
59:15 hmm. Um sometimes let's start to a little bit about the next
59:26 So also mentioned responded in the question think free to ask questions about
59:35 Um, victor. Ah, not . Have any questions on whatever.
59:41 we talked about initializing problems in the video. Yes. So what recovering
59:49 the exam. What what, what would you call that? Chapter
60:03 . Okay. So, so initial . That's what you said. But
60:09 find it, That's part of Chapter six. And um, and
60:17 when will you provide will be done Wednesday. Okay. I'll talk to
60:26 tonight tomorrow so we can look at right. There are questions on
60:38 Santa began. Mm hmm. So I guess said it to do
60:49 So I'll say it one more time terms of cargo summarizing things about for
60:56 normal approximation. So brian said it the form of all and all about
61:02 is just the colonials and good. point of using lines are supposed to
61:14 them. A polynomial Interpolation is having spice polynomial 171 covering the hole in
61:25 wall. The following underground stuff. more points you have hopefully better
61:39 Yes, that's correct. Exactly in points. But what goes on between
61:44 interpolation points test? It's worse than world comes to happen. So less
61:51 more points that are correct. But you look at what goes on between
61:56 points, the exports. Well, that's why if you have many words
62:03 enormous, it tends to behave better the whole interval. Yeah. So
62:08 can use such many points. Forget , but there's a few of them
62:16 als and the second authority points on baseball. So that's some of the
62:26 offs and finds has become very popular many disciplines. The approximations of
62:35 I want to go to graphics that mentioned. My phone probably best to
62:40 the domain for trump for the audience policemen one of them that is
62:49 I'm sorry for work. The whole that becomes and it's the same
62:59 We've dropped us off many kinds of equations or part of the essential questions
63:10 , pressure and whatnot. I just to carve the remains of a small
63:16 . Time to wrap this up solution . That's the name with some form
63:23 despite the approximation in machinery supposed to the proficiency and it was like something
63:32 government for your business but a few just introducing emissions. Okay, so
63:44 not familiar with the initials of I guess. Thanks for that
63:50 So as I just mentioned, you , and science and engineering is very
63:57 to represents a problem by the financial for all of us to try things
64:04 over time. So you try to it can be all the space for
64:07 matter. But many times it's kind easy to think of it. I
64:12 I understand it depends on the the change sharpener and we're supposed to
64:18 resolution and there's kind of all this that will cover and now I don't
64:29 what they are about this thing. order has to be, whether it's
64:35 the first to know everything about involved . The differential equation comes in an
64:46 I guess success function of one independent or something that sends over time,
64:56 for something that is uh that's special dimensions as well. Trying to figure
65:03 the pressure in the box or As part of differential questions. Um
65:14 know more about this, Not such . So we just kind of example
65:26 somebody probably had done in some high physics or something or something, you
65:29 what happens and and the discharge that nerves or something help. It has
65:36 be your son, relate to So if you remember you actually have
65:42 capacity or something to recharge the That's different clubs. Focus the value
65:47 doing an interview. So Ambassador and took place to the resistor, I
65:56 know, take your foot from zero in the experiment. Think it's kind
66:01 exciting. He doesn't get a short to show something. But anyway,
66:07 in that case the capacity that's really discharge immediately. But if you have
66:13 resistor it takes a while. So in this case on this idea that
66:20 the voltage. So as the capacitor its charge and when the voltage current
66:29 it goes down and both pitched across resistor and for six months on the
66:36 but it goes down over time I the compassion of discharge and for whatever
66:43 this guy's that was horrible sight from is seriously kind of click the current
66:49 essentially the magnitude of the currency is The highest one, the voltage is
66:53 highest. And then eventually as this or close to zero because there's no
67:00 left. Then of course the there's no more electrons. So,
67:09 basically I have the zones law writers the voltage across resistance they posted the
67:15 . So there's just Through the Resistor . Um it's just and the current
67:24 is for the change in charge on capacitor. So the challenge is best
67:32 the derivative over time on the change time. So this is just the
67:37 are now and the charge disproportion of capacitance of the statistic times.
67:45 in this case the first order differential that involves on one side and in
67:54 case you have the politics on the side. And it's just all this
67:58 you can usually do that basically in yard. This kind of an impression
68:03 this shows up there are reptiles decrease about it from this source.
68:11 Is um there's no surprises proportion to time processes. It's called r See
68:18 tells you how fast things are And it's a question of the product
68:22 the anyway, so yes, that's example of how you kind of first
68:28 differential equation that you probably have seen is that something fishy about the
68:37 So let's see, synonymous. So value problems attempted slide that the
68:47 the variable you have any question for it depends on same time and maybe
68:54 article itself. So what they had the previous side right there was the
68:59 of the voltage from one side uh rearranged things and on the other side
69:06 have the office itself. Um That a long time. So and then
69:14 this case we're given some starting points have this one was. So you
69:21 what the voltage is when you start experimental causing the local. So this
69:28 kind of the generic initial value problems formulated derivative for the rate of change
69:37 kind of known that this unknown in sense that you have an expression for
69:42 it depends on time and the variable . And then you have some starting
69:47 everyone can start start with developing so so in this case the first thing
69:56 just a single independent variable fee So that makes this kind of an
70:01 differential equation. And in this case only the song the text is just
70:09 single variable sort of. This is a very simple first order for more
70:16 equation but X can also director value not. For instance they get the
70:22 other questions and we'll talk about that well. Thank you. There's some
70:35 oh in terms of differential equations with variable um as this dependence on
70:45 Um in this case is very simple figure out how to do finally variable
70:56 . That has this lifetime delivery some these things you can kind of realistically
71:02 out blocked. They were soft on it because again you want to find
71:10 um behaves over time. This is you found this thing is depending upon
71:18 you are and figuring out what things this kind of derivative, it's affected
71:25 or distract from some of these solution . Then you have these values and
71:31 satisfy also the condition that is the team zero Then you should get
71:39 And the first time here that comes this becomes zero so 1 -1 is
71:44 also? So this solution is satisfied this equation and its initial condition and
71:53 can check the I normally don't you necessarily no from this equation to find
72:05 function whose derivative is the right time that's the reason what it is in
72:10 solutions to ordinary differential equations that it's to find but the functions or other
72:19 still. So that's what I'm certain this one. Okay I have some
72:28 of the stations here in our repeat family start over after. Ah so
72:37 the way anyone goes about this uh can use what we kind of learn
72:47 we talked about the integration. That's way of trying to figure out how
72:50 get the solution to no li ah one thing that can be cast it
72:59 something we know how to work with best recast it as an integration problem
73:05 in some ways it is because you to find that's my opinion over time
73:11 that's access officer that we integration or gravity ah differential or changing next.
73:22 you can take this equation here and that's kind of put the V.
73:27 . On the right hand side and do the integration. Say on a
73:31 interval form The 50 plus h. on the left hand side and corresponding
73:39 all the function that you have for derivative bad times the variable which which
73:48 the derivative restatement. So in this you got this expression now this one
73:54 obvious kind of that 50 and that's X 30 plus H minus X
74:00 So I can write this for the extra T to the right and science
74:06 now we have basically integration problem instead Yeah after solving ordinary differential equations and
74:16 know it's just an initial value You know what that something is supposed
74:19 be and the interval is too Then this integration into evolved. It
74:30 be perhaps pretty crude in terms of clever you are and during the integration
74:37 and that's what but this picture of so here's where we are and what
74:43 accretion tells you this finally um that's for the function that is and is
74:56 and when you're here the issue again T value and this function here then
75:03 essentially the tangent on the curves. you just use dysfunction at the starting
75:11 , you will end up here, is not necessarily where this function
75:17 So, but that's kind of one method is potentially do that. Um
75:24 I think this is here on this , so I can use exactly just
75:30 the attendants at that point. And on the function f it made the
75:35 about approximation. They also have the or drill for doing this thing.
75:42 yeah, and it would take The evaluated three endpoints. And then I
75:48 the average Joyce. Now this has problem. Anyone sees the problem.
76:12 problem is we're trying to find this , my name is lisa here,
76:20 we don't know this guy, we know this guy, so we don't
76:23 the sector. So this is what's as an implicit form. So
76:36 if they were to do this type approach just for today, what it
76:44 , you need to have some form , for instance, an interactive
76:50 ah you get some value and plug in here and you see what you
76:54 and then do something. Well, may eventually converge. But there are
77:02 then methods like this guy, it's because, you know, it's something
77:07 when you start, so if I think this integral is straightforward. So
77:12 you can directly compute X deposits, have this and this is known.
77:18 that's an explicit formula. And this is an interesting for a situation of
77:25 prominence dissolved. So we'll talk about message. But ah, explicit methods
77:33 doesn't worry your direction. That's They prefer to get a choice.
77:43 so this type of notice that will the last one to talk about on
77:48 inflow inflow is both approaches what But the number of inspiration steps to
77:57 Okay, Thank you for today. when you stay home and kind of
78:02 about, I hope to get practice . Mm hmm. Thank you.
78:12 . Thank you. Oh,
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