1 00:00:00 --> 00:00:01 2 00:00:01 --> 00:00:03 The following content is provided under a Creative 3 00:00:03 --> 00:00:03 Commons license. 4 00:00:03 --> 00:00:06 Your support will help MIT OpenCourseWare continue to 5 00:00:06 --> 00:00:09 offer high-quality educational resources for free. 6 00:00:09 --> 00:00:13 To make a donation, or to view additional materials from 7 00:00:13 --> 00:00:16 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:16 --> 00:00:20 at ocw.mit.edu. 9 00:00:20 --> 00:00:24 PROFESSOR STRANG: So this is the first true review 10 00:00:24 --> 00:00:27 session in 18.085. 11 00:00:27 --> 00:00:32 The last Wednesday, the first Wedneday afternoon was a 12 00:00:32 --> 00:00:38 brief review of topics in linear algebra. 13 00:00:38 --> 00:00:40 But now we're into the course. 14 00:00:40 --> 00:00:44 We've done four lectures on the first four sections of the 15 00:00:44 --> 00:00:50 textbook and one homework problem in and back and a 16 00:00:50 --> 00:00:54 second homework set for next Monday. 17 00:00:54 --> 00:00:58 So I'm ready for any questions, including questions that are 18 00:00:58 --> 00:01:00 on the homework if necessary. 19 00:01:00 --> 00:01:03 But anything at all. 20 00:01:03 --> 00:01:08 Or maybe I'll just ask whether the pace, so 21 00:01:08 --> 00:01:09 this is really informal. 22 00:01:09 --> 00:01:13 Is the pace of the course, now today's lecture had a lot in it 23 00:01:13 --> 00:01:17 as I realized when I saw that the board with still full of 24 00:01:17 --> 00:01:22 18.085 and there was a little more still to do because we 25 00:01:22 --> 00:01:26 didn't finish the matrix part. 26 00:01:26 --> 00:01:33 But are you ok with the sort of speed of the course? 27 00:01:33 --> 00:01:34 So that'll be one question. 28 00:01:34 --> 00:01:39 And then, what about specifics? 29 00:01:39 --> 00:01:42 Somebody start off if you will. 30 00:01:42 --> 00:01:42 Anybody. 31 00:01:42 --> 00:01:43 Yes, thanks. 32 00:01:43 --> 00:01:46 AUDIENCE: Well, actually I had a question about the 33 00:01:46 --> 00:01:47 lecture earlier today. 34 00:01:47 --> 00:01:49 PROFESSOR STRANG: Okay, go ahead. 35 00:01:49 --> 00:01:51 AUDIENCE: I was just going to look it up in the text, but. 36 00:01:51 --> 00:01:52 PROFESSOR STRANG: That's alright. 37 00:01:52 --> 00:01:54 AUDIENCE: But haven't had a chance to. 38 00:01:54 --> 00:01:58 But, ok, so I'm not really sure how you take the initial 39 00:01:58 --> 00:02:02 conditions and apply them to the ramp function to 40 00:02:02 --> 00:02:04 actually get a solution. 41 00:02:04 --> 00:02:09 PROFESSOR STRANG: So that's what luckily happens to be 42 00:02:09 --> 00:02:11 still here on the board, right? 43 00:02:11 --> 00:02:14 We've got these boundary, I would call them 44 00:02:14 --> 00:02:16 boundary conditions. 45 00:02:16 --> 00:02:19 So this is definitely, we're in a part of math that's about 46 00:02:19 --> 00:02:23 boundary value problems more than time, isn't 47 00:02:23 --> 00:02:24 in the picture. 48 00:02:24 --> 00:02:28 I mean later time will get into the picture. 49 00:02:28 --> 00:02:37 So with this particular example, the general solution 50 00:02:37 --> 00:02:44 is a standard ramp at the point a where things are happening. 51 00:02:44 --> 00:02:47 Plus the C+Dx, the usual. 52 00:02:47 --> 00:02:50 So that's a particular solution. 53 00:02:50 --> 00:02:54 That particular solution has the right behavior 54 00:02:54 --> 00:02:59 at the impulse. 55 00:02:59 --> 00:03:04 By the right behavior, I mean that the second derivative of 56 00:03:04 --> 00:03:10 the ramp is a delta and when I put the ramp at a, then the 57 00:03:10 --> 00:03:12 delta will show up at a. 58 00:03:12 --> 00:03:16 And when I put on that minus sign it'll mean that the second 59 00:03:16 --> 00:03:19 derivative is minus the delta. 60 00:03:19 --> 00:03:23 So the slope will step down. 61 00:03:23 --> 00:03:24 So that's a particular solution. 62 00:03:24 --> 00:03:31 But that by itself, what does that equal at zero? 63 00:03:31 --> 00:03:36 And what does that equal at one if you remember the ramp? 64 00:03:36 --> 00:03:38 So let me just draw that ramp again. 65 00:03:38 --> 00:03:44 So the ramp was really based on, centered at the point a. 66 00:03:44 --> 00:03:46 And I'll put it with a minus sign. 67 00:03:46 --> 00:03:50 So it came along there and down there. 68 00:03:50 --> 00:03:53 And now suppose this is one end of our interval and 69 00:03:53 --> 00:03:56 that's the other end. 70 00:03:56 --> 00:03:59 So is that ramp the answer to our problem? 71 00:03:59 --> 00:04:00 No. 72 00:04:00 --> 00:04:06 Well it happens to satisfy this boundary condition, happens 73 00:04:06 --> 00:04:08 to start out at zero. 74 00:04:08 --> 00:04:10 But it doesn't end up at zero. 75 00:04:10 --> 00:04:19 So just like every particular solution we need a little more. 76 00:04:19 --> 00:04:22 We have to include more solutions because 77 00:04:22 --> 00:04:24 that was only one. 78 00:04:24 --> 00:04:26 All those are equally good solutions. 79 00:04:26 --> 00:04:30 If I add C+Dx, the second derivative of that is 80 00:04:30 --> 00:04:33 zero, so it doesn't spoil anything at all. 81 00:04:33 --> 00:04:38 On the contrary, it adds more solutions. 82 00:04:38 --> 00:04:42 I mean the great thing we're using here is that our 83 00:04:42 --> 00:04:49 equations are linear and when zero's on the right-hand 84 00:04:49 --> 00:04:51 side-- notice there's no arbitrary constant. 85 00:04:51 --> 00:04:53 I'm not putting an arbitrary constant on 86 00:04:53 --> 00:04:55 that particular solution. 87 00:04:55 --> 00:05:01 Is one particular solution plus a subspace if I 88 00:05:01 --> 00:05:02 use that language. 89 00:05:02 --> 00:05:09 A lot of solutions to the, all the solutions to the problem 90 00:05:09 --> 00:05:10 with zero on the right-hand side. 91 00:05:10 --> 00:05:14 And these are the solutions, these are the null space guys, 92 00:05:14 --> 00:05:19 the ones that have zero on the right-hand side. 93 00:05:19 --> 00:05:22 Now we need those. 94 00:05:22 --> 00:05:27 And the effect of those will be to move that ramp. 95 00:05:27 --> 00:05:29 And effectively, what are they going to do? 96 00:05:29 --> 00:05:33 They're going to swing that ramp around, it'll stay a ramp. 97 00:05:33 --> 00:05:36 But instead of being level here, it'll go up. 98 00:05:36 --> 00:05:41 And instead of going down there, it'll go down there. 99 00:05:41 --> 00:05:44 It'll be the same ramp, with still that slope 100 00:05:44 --> 00:05:46 dropped by one. 101 00:05:46 --> 00:05:52 Slope went down by minus one and that's not 102 00:05:52 --> 00:05:53 going to change here. 103 00:05:53 --> 00:05:59 The slope will still go down by minus one. 104 00:05:59 --> 00:06:03 But now I've just adjusted it so that it goes through 105 00:06:03 --> 00:06:06 the, it satisfies the boundary conditions. 106 00:06:06 --> 00:06:11 And in the second problem with the free end, 107 00:06:11 --> 00:06:14 again, here's my ramp. 108 00:06:14 --> 00:06:16 But now I'm going to adjust it. 109 00:06:16 --> 00:06:17 And what happened? 110 00:06:17 --> 00:06:20 It just needed adjustment upwards. 111 00:06:20 --> 00:06:26 Because this was the zero, this was the u=0 fixed guy. 112 00:06:26 --> 00:06:32 And now if I'm doing u'(0)=0, the free guy, I can lift 113 00:06:32 --> 00:06:33 the whole thing up. 114 00:06:33 --> 00:06:37 So you see, I just lifted it up to the point where it 115 00:06:37 --> 00:06:39 came out right there. 116 00:06:39 --> 00:06:43 So in this case I just needed a C. 117 00:06:43 --> 00:06:45 And in this case I just needed a Dx. 118 00:06:46 --> 00:06:49 And in other cases I might have needed both. 119 00:06:49 --> 00:06:51 A little bit of C and a little of Dx. 120 00:06:51 --> 00:06:59 Anyway that's what was, maybe that's sort of repeating what 121 00:06:59 --> 00:07:02 we did today a little bit. 122 00:07:02 --> 00:07:08 But mechanically it's just, we've got a particular solution 123 00:07:08 --> 00:07:14 and we've got the complete solution and then we just have 124 00:07:14 --> 00:07:18 to choose C and D and we've got two conditions to do it with, 125 00:07:18 --> 00:07:20 the two boundary conditions. 126 00:07:20 --> 00:07:22 Could have other boundary conditions. 127 00:07:22 --> 00:07:26 Now, what about, here's a, yeah. 128 00:07:26 --> 00:07:29 I guess what will often happen in these review sessions is I 129 00:07:29 --> 00:07:33 get on a roll and I just keep, carry on with it. 130 00:07:33 --> 00:07:37 You know, you start me with a question and I can't stop. 131 00:07:37 --> 00:07:42 So I'll go a little longer but then I really will stop, 132 00:07:42 --> 00:07:45 ready for the next one. 133 00:07:45 --> 00:07:50 So we haven't discussed the free-free. u(0) 134 00:07:50 --> 00:07:52 u', sorry, free is u'. 135 00:07:53 --> 00:08:00 That's free at the left and free at the right. 136 00:08:00 --> 00:08:04 What's up with that? 137 00:08:04 --> 00:08:06 What's the solution to that? 138 00:08:06 --> 00:08:13 Again, I'm looking for -u'' equal an impulse. 139 00:08:13 --> 00:08:16 With those two boundary conditions, free-free. 140 00:08:16 --> 00:08:19 And do you know what's going to happen? 141 00:08:19 --> 00:08:22 No solution. 142 00:08:22 --> 00:08:25 Now it'd be interesting to see why not. 143 00:08:25 --> 00:08:28 Why no solution? 144 00:08:28 --> 00:08:32 Well one way to do it is try. 145 00:08:32 --> 00:08:36 Other right-hand sides could have a solution. 146 00:08:36 --> 00:08:41 So it's not just that this is something the matter here. 147 00:08:41 --> 00:08:44 Specifically, that right-hand side and most right-hand 148 00:08:44 --> 00:08:46 sides will fail. 149 00:08:46 --> 00:08:47 But let's just see it with this one. 150 00:08:47 --> 00:08:50 Why does that fail? 151 00:08:50 --> 00:08:57 Well you can see I can't, the slope here is zero and the 152 00:08:57 --> 00:08:59 slope here is minus one. 153 00:08:59 --> 00:09:06 If I adjust those I can't get, this is asking me to get 154 00:09:06 --> 00:09:07 slope zero at both ends. 155 00:09:07 --> 00:09:10 I can't do that, right? 156 00:09:10 --> 00:09:15 Yeah, just not possible for me to. 157 00:09:15 --> 00:09:19 This will add a straight line but it's the same straight. 158 00:09:19 --> 00:09:23 I can't get a ramp that comes flat at both ends because 159 00:09:23 --> 00:09:28 there's, once I say what it's doing at the ends, I've got it. 160 00:09:28 --> 00:09:31 And you see what I mean? 161 00:09:31 --> 00:09:41 I'm looking for one that starts flat and ends flat and that's 162 00:09:41 --> 00:09:45 not in my family of solutions. 163 00:09:45 --> 00:09:49 That problem just doesn't have a solution. 164 00:09:49 --> 00:09:53 So that's one way to do it is look at the solutions and 165 00:09:53 --> 00:09:56 realize you can't satisfy both boundary conditions. 166 00:09:56 --> 00:10:00 Another way might be this. 167 00:10:00 --> 00:10:05 This is a little bit deeper way and it leads to 168 00:10:05 --> 00:10:07 something better. 169 00:10:07 --> 00:10:10 Suppose I take the integral. 170 00:10:10 --> 00:10:13 So this is an equation that's supposed to hold. 171 00:10:13 --> 00:10:18 Let me integrate both sides from zero to one. 172 00:10:18 --> 00:10:21 So there's the idea I'm putting in now. 173 00:10:21 --> 00:10:23 To go a little further, to discover when 174 00:10:23 --> 00:10:25 this has a solution. 175 00:10:25 --> 00:10:28 Or let me take more generally, -u''=f(x). 176 00:10:28 --> 00:10:31 177 00:10:31 --> 00:10:33 So some other load. 178 00:10:33 --> 00:10:37 Not necessarily a point load, not necessarily a uniform load, 179 00:10:37 --> 00:10:39 but maybe some other load. 180 00:10:39 --> 00:10:44 And now my boundary conditions, I'm trying to do free-free. 181 00:10:44 --> 00:10:48 And usually no solution. 182 00:10:48 --> 00:10:52 But let's just see why and when there might be a solution. 183 00:10:52 --> 00:11:02 The key idea is integrate from zero to one. 184 00:11:02 --> 00:11:04 What do I get on the right-hand side? 185 00:11:04 --> 00:11:10 Well, I get the integral of f, whatever it is, and I would 186 00:11:10 --> 00:11:15 call that the total load. 187 00:11:15 --> 00:11:16 Fair enough? 188 00:11:16 --> 00:11:22 The total load of if it was a delta function, the total load 189 00:11:22 --> 00:11:24 would be one and it would be all in one spot. 190 00:11:24 --> 00:11:29 If it was uniformly over the whole interval, well I guess 191 00:11:29 --> 00:11:32 that would also integrate to one, so the total load 192 00:11:32 --> 00:11:33 would be one spread out. 193 00:11:33 --> 00:11:37 But it could be a mixture of the two, could be a few 194 00:11:37 --> 00:11:39 delta functions, whatever. 195 00:11:39 --> 00:11:45 What happens when I integrate the left side from zero to one? 196 00:11:45 --> 00:11:46 Can you do that one? 197 00:11:46 --> 00:11:47 The integral from zero to one. 198 00:11:47 --> 00:11:50 There's that dopey minus sign. u''dx. 199 00:11:52 --> 00:11:55 What do I get? 200 00:11:55 --> 00:11:59 Why do I say that's a good idea? 201 00:11:59 --> 00:12:04 If I integrate the second derivative I get the 202 00:12:04 --> 00:12:05 first derivative. 203 00:12:05 --> 00:12:07 The integral of the second derivative will be the first 204 00:12:07 --> 00:12:11 derivative with the minus. 205 00:12:11 --> 00:12:13 So it's minus the first derivative. 206 00:12:13 --> 00:12:16 And what do I do now? 207 00:12:16 --> 00:12:18 I plug in the end points, right? 208 00:12:18 --> 00:12:19 You integrate. 209 00:12:19 --> 00:12:20 I'm integrating zero to one. 210 00:12:20 --> 00:12:22 So I've found the integral. 211 00:12:22 --> 00:12:26 I've put zero to one in there. 212 00:12:26 --> 00:12:27 So what is that? 213 00:12:27 --> 00:12:33 That's minus the derivative at one plus the 214 00:12:33 --> 00:12:40 derivative at zero. 215 00:12:40 --> 00:12:42 And now, what's that? 216 00:12:42 --> 00:12:44 That's zero. 217 00:12:44 --> 00:12:47 By my boundary conditions, that's zero. 218 00:12:47 --> 00:12:49 So what have I found? 219 00:12:49 --> 00:12:52 I've found that if these are the boundary conditions, then 220 00:12:52 --> 00:12:57 when I integrate the left side it's going to give me zero. 221 00:12:57 --> 00:13:03 So when, what loads could be ok? 222 00:13:03 --> 00:13:14 What loads f(x) could allow me to solve this equation? 223 00:13:14 --> 00:13:19 The condition will be I need, what do I need for the 224 00:13:19 --> 00:13:24 total load to be able to solve this equation? 225 00:13:24 --> 00:13:28 The integral of the left side was zero so the integral of the 226 00:13:28 --> 00:13:31 right side had better be zero. 227 00:13:31 --> 00:13:34 So that's the condition. 228 00:13:34 --> 00:13:36 If I have these boundary conditions, then my 229 00:13:36 --> 00:13:38 problem is singular. 230 00:13:38 --> 00:13:40 Usually no solution. 231 00:13:40 --> 00:13:42 It's like having a singular matrix. 232 00:13:42 --> 00:13:45 It's like having this particular singular 233 00:13:45 --> 00:13:46 matrix, of course. 234 00:13:46 --> 00:13:48 Whoops, not that one. 235 00:13:48 --> 00:13:51 Let me get the plus sign in the right position. 236 00:13:51 --> 00:13:52 That's a plus. 237 00:13:53 --> 00:13:57 -1; -1, 2, -1; -1, 1. 238 00:13:57 --> 00:13:59 Right? 239 00:13:59 --> 00:14:08 This is the discrete version with a zero slope at both ends. 240 00:14:08 --> 00:14:10 It's our T matrix. 241 00:14:10 --> 00:14:12 No, what matrix is it? 242 00:14:12 --> 00:14:14 B. 243 00:14:14 --> 00:14:18 It's our B matrix, both ends. 244 00:14:18 --> 00:14:22 I'll just come back here and then I'll do the discrete one. 245 00:14:22 --> 00:14:26 So tell me a load that we could handle? 246 00:14:26 --> 00:14:30 A load we could handle. 247 00:14:30 --> 00:14:32 So the integral has to be zero. 248 00:14:32 --> 00:14:39 So suppose my load has a delta function at a. 249 00:14:39 --> 00:14:42 Well that integral is one. 250 00:14:42 --> 00:14:46 So can you fix that, change that load or do something, 251 00:14:46 --> 00:14:53 maybe put on another load to get a total load of zero? 252 00:14:53 --> 00:14:55 What shall I do? 253 00:14:55 --> 00:14:59 Add another guy with a minus sign. 254 00:14:59 --> 00:15:05 In other words, maybe this, a delta function 255 00:15:05 --> 00:15:07 at some other point B. 256 00:15:07 --> 00:15:08 Well, that would do it. 257 00:15:08 --> 00:15:11 I believe we could solve that problem. 258 00:15:11 --> 00:15:18 Even with these bad boundary conditions. 259 00:15:18 --> 00:15:20 We could solve that problem. 260 00:15:20 --> 00:15:23 Because the total load would be one from that, minus one from 261 00:15:23 --> 00:15:27 that, the total load would be zero. 262 00:15:27 --> 00:15:30 In other words, what would are solution look like? 263 00:15:30 --> 00:15:32 It has to start with zero slope. 264 00:15:32 --> 00:15:37 So it would buzz alone to a and then after b. 265 00:15:37 --> 00:15:40 And what does it have to do here? 266 00:15:40 --> 00:15:48 If I graph the solution to this guy from zero to one it starts 267 00:15:48 --> 00:15:51 with, it's free, so nothing's happening until I get to a. 268 00:15:51 --> 00:15:55 Then what has to happen? 269 00:15:55 --> 00:16:01 Let's see, if I'm graphing u, it'll be ramp. 270 00:16:01 --> 00:16:01 Right? 271 00:16:01 --> 00:16:02 It'll be a ramp, yeah. 272 00:16:02 --> 00:16:04 Because I've two derivatives. 273 00:16:04 --> 00:16:11 And it has to ramp down by one, so it'll ramp whatever it does. 274 00:16:11 --> 00:16:15 I don't know where it stops. 275 00:16:15 --> 00:16:16 Where does it? 276 00:16:16 --> 00:16:18 Wait a minute. 277 00:16:18 --> 00:16:21 I haven't practiced this. 278 00:16:21 --> 00:16:26 So I start from the other end. 279 00:16:26 --> 00:16:31 The other end is flat. 280 00:16:31 --> 00:16:33 What's up? 281 00:16:33 --> 00:16:36 They gotta meet here. 282 00:16:36 --> 00:16:38 Oh, the other end is flat, but not at zero! 283 00:16:38 --> 00:16:39 Dumb, stupid. 284 00:16:39 --> 00:16:40 Right. 285 00:16:40 --> 00:16:41 Ok. 286 00:16:41 --> 00:16:43 Yes, the other end is flat, right. 287 00:16:43 --> 00:16:45 And it's, oh yeah, look! 288 00:16:45 --> 00:16:46 Oh, wonderful. 289 00:16:46 --> 00:16:47 You see. 290 00:16:47 --> 00:16:53 That slope dropped by one and the slope there increased 291 00:16:53 --> 00:16:55 by one back to zero. 292 00:16:55 --> 00:16:56 Slope was zero. 293 00:16:56 --> 00:17:00 It dropped to minus one because of that load. 294 00:17:00 --> 00:17:03 Now it increased back to zero because of this load 295 00:17:03 --> 00:17:07 with the minus sign and there's a solution. 296 00:17:07 --> 00:17:10 So that's a solvable problem. 297 00:17:10 --> 00:17:12 Well, you say, okay, that was a little surprising to get an 298 00:17:12 --> 00:17:15 answer for a singular problem. 299 00:17:15 --> 00:17:17 And no, it can happen. 300 00:17:17 --> 00:17:21 If we have a total load zero it'll happen. 301 00:17:21 --> 00:17:28 But, there is still a but, that's not the only answer. 302 00:17:28 --> 00:17:31 That picture is a solution. 303 00:17:31 --> 00:17:34 But not the only one. 304 00:17:34 --> 00:17:37 So what my point is going to be, that when the problem is 305 00:17:37 --> 00:17:41 singular, if there's an answer, you say great. 306 00:17:41 --> 00:17:44 But then something has to go wrong and what goes wrong 307 00:17:44 --> 00:17:46 is too many answers. 308 00:17:46 --> 00:17:53 So tell me some more, what other graphs would draw 309 00:17:53 --> 00:18:00 solutions to this problem. 310 00:18:00 --> 00:18:04 I could shift, I could lift the whole thing. 311 00:18:04 --> 00:18:07 Here I've got a plus C that I haven't used. 312 00:18:07 --> 00:18:10 I could just do the whole thing higher up. 313 00:18:10 --> 00:18:11 Any of these. 314 00:18:11 --> 00:18:17 These would all work. 315 00:18:17 --> 00:18:18 It's like temperature. 316 00:18:18 --> 00:18:23 I don't have an absolute temperature here. 317 00:18:23 --> 00:18:30 All I've got is, I would have to, it's not determined because 318 00:18:30 --> 00:18:37 there's a plus C that, the plus C satisfied everything. 319 00:18:37 --> 00:18:41 A plus C, a constant has zero slope, it has zero slope, its 320 00:18:41 --> 00:18:42 second derivative is zero. 321 00:18:42 --> 00:18:48 So it's like, unseen by this equation. 322 00:18:48 --> 00:18:52 And similarly can I just make the analogy as I always like to 323 00:18:52 --> 00:18:56 do with discrete stuff, so suppose I, tell me a right-hand 324 00:18:56 --> 00:19:03 side that we think would probably, is this going to be 325 00:19:03 --> 00:19:05 the same story for this guy? 326 00:19:05 --> 00:19:06 Yes. 327 00:19:06 --> 00:19:11 If I add those, where I integrated there, here I would 328 00:19:11 --> 00:19:13 add and I get zero, zero, zero. 329 00:19:13 --> 00:19:18 So this has to add to zero if there's a solution. 330 00:19:18 --> 00:19:22 So let me put for example, . 331 00:19:22 --> 00:19:26 That would be kind of like our delta function in one direction 332 00:19:26 --> 00:19:28 and our delta function in the other. 333 00:19:28 --> 00:19:31 I believe I can solve that problem. 334 00:19:31 --> 00:19:37 So I'm just carrying, because I always want you to see the 335 00:19:37 --> 00:19:39 discrete one as well as the continuous. 336 00:19:39 --> 00:19:43 Continuous involve this integration. 337 00:19:43 --> 00:19:47 The finite one just involves adding. 338 00:19:47 --> 00:19:50 The left side adds to zero so the right-hand side 339 00:19:50 --> 00:19:52 better add to zero. 340 00:19:52 --> 00:19:53 That right-hand side does. 341 00:19:53 --> 00:19:54 Tell me a solution. 342 00:19:54 --> 00:19:57 Well, let me start out with a seven there. 343 00:19:57 --> 00:20:04 What's the next guy going to be? 344 00:20:04 --> 00:20:05 See, I want seven. 345 00:20:05 --> 00:20:10 Whatever I put there, I better have a seven there, right? 346 00:20:10 --> 00:20:12 Seven, seven, good. 347 00:20:12 --> 00:20:16 Minus seven, plus 14, oh geez. 348 00:20:16 --> 00:20:22 I didn't know this was going to happen. 349 00:20:22 --> 00:20:26 No, I want to get the answer one. 350 00:20:26 --> 00:20:28 What number goes there? 351 00:20:28 --> 00:20:29 Six, is it six? 352 00:20:29 --> 00:20:31 It's six, yeah, good. 353 00:20:31 --> 00:20:34 Minus seven, 14, minus six is that. 354 00:20:34 --> 00:20:36 And now my claim is that we'll come out right 355 00:20:36 --> 00:20:38 on the third equation. 356 00:20:38 --> 00:20:40 So far I've just matched the first two. 357 00:20:40 --> 00:20:44 Now this one gives minus seven, plus six, that's 358 00:20:44 --> 00:20:45 minus one, good. 359 00:20:45 --> 00:20:48 So there's a solution. 360 00:20:48 --> 00:20:52 And I'll leave this problem alone if you tell me the 361 00:20:52 --> 00:20:54 rest, other solutions. 362 00:20:54 --> 00:20:59 That was a solution to a singular problem with a 363 00:20:59 --> 00:21:04 right-hand side that had total load zero, so it was ok. 364 00:21:04 --> 00:21:08 But now that's a solution, but there are more. 365 00:21:08 --> 00:21:13 Tell me another one. 366 00:21:13 --> 00:21:15 I can shift it, right? 367 00:21:15 --> 00:21:18 I could make it . 368 00:21:18 --> 00:21:21 I could add ten to everything. 369 00:21:21 --> 00:21:21 Right? 370 00:21:21 --> 00:21:24 That's the plus C that I could do over there. 371 00:21:24 --> 00:21:30 That can't change because 17 - 17 is still zero. 372 00:21:30 --> 00:21:35 -17 + 16 will still be minus one. all good. 373 00:21:35 --> 00:21:39 So actually that just like helps our intuition 374 00:21:39 --> 00:21:42 and physically my intuition is this. 375 00:21:42 --> 00:21:49 That I've got this bar and nothing's holding it. 376 00:21:49 --> 00:21:57 So if I put a weight on it, nothing to hold it, it'll just, 377 00:21:57 --> 00:22:00 rigid motion will take it out of sight, no good. 378 00:22:00 --> 00:22:02 But if I put another equal weight on it, 379 00:22:02 --> 00:22:03 no it's not a weight. 380 00:22:03 --> 00:22:04 What do I call it? 381 00:22:04 --> 00:22:09 If I lift it at that point, that's the other delta function 382 00:22:09 --> 00:22:13 that's going the other way, then it will sit there. 383 00:22:13 --> 00:22:18 But it would still be in equilibrium if I just 384 00:22:18 --> 00:22:21 moved it up to there or moved it as I like. 385 00:22:21 --> 00:22:26 I don't know if that is kind of a dumb picture. 386 00:22:26 --> 00:22:31 But it's saying what we've said from math. 387 00:22:31 --> 00:22:33 Well, you see where you're question lead. 388 00:22:33 --> 00:22:44 Yeah, thanks. 389 00:22:44 --> 00:22:49 No, the integral, it was-- Watch what we integrated. 390 00:22:49 --> 00:22:50 We integrated u''. 391 00:22:50 --> 00:22:53 392 00:22:53 --> 00:22:55 So that's not the area. 393 00:22:55 --> 00:22:59 We integrated u'' and got, it's integral was u'. 394 00:22:59 --> 00:23:03 So that just told us that a difference in slopes 395 00:23:03 --> 00:23:05 at the ends, yeah. 396 00:23:05 --> 00:23:09 Good, because our intuition automatically is if we're 397 00:23:09 --> 00:23:11 integrating something, we're finding an area. 398 00:23:11 --> 00:23:15 But here, if it was u , then I'd be finding the area 399 00:23:15 --> 00:23:19 under u, but we integrated the second derivative. 400 00:23:19 --> 00:23:23 Right, good. 401 00:23:23 --> 00:23:24 Now let's change the subject. 402 00:23:24 --> 00:23:33 Yes, please. 403 00:23:33 --> 00:23:34 Yeah, I guess so. 404 00:23:34 --> 00:23:44 I'll try. 405 00:23:44 --> 00:23:45 Let's see. 406 00:23:45 --> 00:23:53 So my discrete equation was, like -u. 407 00:23:53 --> 00:23:57 Yeah, so let's back up to the beginning. 408 00:23:57 --> 00:23:59 We've got this minus sign and we're using 409 00:23:59 --> 00:24:00 a second difference. 410 00:24:00 --> 00:24:03 So second differences have coefficients one, 411 00:24:03 --> 00:24:06 minus two and one. 412 00:24:06 --> 00:24:09 Now I'm reversing the signs because of my minus. 413 00:24:09 --> 00:24:13 So I have -u at some point. 414 00:24:13 --> 00:24:17 Let's take that as the point to the left. 415 00:24:17 --> 00:24:21 Two u's at what I'll think of as the center point. 416 00:24:22 --> 00:24:29 -u_(i-1) is the load at that center. 417 00:24:29 --> 00:24:37 That center point is i times delta x. 418 00:24:37 --> 00:24:38 That's where I would be looking. 419 00:24:38 --> 00:24:44 So now I'm using subscript. 420 00:24:44 --> 00:24:50 It's a little bit of practice then to take subscripts, take 421 00:24:50 --> 00:24:54 this way of writing the equation and convert 422 00:24:54 --> 00:24:57 it to a matrix way. 423 00:24:57 --> 00:25:01 It's usually clearer once you see it as a matrix. 424 00:25:01 --> 00:25:04 Now this is happening at all the points. 425 00:25:04 --> 00:25:24 At i=1, let's say, I have -u_0+2u_1-u_2 is f at, 426 00:25:24 --> 00:25:30 agrees with the load at the first mesh point. 427 00:25:30 --> 00:25:36 That's the center, the point h, delta x. 428 00:25:36 --> 00:25:42 And then if I want to back up further, I would have -u_-1, 429 00:25:42 --> 00:25:49 but that doesn't really exist, plus 2u_0-u_1 should 430 00:25:49 --> 00:25:55 match the load at zero. 431 00:25:55 --> 00:25:57 And so on forward. 432 00:25:57 --> 00:26:00 But now I want to put in the boundary condition. 433 00:26:00 --> 00:26:01 That's what you want me to do, right? 434 00:26:01 --> 00:26:03 Put in this boundary condition. 435 00:26:03 --> 00:26:11 So what am I going to take as boundary condition? 436 00:26:11 --> 00:26:14 It has to be some approximation to u'(0)=0. 437 00:26:14 --> 00:26:18 438 00:26:18 --> 00:26:20 Maybe I'm never going to get to minus one. 439 00:26:20 --> 00:26:33 Maybe I don't need minus one. 440 00:26:33 --> 00:26:38 That's right, yeah, exactly. 441 00:26:38 --> 00:26:41 We did. 442 00:26:41 --> 00:26:42 That's what we knew about it. 443 00:26:42 --> 00:26:45 Sending it forward, we knew about forward difference, 444 00:26:45 --> 00:26:51 so I chose to do it. 445 00:26:51 --> 00:26:53 But then I think better of it. 446 00:26:53 --> 00:26:57 I chose to do it because it made the point that we, that at 447 00:26:57 --> 00:27:02 that boundary we were introducing a higher order 448 00:27:02 --> 00:27:08 error, first order error that's going to wreck things. 449 00:27:08 --> 00:27:11 I mean, it's going to spoil the, this is 450 00:27:11 --> 00:27:13 second order accuracy. 451 00:27:13 --> 00:27:17 And, but let me do that first order. 452 00:27:17 --> 00:27:19 So what shall I take? 453 00:27:19 --> 00:27:23 I'm going to approximate that by u-- Shall I take this 454 00:27:23 --> 00:27:33 one as I did in class? 455 00:27:33 --> 00:27:35 Yes. 456 00:27:35 --> 00:27:39 Ah, plus one, thanks, plus one, right, thank you. 457 00:27:39 --> 00:27:42 Thank you, good. 458 00:27:42 --> 00:27:44 Okeydoke. 459 00:27:44 --> 00:27:47 Alright. 460 00:27:47 --> 00:27:50 This is how we got to that equation. 461 00:27:50 --> 00:27:54 If I now bring in this boundary condition-- I guess I don't 462 00:27:54 --> 00:27:58 have to, let me take your eye off of that guy for 463 00:27:58 --> 00:28:02 the moment, I think. 464 00:28:02 --> 00:28:05 We're getting beautiful music here. 465 00:28:05 --> 00:28:07 Is it coming out of this box or? 466 00:28:07 --> 00:28:08 No. 467 00:28:08 --> 00:28:18 Anyway. so I'm going to use this boundary condition to say, 468 00:28:18 --> 00:28:22 well ok, if u_1 is u_0, I'm going to replace 469 00:28:22 --> 00:28:23 this u_0 by u_1. 470 00:28:23 --> 00:28:26 471 00:28:26 --> 00:28:28 This is the direct way. 472 00:28:28 --> 00:28:33 I replaced that u_0 by u_1 in that first equation. 473 00:28:33 --> 00:28:37 And then what I have is -u_1 and 2u_1, so that's the 474 00:28:37 --> 00:28:39 one and I have the -u_2. 475 00:28:40 --> 00:28:44 So do you see that that equation, when I put those 476 00:28:44 --> 00:28:54 together into a one, is going to, if this is u_1, this is 477 00:28:54 --> 00:29:01 u_2, this is u_3 onwards, that first equation is u_1-u_2 478 00:29:01 --> 00:29:03 and that's what I've got. 479 00:29:03 --> 00:29:05 This is u_1-u_2. 480 00:29:05 --> 00:29:09 481 00:29:09 --> 00:29:10 So I did it. 482 00:29:10 --> 00:29:12 I got to that matrix. 483 00:29:12 --> 00:29:15 The matrix is actually quite an important matrix. 484 00:29:15 --> 00:29:20 But from the point of view of accuracy in solving this 485 00:29:20 --> 00:29:24 differential equation, it's not the greatest. 486 00:29:24 --> 00:29:27 It's lost accuracy at that point. 487 00:29:27 --> 00:29:31 But the way to recover it turned out to be just a small 488 00:29:31 --> 00:29:35 adjustment at the boundary, so not a problem. 489 00:29:35 --> 00:29:35 Thanks. 490 00:29:35 --> 00:29:36 That's good. 491 00:29:36 --> 00:29:41 Yes, thanks. 492 00:29:41 --> 00:29:49 Sorry? 493 00:29:49 --> 00:29:49 When the boundary-- sorry. 494 00:29:49 --> 00:29:55 Two boundary conditions at the same point? 495 00:29:55 --> 00:29:57 That's a good question. 496 00:29:57 --> 00:29:59 So when would we have two? 497 00:29:59 --> 00:30:02 So instead of a boundary condition at zero and a 498 00:30:02 --> 00:30:05 boundary condition at one, you're putting them, 499 00:30:05 --> 00:30:06 is that what you mean? 500 00:30:06 --> 00:30:08 Put both boundary conditions at the end. 501 00:30:08 --> 00:30:10 Ok. 502 00:30:10 --> 00:30:14 So that would be, that would happen, I would think that 503 00:30:14 --> 00:30:19 would be more, it would be very typical in a, let me see if 504 00:30:19 --> 00:30:22 there's some space here, yeah. 505 00:30:22 --> 00:30:28 That would be very typical and we will do it, 506 00:30:28 --> 00:30:30 can I change x to t? 507 00:30:30 --> 00:30:38 Because that's what, if I have some. 508 00:30:38 --> 00:30:39 What does this problem look like? 509 00:30:39 --> 00:30:43 And u(0)=0 and u'(0)=0. 510 00:30:45 --> 00:30:52 Both at the same, at the start equals zero or whatever. 511 00:30:52 --> 00:30:56 So what kind of a problem is that? 512 00:30:56 --> 00:30:59 Now these are, I would say, initial values. 513 00:30:59 --> 00:31:03 Initial values instead of boundary values, I now 514 00:31:03 --> 00:31:05 have initial values. 515 00:31:05 --> 00:31:07 And can I solve it? 516 00:31:07 --> 00:31:10 Yes. 517 00:31:10 --> 00:31:11 So I'm starting at time zero. 518 00:31:11 --> 00:31:12 This is t=0. 519 00:31:14 --> 00:31:18 I'm starting at rest. 520 00:31:18 --> 00:31:23 No velocity and actually no displacement and just 521 00:31:23 --> 00:31:25 going forward in time. 522 00:31:25 --> 00:31:30 So I could solve that differential equation. 523 00:31:30 --> 00:31:32 I'd be interested in the corresponding 524 00:31:32 --> 00:31:36 difference equation. 525 00:31:36 --> 00:31:37 All fine. 526 00:31:37 --> 00:31:39 It's a different category of problem. 527 00:31:39 --> 00:31:49 This is an initial value problem. 528 00:31:49 --> 00:32:00 It's like tracking some mass that's, some satellite. 529 00:32:00 --> 00:32:04 So that's what you're doing in tracking a satellite 530 00:32:04 --> 00:32:09 or a planet or something. 531 00:32:09 --> 00:32:13 Yeah, tracking a planet or a satellite. 532 00:32:13 --> 00:32:16 You're solving equations like this. 533 00:32:16 --> 00:32:18 Forward in time. 534 00:32:18 --> 00:32:22 You know the initial position and you know 535 00:32:22 --> 00:32:24 the forces acting on it. 536 00:32:24 --> 00:32:26 Probably gravity. 537 00:32:26 --> 00:32:28 And you go forward in time. 538 00:32:28 --> 00:32:31 Yes. 539 00:32:31 --> 00:32:33 What would the matrix be? 540 00:32:33 --> 00:32:34 Good question. 541 00:32:34 --> 00:32:41 What would the matrix look like? 542 00:32:41 --> 00:32:47 So an electrical engineer would call a problem like this, and 543 00:32:47 --> 00:32:50 the kind of matrix that I'm going to write down, I think, 544 00:32:50 --> 00:32:53 would be called causal. 545 00:32:53 --> 00:32:57 That word just popped into my head, so let me mention it. 546 00:32:57 --> 00:33:01 You know, part of science and engineering, a big part of it 547 00:33:01 --> 00:33:04 is learning language, learning words. 548 00:33:04 --> 00:33:10 And you have to learn sort of the math language and the 549 00:33:10 --> 00:33:14 engineering language for whatever you're focusing on. 550 00:33:14 --> 00:33:19 But it's good to also to know a few other languages. 551 00:33:19 --> 00:33:23 Electrical engineering languages of filters and 552 00:33:23 --> 00:33:28 causal and other things that we'll see are important. 553 00:33:28 --> 00:33:32 What would the matrix look like? 554 00:33:32 --> 00:33:35 Here's what I think it would be. 555 00:33:35 --> 00:33:40 I've made this a plus there just so I'll 556 00:33:40 --> 00:33:42 have to remember that. 557 00:33:42 --> 00:33:46 I think, so I'm looking at u_0, u_1, no. 558 00:33:46 --> 00:33:49 Well, u_0 I actually know, so let me start with 559 00:33:49 --> 00:33:50 u_1, u_2, u_3, u_4. 560 00:33:50 --> 00:33:55 561 00:33:55 --> 00:33:59 What would a typical equal sum, right side, f_1, f_2, f_3, f_4. 562 00:33:59 --> 00:34:06 1, What do you think, what kind of a matrix am I going to get? 563 00:34:06 --> 00:34:09 Before I put it in there. 564 00:34:09 --> 00:34:11 This is a good question. 565 00:34:11 --> 00:34:14 What's the shape of this matrix? 566 00:34:14 --> 00:34:19 It's going to be triangular. 567 00:34:19 --> 00:34:22 Instead of being symmetric it's going to be triangular. 568 00:34:22 --> 00:34:27 I'm going to find, let's see, a typical value would be, say, 569 00:34:27 --> 00:34:35 u_3 because I've used a plus sign, oops! 570 00:34:35 --> 00:34:39 I can't make myself do it right. 571 00:34:39 --> 00:34:44 1, -2, 1. 572 00:34:44 --> 00:34:54 That would say u_3-2u_2+u_1 would be the new force. 573 00:34:54 --> 00:34:56 This is the kind of thing we're going to get. 574 00:34:56 --> 00:34:59 One, maybe one something. 575 00:34:59 --> 00:35:00 I don't know what this is. 576 00:35:00 --> 00:35:03 This is up in the boundary, in the initial values. 577 00:35:03 --> 00:35:10 But from now on it'll be below the diagonal. 578 00:35:10 --> 00:35:15 It'll be 1, -2, 1. 579 00:35:15 --> 00:35:17 Do you see? 580 00:35:17 --> 00:35:20 We're marching. 581 00:35:20 --> 00:35:25 We're marching forward. 582 00:35:25 --> 00:35:28 We start by knowing these and then the equation 583 00:35:28 --> 00:35:29 tells us the next one. 584 00:35:29 --> 00:35:31 Then the equation tells us the next one. 585 00:35:31 --> 00:35:34 That's what initial value problems do. 586 00:35:34 --> 00:35:39 You're told how you begin and you take a step, you take a 587 00:35:39 --> 00:35:41 step, take a step every time. 588 00:35:41 --> 00:35:47 And the new value just needs to know the older values. 589 00:35:47 --> 00:35:50 Do you see the big difference between that 590 00:35:50 --> 00:35:53 and our problems here? 591 00:35:53 --> 00:35:58 Our problem is looking left and right looking for 592 00:35:58 --> 00:36:00 back and forward. 593 00:36:00 --> 00:36:02 Back for one condition, forward for another. 594 00:36:02 --> 00:36:07 We start with one, but we're, it's more of a, it's 595 00:36:07 --> 00:36:10 like a hitting problem. 596 00:36:10 --> 00:36:15 We start forward, marching forward in our problems, 597 00:36:15 --> 00:36:19 but we have to hit the other end correctly. 598 00:36:19 --> 00:36:22 We don't know the slope, we don't know the starting slope, 599 00:36:22 --> 00:36:24 we know what we want to hit. 600 00:36:24 --> 00:36:31 Whereas these problems, we're told how we start and we 601 00:36:31 --> 00:36:34 just follow it in time. 602 00:36:34 --> 00:36:40 So that's the difference here. 603 00:36:40 --> 00:36:49 Yeah, sure, okay. 604 00:36:49 --> 00:36:51 That's true. 605 00:36:51 --> 00:36:54 So this'll be known. 606 00:36:54 --> 00:36:56 Yeah, that'll be known. 607 00:36:56 --> 00:36:58 Yeah. u_1 will also be known. 608 00:36:58 --> 00:37:03 Yeah, and really, maybe I should have got, let me put 609 00:37:03 --> 00:37:06 even the other known one. 610 00:37:06 --> 00:37:08 So we know this, we know this. 611 00:37:08 --> 00:37:12 So those are sort of not in our, yeah, that shouldn't 612 00:37:12 --> 00:37:23 be in our problem somehow. 613 00:37:23 --> 00:37:28 No, I think, what would we get in the end? 614 00:37:28 --> 00:37:28 You're always looking backwards. 615 00:37:28 --> 00:37:29 That's the point. 616 00:37:29 --> 00:37:33 Lower triangular matrices are always looking, they only look 617 00:37:33 --> 00:37:35 backwards for earlier values and then they give you 618 00:37:35 --> 00:37:37 the current value. 619 00:37:37 --> 00:37:41 So that's why lower triangular matrices are so easy to invert. 620 00:37:41 --> 00:37:42 No problem. 621 00:37:42 --> 00:37:46 If it's lower triangular, you just, like, march forward. 622 00:37:46 --> 00:37:50 And if it's upper triangular, which way do you march? 623 00:37:50 --> 00:37:54 So if you have an upper triangular problem, suppose I 624 00:37:54 --> 00:37:59 gave you the problem, let me make it upper triangular. 625 00:37:59 --> 00:37:59 So x+y+z=7. 626 00:37:59 --> 00:38:03 627 00:38:03 --> 00:38:08 2y+3z=12 and z=17. 628 00:38:08 --> 00:38:11 629 00:38:11 --> 00:38:13 So that's upper triangular. 630 00:38:13 --> 00:38:17 Where do we start in solving that one? 631 00:38:17 --> 00:38:18 From the bottom. 632 00:38:18 --> 00:38:20 From the right-hand end, the bottom. 633 00:38:20 --> 00:38:25 And we march backwards in time. 634 00:38:25 --> 00:38:29 And what I was saying about A, well L times U, yeah, 635 00:38:29 --> 00:38:31 this is worth seeing. 636 00:38:31 --> 00:38:38 What I was saying about A=LU, it was, you remember that? 637 00:38:38 --> 00:38:40 Those letters? 638 00:38:40 --> 00:38:45 What that was saying was that this matrix that's looking both 639 00:38:45 --> 00:38:52 ways can be written as a product of a matrix L that 640 00:38:52 --> 00:38:59 looks behind for old values and you can go forward with it. 641 00:38:59 --> 00:39:05 And a matrix U, like this one, this upper triangular, 1, 1, 1, 642 00:39:05 --> 00:39:10 zeroes below that diagonal, that you go backward with. 643 00:39:10 --> 00:39:12 Somehow that's appealing. 644 00:39:12 --> 00:39:19 That's like aesthetic to break up a two-way problem into a 645 00:39:19 --> 00:39:24 problem like marches one way and then the other. 646 00:39:24 --> 00:39:29 And of course, that's what elimination aims for, is this 647 00:39:29 --> 00:39:33 problem that it can solve by, the words would be 648 00:39:33 --> 00:39:35 back substitution. 649 00:39:35 --> 00:39:37 When you've started with your original problem, got to this 650 00:39:37 --> 00:39:44 one, then you just have a, back substitution, you go backwards. 651 00:39:44 --> 00:39:49 Oh, so much, I'll mention the Kalman filter. 652 00:39:49 --> 00:39:53 That's a similar process of going forward, that's 653 00:39:53 --> 00:39:55 called prediction. 654 00:39:55 --> 00:39:57 Going backward, that's called smoothing. 655 00:39:57 --> 00:40:04 And so, Kalman had the great idea that he could break these 656 00:40:04 --> 00:40:14 problems that were fundamental in space computations for 657 00:40:14 --> 00:40:19 prediction and smoothing. 658 00:40:19 --> 00:40:20 Once again, we've got off. 659 00:40:20 --> 00:40:25 Yes? 660 00:40:25 --> 00:40:28 Oh, the beam. 661 00:40:28 --> 00:40:33 Let me help you even more before the question. 662 00:40:33 --> 00:40:39 I said it's better to draw the beam this way. 663 00:40:39 --> 00:40:41 I like the beam better this way. 664 00:40:41 --> 00:40:46 Because the point of the beam problem is loads are acting, 665 00:40:46 --> 00:40:51 and we'll see this, of course later, loads are acting 666 00:40:51 --> 00:40:54 perpendicular to the direction of the beam. 667 00:40:54 --> 00:40:57 That's why the beam bends. 668 00:40:57 --> 00:41:00 So it'll bend a little, right? 669 00:41:00 --> 00:41:04 And that is what leads us, it's bending moments 670 00:41:04 --> 00:41:05 and other stuff. 671 00:41:05 --> 00:41:10 If you haven't met beams, well, it'll be great to just have a 672 00:41:10 --> 00:41:15 very, half a lecture about, or maybe a lecture about beams. 673 00:41:15 --> 00:41:20 That gives a fourth order equation that I'll 674 00:41:20 --> 00:41:21 write down again. 675 00:41:21 --> 00:41:24 Fourth derivative equal the load. 676 00:41:24 --> 00:41:39 Now, ready. 677 00:41:39 --> 00:41:42 Yeah, now here I don't have the negative sign. 678 00:41:42 --> 00:41:47 Because once I've got second derivatives twice, so the 679 00:41:47 --> 00:41:52 second derivative is, in some way, negative. 680 00:41:52 --> 00:41:55 I'll complete that sentence in a second. 681 00:41:55 --> 00:41:59 Somehow the second derivative, which is the guy that has the 682 00:41:59 --> 00:42:04 1, -2, 1, somehow that's a negative thing. 683 00:42:04 --> 00:42:07 But fourth derivative is second derivative of 684 00:42:07 --> 00:42:08 the second derivative. 685 00:42:08 --> 00:42:12 Yeah, do you want to tell me what the numbers would be? 686 00:42:12 --> 00:42:17 As long as we're wildly looking forward to fourth 687 00:42:17 --> 00:42:19 derivatives, just, it helps. 688 00:42:19 --> 00:42:28 Do you want to guess what will a typical row of the matrix B 689 00:42:28 --> 00:42:30 when I go to finite differences, fourth 690 00:42:30 --> 00:42:32 differences? 691 00:42:32 --> 00:42:36 Probably you've never seen a fourth difference. 692 00:42:36 --> 00:42:38 You may not have seen second differences before. 693 00:42:38 --> 00:42:43 That was a big deal, then, to introduce second differences. 694 00:42:43 --> 00:42:46 Those 1, -2, 1's. 695 00:42:46 --> 00:42:48 That was second differences. 696 00:42:48 --> 00:42:50 Fourth? 697 00:42:50 --> 00:42:53 Yeah, 1, 4, 6, 4, 1 with minus sign. 698 00:42:53 --> 00:42:58 1, -4, 6, -4, and 1. 699 00:42:58 --> 00:43:06 In some way, I would get that by squaring this guy. 700 00:43:06 --> 00:43:17 So that would be a fourth difference. 701 00:43:17 --> 00:43:19 Oh, what's the deal with boundary conditions? 702 00:43:19 --> 00:43:25 What are you figuring on beams, beam problems for a fourth 703 00:43:25 --> 00:43:30 order equation and a matrix that's stretching out further. 704 00:43:30 --> 00:43:35 What's going to happen at the left-hand boundary? 705 00:43:35 --> 00:43:39 I guess my specific question is, How many boundary 706 00:43:39 --> 00:43:41 conditions do I now need? 707 00:43:41 --> 00:43:42 Four. 708 00:43:42 --> 00:43:44 And the typical is two at each end. 709 00:43:44 --> 00:43:46 That's the balanced way. 710 00:43:46 --> 00:43:50 That's the way that would make this matrix sort of symmetric. 711 00:43:50 --> 00:43:57 So I have maybe at this end I say it's held at zero and 712 00:43:57 --> 00:44:01 maybe it's just sitting on a log there. 713 00:44:01 --> 00:44:02 Right? 714 00:44:02 --> 00:44:06 That boundary condition I would call simply supported. 715 00:44:06 --> 00:44:09 That boundary condition says that u(0)=0. 716 00:44:11 --> 00:44:13 Because it's sitting there. 717 00:44:13 --> 00:44:17 And but the slope doesn't have to be zero. 718 00:44:17 --> 00:44:20 What does have to be zero there? 719 00:44:20 --> 00:44:23 Yeah, sort of the bending moment. 720 00:44:23 --> 00:44:27 Nobody's here twisting it, right? 721 00:44:27 --> 00:44:32 So the other condition in that picture would be second 722 00:44:32 --> 00:44:36 derivative equal zero. 723 00:44:36 --> 00:44:43 Maybe my point is that now you see what I said before, that 724 00:44:43 --> 00:44:46 the getting the boundary conditions into the problem 725 00:44:46 --> 00:44:49 is often the hardest part. 726 00:44:49 --> 00:44:52 Because I have to replace u(0)=0, that shouldn't 727 00:44:52 --> 00:44:53 be too hard to do. 728 00:44:53 --> 00:44:57 But I have to use this other condition somehow, 729 00:44:57 --> 00:45:01 it's going to screw up the 1, -4, 6, -4, 1. 730 00:45:01 --> 00:45:06 I'll have two boundary rows at the top, two boundary 731 00:45:06 --> 00:45:08 rows at the bottom. 732 00:45:08 --> 00:45:11 I don't want to go further today. 733 00:45:11 --> 00:45:15 But I think maybe just mentioning this gives you 734 00:45:15 --> 00:45:23 the picture of sort of the how things fit together. 735 00:45:23 --> 00:45:28 We would still have some nice constant diagonals in the 736 00:45:28 --> 00:45:33 middle, but now we'll have two boundary rows at each end. 737 00:45:33 --> 00:45:36 So that's something to come. 738 00:45:36 --> 00:45:44 Yes, now back to reality which is any questions. 739 00:45:44 --> 00:45:52 Lower triangular guy, yeah. 740 00:45:52 --> 00:45:56 What do I mean by marching forward? 741 00:45:56 --> 00:46:01 So let's see. 742 00:46:01 --> 00:46:04 I'll replace this. 743 00:46:04 --> 00:46:06 Let's see it better. 744 00:46:06 --> 00:46:11 I would replace this by maybe u, I'll use a different letter, 745 00:46:11 --> 00:46:26 n+1 at u_(n+1)-2u_n+u(n-1) is some right-hand side if there's 746 00:46:26 --> 00:46:28 a force acting on my thing. 747 00:46:28 --> 00:46:32 So f_n maybe. 748 00:46:32 --> 00:46:37 By marching forward, I just mean that this equation, 749 00:46:37 --> 00:46:40 that I can go in order. 750 00:46:40 --> 00:46:42 I can start with u_0 and u_1. 751 00:46:43 --> 00:46:47 They come from the boundary conditions. 752 00:46:47 --> 00:46:49 Then this equation will tell me u_2. 753 00:46:51 --> 00:46:53 I use the equation. 754 00:46:53 --> 00:46:54 With n as one. 755 00:46:54 --> 00:47:01 This says u_2, some u_1's, some u_0's, f_1's, all that I know. 756 00:47:01 --> 00:47:07 In other words, once I get started, I'm on a roll. 757 00:47:07 --> 00:47:10 If I have two boundary conditions to get me started, 758 00:47:10 --> 00:47:13 then the equation tells me u_2. 759 00:47:14 --> 00:47:16 And then the next time, u_3-2u_2+u_1. 760 00:47:19 --> 00:47:20 I can find u_3. 761 00:47:21 --> 00:47:24 So I can get those, I can go forever. 762 00:47:24 --> 00:47:28 If you give me enough to start on, two things to start, 763 00:47:28 --> 00:47:32 then I march forward. 764 00:47:32 --> 00:47:37 Whereas in our problems, we've only got one thing to start on 765 00:47:37 --> 00:47:39 and we've got one goal to hit. 766 00:47:39 --> 00:47:44 And that's why we have to solve the whole system together. 767 00:47:44 --> 00:47:48 This is, we can solve it step-by-step. 768 00:47:48 --> 00:47:51 This is way faster of course. 769 00:47:51 --> 00:47:57 To be able to just go forward in time. 770 00:47:57 --> 00:48:01 I'll mention that the topic of initial value problems and 771 00:48:01 --> 00:48:07 finite differences for them, we can't get to that. 772 00:48:07 --> 00:48:13 So we're seeing a little bit here, but it's done properly in 773 00:48:13 --> 00:48:18 18.086 in the second semester is the initial value 774 00:48:18 --> 00:48:20 problem start part. 775 00:48:20 --> 00:48:27 And that has it's own interesting questions. 776 00:48:27 --> 00:48:33 Somehow we've talked about fourth order equations, 777 00:48:33 --> 00:48:39 initial value problems. 778 00:48:39 --> 00:48:42 But no homework problems. 779 00:48:42 --> 00:48:46 So I'm ready for, or even related. 780 00:48:46 --> 00:48:51 But that's fine with me. 781 00:48:51 --> 00:48:53 Is there a question? 782 00:48:53 --> 00:48:54 Yeah, thanks. 783 00:48:54 --> 00:48:56 Or it doesn't have to be a homework question, 784 00:48:56 --> 00:49:05 another question. 785 00:49:05 --> 00:49:09 Oh, good question. 786 00:49:09 --> 00:49:11 You mean I should just send the homework out to 787 00:49:11 --> 00:49:13 Natick where MATLAB is. 788 00:49:13 --> 00:49:16 Do you know that MATLAB is just 15 miles away? 789 00:49:16 --> 00:49:22 I almost get there, I live 2/3 of the way there. 790 00:49:22 --> 00:49:24 Yeah, so we could just send the whole thing out 791 00:49:24 --> 00:49:27 there and get it back. 792 00:49:27 --> 00:49:36 That would save a lot of work. 793 00:49:36 --> 00:49:38 I suppose, I'm ok. 794 00:49:38 --> 00:49:40 Why should I say no? 795 00:49:40 --> 00:49:42 Anything MATLAB can do and you can make it 796 00:49:42 --> 00:49:45 do, I'm ok with that. 797 00:49:45 --> 00:49:48 I don't see that you have to do things by hand if 798 00:49:48 --> 00:49:50 you've got a better way. 799 00:49:50 --> 00:49:50 That's ok. 800 00:49:50 --> 00:49:53 And then probably the answer gets printed 801 00:49:53 --> 00:49:55 and you can graph it. 802 00:49:55 --> 00:49:57 So that's fine. 803 00:49:57 --> 00:50:00 So I mean, somehow a course like this has 804 00:50:00 --> 00:50:03 got two parts to it. 805 00:50:03 --> 00:50:05 Applied math has two parts to it. 806 00:50:05 --> 00:50:08 The modeling part, set up the equation, think, what is 807 00:50:08 --> 00:50:10 it you're supposed to do. 808 00:50:10 --> 00:50:13 And then, step two is do it. 809 00:50:13 --> 00:50:15 The numerical part, the computing part. 810 00:50:15 --> 00:50:19 And that's where MATLAB, Python, Fortran, whatever, 811 00:50:19 --> 00:50:25 is going to do a lot of the heavy lifting. 812 00:50:25 --> 00:50:27 Was there another question? 813 00:50:27 --> 00:50:35 So that first homework was certainly very general 814 00:50:35 --> 00:50:35 intentionally. 815 00:50:35 --> 00:50:40 Because I'm hoping you will read the book. 816 00:50:40 --> 00:50:47 The lecture, you'll be able to match the lectures with the 817 00:50:47 --> 00:50:51 book even later on when they separate a little 818 00:50:51 --> 00:50:54 or separate more. 819 00:50:54 --> 00:50:55 You'll see what we're doing. 820 00:50:55 --> 00:50:59 And those, the homework problems, you should look at 821 00:50:59 --> 00:51:01 some of the others just to see. 822 00:51:01 --> 00:51:02 Do I know how to do that? 823 00:51:02 --> 00:51:05 Right. 824 00:51:05 --> 00:51:08 Let's stop here for this first review. 825 00:51:08 --> 00:51:10 I'm sure we'll have more, questions will build up 826 00:51:10 --> 00:51:12 for the second week.