1 00:00:00,000 --> 00:00:00,499 2 00:00:00,499 --> 00:00:02,944 The following content is provided under a Creative 3 00:00:02,944 --> 00:00:03,610 Commons license. 4 00:00:03,610 --> 00:00:05,770 Your support will help MIT OpenCourseWare 5 00:00:05,770 --> 00:00:09,930 continue to offer high quality educational resources for free. 6 00:00:09,930 --> 00:00:12,590 To make a donation, or to view additional materials 7 00:00:12,590 --> 00:00:16,150 from hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:16,150 --> 00:00:20,160 at ocw.mit.edu. 9 00:00:20,160 --> 00:00:24,940 PROFESSOR STRANG: So this is the first true review 10 00:00:24,940 --> 00:00:27,860 session in 18.085. 11 00:00:27,860 --> 00:00:31,820 The last Wednesday, the first Wednesday afternoon, 12 00:00:31,820 --> 00:00:38,450 was a brief review of topics in linear algebra. 13 00:00:38,450 --> 00:00:40,720 But now we're into the course. 14 00:00:40,720 --> 00:00:44,500 We've done four lectures on the first four sections 15 00:00:44,500 --> 00:00:50,210 of the textbook and one homework problem in and back 16 00:00:50,210 --> 00:00:54,530 and a second homework set for next Monday. 17 00:00:54,530 --> 00:00:58,490 So I'm ready for any questions, including questions that are 18 00:00:58,490 --> 00:01:00,090 on the homework if necessary. 19 00:01:00,090 --> 00:01:03,020 But anything at all. 20 00:01:03,020 --> 00:01:08,270 Or maybe I'll just ask whether the pace, 21 00:01:08,270 --> 00:01:09,740 so this is really informal. 22 00:01:09,740 --> 00:01:13,580 Is the pace of the course, now today's lecture had a lot in it 23 00:01:13,580 --> 00:01:16,490 as I realized when I saw that the board was still 24 00:01:16,490 --> 00:01:21,330 full of 18.085 and there was a little more 25 00:01:21,330 --> 00:01:26,050 still to do because we didn't finish the matrix part. 26 00:01:26,050 --> 00:01:33,000 But are you okay with the sort of speed of the course? 27 00:01:33,000 --> 00:01:34,890 So that'll be one question. 28 00:01:34,890 --> 00:01:39,770 And then, what about specifics? 29 00:01:39,770 --> 00:01:42,461 Somebody start off if you will. 30 00:01:42,461 --> 00:01:42,960 Anybody. 31 00:01:42,960 --> 00:01:43,620 Yes, thanks. 32 00:01:43,620 --> 00:01:46,617 AUDIENCE: Well, actually I had a question 33 00:01:46,617 --> 00:01:47,950 about the lecture earlier today. 34 00:01:47,950 --> 00:01:49,325 PROFESSOR STRANG: Okay, go ahead. 35 00:01:49,325 --> 00:01:51,760 AUDIENCE: I was just going to look it up in the text, but. 36 00:01:51,760 --> 00:01:53,135 PROFESSOR STRANG: That's alright. 37 00:01:53,135 --> 00:01:54,970 AUDIENCE: But haven't had a chance to. 38 00:01:54,970 --> 00:01:57,260 But, okay, so I'm not really sure 39 00:01:57,260 --> 00:01:58,950 how you take the initial conditions 40 00:01:58,950 --> 00:02:01,300 and apply them to the ramp function 41 00:02:01,300 --> 00:02:04,070 to actually get a solution. 42 00:02:04,070 --> 00:02:09,720 PROFESSOR STRANG: So that's what luckily happens to be still 43 00:02:09,720 --> 00:02:11,620 here on the board, right? 44 00:02:11,620 --> 00:02:14,450 We've got these boundary-- I would call them 45 00:02:14,450 --> 00:02:16,160 boundary conditions. 46 00:02:16,160 --> 00:02:18,940 So this is definitely, we're in a part of math 47 00:02:18,940 --> 00:02:21,400 that's about boundary value problems 48 00:02:21,400 --> 00:02:24,470 more than-- time isn't in the picture. 49 00:02:24,470 --> 00:02:28,350 I mean later time will get into the picture. 50 00:02:28,350 --> 00:02:37,540 So with this particular example, the general solution 51 00:02:37,540 --> 00:02:44,540 is a standard ramp at the point a where things are happening. 52 00:02:44,540 --> 00:02:47,360 Plus the C+Dx, the usual. 53 00:02:47,360 --> 00:02:50,270 So that's a particular solution. 54 00:02:50,270 --> 00:02:59,060 That particular solution has the right behavior at the impulse. 55 00:02:59,060 --> 00:03:03,940 By the right behavior, I mean that the second derivative 56 00:03:03,940 --> 00:03:10,500 of the ramp is a delta and when I put the ramp at a, 57 00:03:10,500 --> 00:03:12,760 then the delta will show up at a. 58 00:03:12,760 --> 00:03:14,850 And when I put on that minus sign 59 00:03:14,850 --> 00:03:19,950 it'll mean that the second derivative is minus the delta. 60 00:03:19,950 --> 00:03:23,140 So the slope will step down. 61 00:03:23,140 --> 00:03:24,940 So that's a particular solution. 62 00:03:24,940 --> 00:03:31,220 But that by itself, what does that equal at zero? 63 00:03:31,220 --> 00:03:36,420 And what does that equal at one if you remember the ramp? 64 00:03:36,420 --> 00:03:38,770 So let me just draw that ramp again. 65 00:03:38,770 --> 00:03:44,680 So the ramp was really based on, centered at the point a. 66 00:03:44,680 --> 00:03:46,760 And I'll put it with a minus sign. 67 00:03:46,760 --> 00:03:50,120 So it came along there and down there. 68 00:03:50,120 --> 00:03:53,330 And now suppose this is one end of our interval 69 00:03:53,330 --> 00:03:56,480 and that's the other end. 70 00:03:56,480 --> 00:03:59,420 So is that ramp the answer to our problem? 71 00:03:59,420 --> 00:04:00,950 No. 72 00:04:00,950 --> 00:04:05,730 Well it happens to satisfy this boundary condition, 73 00:04:05,730 --> 00:04:08,130 happens to start out at zero. 74 00:04:08,130 --> 00:04:10,710 But it doesn't end up at zero. 75 00:04:10,710 --> 00:04:19,150 So just like every particular solution we need a little more. 76 00:04:19,150 --> 00:04:22,630 We have to include more solutions 77 00:04:22,630 --> 00:04:24,010 because that was only one. 78 00:04:24,010 --> 00:04:26,100 All those are equally good solutions. 79 00:04:26,100 --> 00:04:31,050 If I add C+Dx, the second derivative of that is zero, 80 00:04:31,050 --> 00:04:33,100 so it doesn't spoil anything at all. 81 00:04:33,100 --> 00:04:38,750 On the contrary, it adds more solutions. 82 00:04:38,750 --> 00:04:41,380 I mean the great thing we're using here 83 00:04:41,380 --> 00:04:47,630 is that our equations are linear and when 84 00:04:47,630 --> 00:04:49,560 zero's on the right-hand side-- notice 85 00:04:49,560 --> 00:04:51,300 there's no arbitrary constant. 86 00:04:51,300 --> 00:04:53,220 I'm not putting an arbitrary constant 87 00:04:53,220 --> 00:04:55,320 on that particular solution. 88 00:04:55,320 --> 00:05:00,110 Is one particular solution plus a 89 00:05:00,110 --> 00:05:02,890 subspace if I use that language. 90 00:05:02,890 --> 00:05:07,980 A lot of solutions to the-- All the solutions 91 00:05:07,980 --> 00:05:10,840 to the problem with zero on the right-hand side. 92 00:05:10,840 --> 00:05:13,230 And these are the solutions, these 93 00:05:13,230 --> 00:05:17,350 are the null space guys, the ones that have zero 94 00:05:17,350 --> 00:05:19,410 on the right-hand side. 95 00:05:19,410 --> 00:05:22,410 Now we need those. 96 00:05:22,410 --> 00:05:27,190 And the effect of those will be to move that ramp. 97 00:05:27,190 --> 00:05:29,880 And effectively, what are they going to do? 98 00:05:29,880 --> 00:05:33,620 They're going to swing that ramp around, it'll stay a ramp. 99 00:05:33,620 --> 00:05:36,750 But instead of being level here, it'll go up. 100 00:05:36,750 --> 00:05:41,810 And instead of going down there, it'll go down there. 101 00:05:41,810 --> 00:05:44,020 It'll be the same ramp, with still 102 00:05:44,020 --> 00:05:46,810 that slope dropped by one. 103 00:05:46,810 --> 00:05:52,460 Slope went down by minus one and that's not 104 00:05:52,460 --> 00:05:53,920 going to change here. 105 00:05:53,920 --> 00:05:59,380 The slope will still go down by minus one. 106 00:05:59,380 --> 00:06:03,450 But now I've just adjusted it so that it goes through 107 00:06:03,450 --> 00:06:06,090 the-- it satisfies the boundary conditions. 108 00:06:06,090 --> 00:06:11,690 And in the second problem with the free end, 109 00:06:11,690 --> 00:06:14,630 again, here's my ramp. 110 00:06:14,630 --> 00:06:16,240 But now I'm going to adjust it. 111 00:06:16,240 --> 00:06:17,410 And what happened? 112 00:06:17,410 --> 00:06:20,470 It just needed adjustment upwards. 113 00:06:20,470 --> 00:06:26,890 Because this was the zero, this was the u=0 fixed guy. 114 00:06:26,890 --> 00:06:31,740 And now if I'm doing u'(0)=0, the free guy, 115 00:06:31,740 --> 00:06:33,320 I can lift the whole thing up. 116 00:06:33,320 --> 00:06:36,290 So you see, I just lifted it up to the point 117 00:06:36,290 --> 00:06:39,390 where it came out right there. 118 00:06:39,390 --> 00:06:44,200 So in this case I just needed a C. And in this case 119 00:06:44,200 --> 00:06:46,330 I just needed a Dx. 120 00:06:46,330 --> 00:06:49,170 And in other cases I might have needed both. 121 00:06:49,170 --> 00:06:51,910 A little bit of C and a little of Dx. 122 00:06:51,910 --> 00:06:58,000 Anyway that's what was-- maybe that's sort of repeating 123 00:06:58,000 --> 00:07:02,270 what we did today a little bit. 124 00:07:02,270 --> 00:07:08,670 But mechanically it's just, we've got a particular solution 125 00:07:08,670 --> 00:07:12,790 and we've got the complete solution 126 00:07:12,790 --> 00:07:15,840 and then we just have to choose C and D 127 00:07:15,840 --> 00:07:18,330 and we've got two conditions to do it with, 128 00:07:18,330 --> 00:07:20,790 the two boundary conditions. 129 00:07:20,790 --> 00:07:22,470 Could have other boundary conditions. 130 00:07:22,470 --> 00:07:26,040 Now, what about, here's a, yeah. 131 00:07:26,040 --> 00:07:29,560 I guess what will often happen in these review sessions is 132 00:07:29,560 --> 00:07:33,410 I get on a roll and I just keep, carry on with it. 133 00:07:33,410 --> 00:07:37,210 You know, you start me with a question and I can't stop. 134 00:07:37,210 --> 00:07:42,890 So I'll go a little longer but then I really will stop, ready 135 00:07:42,890 --> 00:07:45,050 for the next one. 136 00:07:45,050 --> 00:07:50,360 So we haven't discussed the free-free. u(0) equal-- 137 00:07:50,360 --> 00:07:53,260 u', sorry, free is u'. 138 00:07:53,260 --> 00:08:00,710 That's free at the left and free at the right. 139 00:08:00,710 --> 00:08:04,580 What's up with that? 140 00:08:04,580 --> 00:08:06,070 What's the solution to that? 141 00:08:06,070 --> 00:08:13,490 Again, I'm looking for -u'' equal an impulse. 142 00:08:13,490 --> 00:08:16,920 With those two boundary conditions, free-free. 143 00:08:16,920 --> 00:08:19,510 And do you know what's going to happen? 144 00:08:19,510 --> 00:08:22,150 No solution. 145 00:08:22,150 --> 00:08:25,740 Now it'd be interesting to see why not. 146 00:08:25,740 --> 00:08:28,130 Why no solution? 147 00:08:28,130 --> 00:08:32,030 Well one way to do it is try. 148 00:08:32,030 --> 00:08:36,000 Other right-hand sides could have a solution. 149 00:08:36,000 --> 00:08:41,280 So it's not just that this is-- something's the matter here. 150 00:08:41,280 --> 00:08:44,940 Specifically, that right-hand side and most right-hand sides 151 00:08:44,940 --> 00:08:46,220 will fail. 152 00:08:46,220 --> 00:08:47,850 But let's just see it with this one. 153 00:08:47,850 --> 00:08:50,620 Why does that fail? 154 00:08:50,620 --> 00:08:56,880 Well you can see I can't, the slope here is zero 155 00:08:56,880 --> 00:08:59,570 and the slope here is minus one. 156 00:08:59,570 --> 00:09:04,540 If I adjust those I can't get, this 157 00:09:04,540 --> 00:09:07,440 is asking me to get slope zero at both ends. 158 00:09:07,440 --> 00:09:10,610 I can't do that, right? 159 00:09:10,610 --> 00:09:17,130 Yeah, just not possible for me to-- This will add a straight 160 00:09:17,130 --> 00:09:19,160 line but it's the same straight-- 161 00:09:19,160 --> 00:09:23,210 I can't get a ramp that comes flat at both ends 162 00:09:23,210 --> 00:09:27,050 because there's-- once I say what it's doing at the ends, 163 00:09:27,050 --> 00:09:28,160 I've got it. 164 00:09:28,160 --> 00:09:31,530 And you see what I mean? 165 00:09:31,530 --> 00:09:37,340 I'm looking for one that starts flat and ends flat 166 00:09:37,340 --> 00:09:45,620 and that's not in my family of solutions. 167 00:09:45,620 --> 00:09:49,480 That problem just doesn't have a solution. 168 00:09:49,480 --> 00:09:53,440 So that's one way to do it, is look at the solutions 169 00:09:53,440 --> 00:09:56,720 and realize you can't satisfy both boundary conditions. 170 00:09:56,720 --> 00:10:00,150 Another way might be this. 171 00:10:00,150 --> 00:10:06,250 This is a little bit deeper way and it leads to something 172 00:10:06,250 --> 00:10:07,540 better. 173 00:10:07,540 --> 00:10:10,540 Suppose I take the integral. 174 00:10:10,540 --> 00:10:13,360 So this is an equation that's supposed to hold. 175 00:10:13,360 --> 00:10:18,160 Let me integrate both sides from zero to one. 176 00:10:18,160 --> 00:10:21,040 So there's the idea I'm putting in now. 177 00:10:21,040 --> 00:10:23,380 To go a little further, to discover 178 00:10:23,380 --> 00:10:25,430 when this has a solution. 179 00:10:25,430 --> 00:10:28,080 Or let me take more generally, -u''=f(x). 180 00:10:28,080 --> 00:10:31,300 181 00:10:31,300 --> 00:10:33,240 So some other load. 182 00:10:33,240 --> 00:10:37,210 Not necessarily a point load, not necessarily a uniform load, 183 00:10:37,210 --> 00:10:39,030 but maybe some other load. 184 00:10:39,030 --> 00:10:44,780 And now my boundary conditions, I'm trying to do free-free. 185 00:10:44,780 --> 00:10:48,400 And usually no solution. 186 00:10:48,400 --> 00:10:52,540 But let's just see why and when there might be a solution. 187 00:10:52,540 --> 00:11:02,530 The key idea is integrate from zero to one. 188 00:11:02,530 --> 00:11:04,140 What do I get on the right-hand side? 189 00:11:04,140 --> 00:11:09,280 Well, I get the integral of f, whatever it is, 190 00:11:09,280 --> 00:11:15,260 and I would call that the total load. 191 00:11:15,260 --> 00:11:16,650 Fair enough? 192 00:11:16,650 --> 00:11:21,300 The total load of-- If it was a delta function, 193 00:11:21,300 --> 00:11:24,900 the total load would be one and it would be all in one spot. 194 00:11:24,900 --> 00:11:28,570 If it was uniformly over the whole interval, 195 00:11:28,570 --> 00:11:31,340 well I guess that would also integrate to one, 196 00:11:31,340 --> 00:11:33,900 so the total load would be one spread out. 197 00:11:33,900 --> 00:11:37,560 But it could be a mixture of the two, could be a few delta 198 00:11:37,560 --> 00:11:39,530 functions, whatever. 199 00:11:39,530 --> 00:11:45,070 What happens when I integrate the left side from zero to one? 200 00:11:45,070 --> 00:11:46,510 Can you do that one? 201 00:11:46,510 --> 00:11:47,980 The integral from zero to one. 202 00:11:47,980 --> 00:11:52,290 There's that dopey minus sign. u''dx. 203 00:11:52,290 --> 00:11:55,220 What do I get? 204 00:11:55,220 --> 00:11:59,190 Why do I say that's a good idea? 205 00:11:59,190 --> 00:12:02,370 If I integrate the second derivative 206 00:12:02,370 --> 00:12:05,549 I get the first derivative. 207 00:12:05,549 --> 00:12:07,090 The integral of the second derivative 208 00:12:07,090 --> 00:12:11,020 will be the first derivative with the minus. 209 00:12:11,020 --> 00:12:13,740 So it's minus the first derivative. 210 00:12:13,740 --> 00:12:16,350 And what do I do now? 211 00:12:16,350 --> 00:12:18,970 I plug in the end points, right? 212 00:12:18,970 --> 00:12:19,640 You integrate. 213 00:12:19,640 --> 00:12:20,990 I'm integrating zero to one. 214 00:12:20,990 --> 00:12:22,650 So I've found the integral. 215 00:12:22,650 --> 00:12:26,100 I've put zero to one in there. 216 00:12:26,100 --> 00:12:27,770 So what is that? 217 00:12:27,770 --> 00:12:33,210 That's minus the derivative at one 218 00:12:33,210 --> 00:12:40,820 plus the derivative at zero. 219 00:12:40,820 --> 00:12:42,870 And now, what's that? 220 00:12:42,870 --> 00:12:44,010 That's zero. 221 00:12:44,010 --> 00:12:47,610 By my boundary conditions, that's zero. 222 00:12:47,610 --> 00:12:49,890 So what have I found? 223 00:12:49,890 --> 00:12:52,980 I've found that if these are the boundary conditions, then when 224 00:12:52,980 --> 00:12:57,120 I integrate the left side it's going to give me zero. 225 00:12:57,120 --> 00:13:03,740 So when, what loads could be okay? 226 00:13:03,740 --> 00:13:14,480 What loads f(x) could allow me to solve this equation? 227 00:13:14,480 --> 00:13:18,550 The condition will be I need, what do I 228 00:13:18,550 --> 00:13:24,260 need for the total load to be able to solve this equation? 229 00:13:24,260 --> 00:13:27,500 The integral of the left side was zero 230 00:13:27,500 --> 00:13:31,710 so the integral of the right side had better be zero. 231 00:13:31,710 --> 00:13:34,740 So that's the condition. 232 00:13:34,740 --> 00:13:36,590 If I have these boundary conditions, 233 00:13:36,590 --> 00:13:38,430 then my problem is singular. 234 00:13:38,430 --> 00:13:40,090 Usually no solution. 235 00:13:40,090 --> 00:13:42,580 It's like having a singular matrix. 236 00:13:42,580 --> 00:13:46,920 It's like having this particular singular matrix, of course. 237 00:13:46,920 --> 00:13:48,530 Whoops, not that one. 238 00:13:48,530 --> 00:13:51,700 Let me get the plus sign in the right position. 239 00:13:51,700 --> 00:13:53,330 That's a plus. 240 00:13:53,330 --> 00:13:57,400 -1; -1, 2, -1; -1, 1. 241 00:13:57,400 --> 00:13:59,380 Right? 242 00:13:59,380 --> 00:14:08,080 This is the discrete version with a zero slope at both ends. 243 00:14:08,080 --> 00:14:10,540 It's our T matrix. 244 00:14:10,540 --> 00:14:12,880 No, what matrix is it? 245 00:14:12,880 --> 00:14:18,880 B. It's our B matrix, both ends. 246 00:14:18,880 --> 00:14:22,120 I'll just come back here and then I'll do the discrete one. 247 00:14:22,120 --> 00:14:26,930 So tell me a load that we could handle? 248 00:14:26,930 --> 00:14:30,060 A load we could handle. 249 00:14:30,060 --> 00:14:32,700 So the integral has to be zero. 250 00:14:32,700 --> 00:14:39,640 So suppose my load has a delta function at a. 251 00:14:39,640 --> 00:14:42,030 Well that integral is one. 252 00:14:42,030 --> 00:14:45,860 So can you fix that, change that load 253 00:14:45,860 --> 00:14:49,590 or do something, maybe put on another load 254 00:14:49,590 --> 00:14:53,770 to get a total load of zero? 255 00:14:53,770 --> 00:14:55,980 What shall I do? 256 00:14:55,980 --> 00:14:59,650 Add another guy with a minus sign. 257 00:14:59,650 --> 00:15:04,710 In other words, maybe this, a delta 258 00:15:04,710 --> 00:15:07,380 function at some other point b. 259 00:15:07,380 --> 00:15:08,890 Well, that would do it. 260 00:15:08,890 --> 00:15:11,190 I believe we could solve that problem. 261 00:15:11,190 --> 00:15:18,470 Even with these bad boundary conditions. 262 00:15:18,470 --> 00:15:20,170 We could solve that problem. 263 00:15:20,170 --> 00:15:23,530 Because the total load would be, one from that, minus one 264 00:15:23,530 --> 00:15:27,290 from that, the total load would be zero. 265 00:15:27,290 --> 00:15:30,290 In other words, what would are solution look like? 266 00:15:30,290 --> 00:15:32,810 It has to start with zero slope. 267 00:15:32,810 --> 00:15:37,520 So it would buzz alone to a and then after b. 268 00:15:37,520 --> 00:15:40,890 And what does it have to do here? 269 00:15:40,890 --> 00:15:44,050 If I graph the solution to this guy 270 00:15:44,050 --> 00:15:48,570 from zero to one it starts with, it's 271 00:15:48,570 --> 00:15:51,480 free, so nothing's happening until I get to a. 272 00:15:51,480 --> 00:15:55,940 Then what has to happen? 273 00:15:55,940 --> 00:16:00,980 Let's see, if I'm graphing u, it'll be ramp. 274 00:16:00,980 --> 00:16:01,480 Right? 275 00:16:01,480 --> 00:16:02,690 It'll be a ramp, yeah. 276 00:16:02,690 --> 00:16:04,320 Because I've two derivatives. 277 00:16:04,320 --> 00:16:11,110 And it has to ramp down by one, so it'll ramp whatever it does. 278 00:16:11,110 --> 00:16:15,540 I don't know where it stops. 279 00:16:15,540 --> 00:16:16,190 Where does it? 280 00:16:16,190 --> 00:16:18,150 Wait a minute. 281 00:16:18,150 --> 00:16:21,150 I haven't practiced this. 282 00:16:21,150 --> 00:16:26,000 So I start from the other end. 283 00:16:26,000 --> 00:16:31,120 The other end is flat. 284 00:16:31,120 --> 00:16:33,580 What's up? 285 00:16:33,580 --> 00:16:36,830 They gotta meet here. 286 00:16:36,830 --> 00:16:38,790 Oh, the other end is flat, but not at zero! 287 00:16:38,790 --> 00:16:39,780 Dumb, stupid. 288 00:16:39,780 --> 00:16:40,280 Right. 289 00:16:40,280 --> 00:16:41,040 Okay. 290 00:16:41,040 --> 00:16:43,340 Yes, the other end is flat, right. 291 00:16:43,340 --> 00:16:45,050 And it's, oh yeah, look! 292 00:16:45,050 --> 00:16:46,500 Oh, wonderful. 293 00:16:46,500 --> 00:16:47,850 You see. 294 00:16:47,850 --> 00:16:52,720 That slope dropped by one and the slope there 295 00:16:52,720 --> 00:16:55,130 increased by one back to zero. 296 00:16:55,130 --> 00:16:56,490 Slope was zero. 297 00:16:56,490 --> 00:17:00,460 It dropped to minus one because of that load. 298 00:17:00,460 --> 00:17:03,030 Now it increased back to zero because 299 00:17:03,030 --> 00:17:07,080 of this load with the minus sign and there's a solution. 300 00:17:07,080 --> 00:17:10,020 So that's a solvable problem. 301 00:17:10,020 --> 00:17:12,330 Well, you say, okay, that was a little surprising 302 00:17:12,330 --> 00:17:15,020 to get an answer for a singular problem. 303 00:17:15,020 --> 00:17:17,990 And no, it can happen. 304 00:17:17,990 --> 00:17:21,850 If we have a total load zero it'll happen. 305 00:17:21,850 --> 00:17:28,900 But, there is still a but, that's not the only answer. 306 00:17:28,900 --> 00:17:31,780 That picture is a solution. 307 00:17:31,780 --> 00:17:34,310 But not the only one. 308 00:17:34,310 --> 00:17:37,090 So what my point is going to be, that when 309 00:17:37,090 --> 00:17:40,140 the problem is singular, if there's an answer, 310 00:17:40,140 --> 00:17:41,390 you say great. 311 00:17:41,390 --> 00:17:44,710 But then something has to go wrong and what goes wrong 312 00:17:44,710 --> 00:17:46,140 is too many answers. 313 00:17:46,140 --> 00:17:54,440 So tell me some more, what other graphs would draw solutions 314 00:17:54,440 --> 00:18:00,350 to this problem. 315 00:18:00,350 --> 00:18:04,430 I could shift, I could lift the whole thing. 316 00:18:04,430 --> 00:18:07,430 Here I've got a plus C that I haven't used. 317 00:18:07,430 --> 00:18:10,480 I could just do the whole thing higher up. 318 00:18:10,480 --> 00:18:11,260 Any of these. 319 00:18:11,260 --> 00:18:17,160 These would all work. 320 00:18:17,160 --> 00:18:18,470 It's like temperature. 321 00:18:18,470 --> 00:18:23,500 I don't have an absolute temperature here. 322 00:18:23,500 --> 00:18:29,470 All I've got is, I would have to-- It's not determined 323 00:18:29,470 --> 00:18:35,800 because there's a plus C that-- the plus C satisfied 324 00:18:35,800 --> 00:18:37,050 everything. 325 00:18:37,050 --> 00:18:41,270 A plus C, a constant has zero slope, it has zero slope, 326 00:18:41,270 --> 00:18:42,750 its second derivative is zero. 327 00:18:42,750 --> 00:18:48,330 So it's like, unseen by this equation. 328 00:18:48,330 --> 00:18:50,890 And similarly can I just make the analogy 329 00:18:50,890 --> 00:18:55,440 as I always like to do with discrete stuff, so suppose I-- 330 00:18:55,440 --> 00:19:02,100 Tell me a right-hand side that we think would probably, 331 00:19:02,100 --> 00:19:05,320 is this going to be the same story for this guy? 332 00:19:05,320 --> 00:19:06,250 Yes. 333 00:19:06,250 --> 00:19:10,750 If I add those, where I integrated there, 334 00:19:10,750 --> 00:19:13,940 here I would add and I get zero, zero, zero. 335 00:19:13,940 --> 00:19:18,740 So this has to add to zero if there's a solution. 336 00:19:18,740 --> 00:19:22,270 So let me put for example, [0, 1, -1]. 337 00:19:22,270 --> 00:19:26,800 That would be kind of like our delta function in one direction 338 00:19:26,800 --> 00:19:28,550 and our delta function in the other. 339 00:19:28,550 --> 00:19:31,860 I believe I can solve that problem. 340 00:19:31,860 --> 00:19:34,540 So I'm just carrying, because I always 341 00:19:34,540 --> 00:19:39,970 want you to see the discrete one as well as the continuous. 342 00:19:39,970 --> 00:19:43,430 Continuous involved this integration. 343 00:19:43,430 --> 00:19:47,310 The finite one just involves adding. 344 00:19:47,310 --> 00:19:50,870 The left side adds to zero so the right-hand side better 345 00:19:50,870 --> 00:19:52,140 add to zero. 346 00:19:52,140 --> 00:19:53,540 That right-hand side does. 347 00:19:53,540 --> 00:19:54,770 Tell me a solution. 348 00:19:54,770 --> 00:19:57,620 Well, let me start out with a seven there. 349 00:19:57,620 --> 00:20:04,250 What's the next guy going to be? 350 00:20:04,250 --> 00:20:05,880 See, I want seven. 351 00:20:05,880 --> 00:20:10,540 Whatever I put there, I better have a seven there, right? 352 00:20:10,540 --> 00:20:12,680 Seven, seven, good. 353 00:20:12,680 --> 00:20:16,150 Minus seven, plus 14, oh geez. 354 00:20:16,150 --> 00:20:22,180 I didn't know this was going to happen. 355 00:20:22,180 --> 00:20:26,160 No, I want to get the answer one. 356 00:20:26,160 --> 00:20:28,250 What number goes there? 357 00:20:28,250 --> 00:20:29,940 Six, is it six? 358 00:20:29,940 --> 00:20:31,780 It's six, yeah, good. 359 00:20:31,780 --> 00:20:34,610 Minus seven, 14, minus six is that. 360 00:20:34,610 --> 00:20:36,540 And now my claim is that we'll come out 361 00:20:36,540 --> 00:20:38,440 right on the third equation. 362 00:20:38,440 --> 00:20:40,920 So far I've just matched the first two. 363 00:20:40,920 --> 00:20:43,780 Now this one gives minus seven, plus six, 364 00:20:43,780 --> 00:20:45,750 that's minus one, good. 365 00:20:45,750 --> 00:20:48,780 So there's a solution. 366 00:20:48,780 --> 00:20:51,310 And I'll leave this problem alone 367 00:20:51,310 --> 00:20:54,670 if you tell me the rest, other solutions. 368 00:20:54,670 --> 00:20:59,670 That [7, 7, 6] was a solution to a singular problem 369 00:20:59,670 --> 00:21:02,690 with a right-hand side that had total load zero, 370 00:21:02,690 --> 00:21:04,380 so it was okay. 371 00:21:04,380 --> 00:21:08,530 But now that's a solution, but there are more. 372 00:21:08,530 --> 00:21:13,550 Tell me another one. 373 00:21:13,550 --> 00:21:15,720 I can shift it, right? 374 00:21:15,720 --> 00:21:18,330 I could make it [17, 17, 16]. 375 00:21:18,330 --> 00:21:21,091 I could add ten to everything. 376 00:21:21,091 --> 00:21:21,590 Right? 377 00:21:21,590 --> 00:21:24,830 That's the plus C that I could do over there. 378 00:21:24,830 --> 00:21:30,590 That can't change because 17 - 17 is still zero. 379 00:21:30,590 --> 00:21:33,400 -17 + 16 will still be minus one. 380 00:21:33,400 --> 00:21:35,370 All good. 381 00:21:35,370 --> 00:21:39,280 So actually that just like helps our intuition, 382 00:21:39,280 --> 00:21:42,170 and physically my intuition is this. 383 00:21:42,170 --> 00:21:49,180 That I've got this bar and nothing's holding it. 384 00:21:49,180 --> 00:21:57,210 So if I put a weight on it, nothing to hold it, it'll just, 385 00:21:57,210 --> 00:22:00,030 rigid motion will take it out of sight, no good. 386 00:22:00,030 --> 00:22:02,380 But if I put another equal weight on it, 387 00:22:02,380 --> 00:22:03,470 no it's not a weight. 388 00:22:03,470 --> 00:22:04,220 What do I call it? 389 00:22:04,220 --> 00:22:07,870 If I lift it at that point, that's 390 00:22:07,870 --> 00:22:10,940 the other delta function that's going the other way, 391 00:22:10,940 --> 00:22:13,640 then it will sit there. 392 00:22:13,640 --> 00:22:17,310 But it would still be in equilibrium 393 00:22:17,310 --> 00:22:21,310 if I just moved it up to there or moved it as I like. 394 00:22:21,310 --> 00:22:26,690 I don't know if that is kind of a dumb picture. 395 00:22:26,690 --> 00:22:31,780 But it's saying what we've said from math. 396 00:22:31,780 --> 00:22:33,940 Well, you see where your question led. 397 00:22:33,940 --> 00:22:44,440 Yeah, thanks. 398 00:22:44,440 --> 00:22:49,300 No, the integral, it was-- Watch what we integrated. 399 00:22:49,300 --> 00:22:50,460 We integrated u''. 400 00:22:50,460 --> 00:22:53,280 401 00:22:53,280 --> 00:22:55,940 So that's not the area. 402 00:22:55,940 --> 00:22:59,580 We integrated u'' and got its integral was u'. 403 00:22:59,580 --> 00:23:02,150 So that just told us that a difference 404 00:23:02,150 --> 00:23:05,270 in slopes at the ends, yeah. 405 00:23:05,270 --> 00:23:08,904 Good, because our intuition automatically 406 00:23:08,904 --> 00:23:10,320 is if we're integrating something, 407 00:23:10,320 --> 00:23:11,740 we're finding an area. 408 00:23:11,740 --> 00:23:16,210 But here, if it was u , then I'd be finding the area under u, 409 00:23:16,210 --> 00:23:19,910 but we integrated the second derivative. 410 00:23:19,910 --> 00:23:23,420 Right, good. 411 00:23:23,420 --> 00:23:24,810 Now let's change the subject. 412 00:23:24,810 --> 00:23:33,520 Yes, please. 413 00:23:33,520 --> 00:23:34,920 Yeah, I guess so. 414 00:23:34,920 --> 00:23:44,090 I'll try. 415 00:23:44,090 --> 00:23:45,100 Let's see. 416 00:23:45,100 --> 00:23:53,080 So my discrete equation was like -u... 417 00:23:53,080 --> 00:23:57,160 Yeah, so let's back up to the beginning. 418 00:23:57,160 --> 00:23:59,510 We've got this minus sign and we're 419 00:23:59,510 --> 00:24:00,720 using a second difference. 420 00:24:00,720 --> 00:24:02,880 So second differences have coefficients 421 00:24:02,880 --> 00:24:06,360 one, minus two and one. 422 00:24:06,360 --> 00:24:09,120 Now I'm reversing the signs because of my minus. 423 00:24:09,120 --> 00:24:13,650 So I have -u at some point. 424 00:24:13,650 --> 00:24:17,020 Let's take that as the point to the left. 425 00:24:17,020 --> 00:24:22,000 Two u's at what I'll think of as the center point, 426 00:24:22,000 --> 00:24:24,160 minus u_(i-1). 427 00:24:24,160 --> 00:24:29,680 Is the load at that center. 428 00:24:29,680 --> 00:24:37,180 That center point is i times delta x. 429 00:24:37,180 --> 00:24:38,820 That's where I would be looking. 430 00:24:38,820 --> 00:24:44,320 So now I'm using subscript. 431 00:24:44,320 --> 00:24:50,190 It's a little bit of practice then to take subscripts, take 432 00:24:50,190 --> 00:24:57,090 this way of writing the equation and convert it to a matrix way. 433 00:24:57,090 --> 00:25:01,010 It's usually clearer once you see it as a matrix. 434 00:25:01,010 --> 00:25:04,850 Now this is happening at all the points. 435 00:25:04,850 --> 00:25:24,610 At i=1, let's say, I have -u_0+2u_1-u_2 is f at, agrees 436 00:25:24,610 --> 00:25:30,010 with the load at the first mesh point. 437 00:25:30,010 --> 00:25:36,060 That's the center, the point h delta x. 438 00:25:36,060 --> 00:25:42,330 And then if I want to back up further, I would have -u_(-1), 439 00:25:42,330 --> 00:25:49,860 but that doesn't really exist, plus 2u_0-u_1 should match 440 00:25:49,860 --> 00:25:55,750 the load at zero. 441 00:25:55,750 --> 00:25:57,980 And so on forward. 442 00:25:57,980 --> 00:26:00,299 But now I want to put in the boundary condition. 443 00:26:00,299 --> 00:26:01,840 That's what you want me to do, right? 444 00:26:01,840 --> 00:26:03,840 Put in this boundary condition. 445 00:26:03,840 --> 00:26:11,720 So what am I going to take as boundary condition? 446 00:26:11,720 --> 00:26:14,000 It has to be some approximation to u'(0)=0. 447 00:26:14,000 --> 00:26:18,320 448 00:26:18,320 --> 00:26:20,510 Maybe I'm never going to get to minus one. 449 00:26:20,510 --> 00:26:33,870 Maybe I don't need minus one. 450 00:26:33,870 --> 00:26:38,020 That's right, yeah, exactly. 451 00:26:38,020 --> 00:26:41,530 We did. 452 00:26:41,530 --> 00:26:42,790 That's what we knew about it. 453 00:26:42,790 --> 00:26:45,220 Sending it forward, we knew about forward difference, 454 00:26:45,220 --> 00:26:51,360 so I chose to do it. 455 00:26:51,360 --> 00:26:53,700 But then I think better of it. 456 00:26:53,700 --> 00:26:57,650 I chose to do it because it made the point that we, that 457 00:26:57,650 --> 00:27:02,140 at that boundary we were introducing a higher order 458 00:27:02,140 --> 00:27:08,020 error, first order error that's going to wreck things. 459 00:27:08,020 --> 00:27:13,350 I mean, it's going to spoil the, this is second order accuracy. 460 00:27:13,350 --> 00:27:17,520 And, but let me do that first order. 461 00:27:17,520 --> 00:27:19,480 So what shall I take? 462 00:27:19,480 --> 00:27:22,400 I'm going to approximate that by u-- Shall 463 00:27:22,400 --> 00:27:33,260 I take this one as I did in class? 464 00:27:33,260 --> 00:27:35,850 Yes. 465 00:27:35,850 --> 00:27:39,560 Ah, plus one, thanks, plus one, right, thank you. 466 00:27:39,560 --> 00:27:42,370 Thank you, good. 467 00:27:42,370 --> 00:27:44,940 Okeydoke. 468 00:27:44,940 --> 00:27:47,200 Alright. 469 00:27:47,200 --> 00:27:50,000 This is how we got to that equation. 470 00:27:50,000 --> 00:27:53,390 If I now bring in this boundary condition-- 471 00:27:53,390 --> 00:27:57,110 I guess I don't have to, let me take your eye off 472 00:27:57,110 --> 00:28:02,530 of that guy for the moment, I think. 473 00:28:02,530 --> 00:28:05,140 We're getting beautiful music here. 474 00:28:05,140 --> 00:28:07,210 Is it coming out of this box or? 475 00:28:07,210 --> 00:28:08,420 No. 476 00:28:08,420 --> 00:28:14,100 Anyway. 477 00:28:14,100 --> 00:28:17,730 So I'm going to use this boundary condition 478 00:28:17,730 --> 00:28:21,280 to say, well okay, if u_1 is u_0, 479 00:28:21,280 --> 00:28:23,800 I'm going to replace this u_0 by u_1. 480 00:28:23,800 --> 00:28:26,630 481 00:28:26,630 --> 00:28:28,680 This is the direct way. 482 00:28:28,680 --> 00:28:33,290 I replaced that u_0 by u_1 in that first equation. 483 00:28:33,290 --> 00:28:38,130 And then what I have is -u_1 and 2u_1, so that's the one 484 00:28:38,130 --> 00:28:40,660 and I have the -u_2. 485 00:28:40,660 --> 00:28:44,640 So do you see that that equation, when I put those 486 00:28:44,640 --> 00:28:53,650 together into a one, is going to-- If this is u_1, 487 00:28:53,650 --> 00:28:58,190 this is u_2, this is u_3 onwards, that first equation is 488 00:28:58,190 --> 00:29:03,600 u_1-u_2 and that's what I've got. 489 00:29:03,600 --> 00:29:05,190 This is u_1-u_2. 490 00:29:05,190 --> 00:29:09,070 491 00:29:09,070 --> 00:29:10,760 So I did it. 492 00:29:10,760 --> 00:29:12,150 I got to that matrix. 493 00:29:12,150 --> 00:29:15,530 The matrix is actually quite an important matrix. 494 00:29:15,530 --> 00:29:19,440 But from the point of view of accuracy 495 00:29:19,440 --> 00:29:22,120 in solving this differential equation, 496 00:29:22,120 --> 00:29:24,220 it's not the greatest. 497 00:29:24,220 --> 00:29:27,080 It's lost accuracy at that point. 498 00:29:27,080 --> 00:29:30,070 But the way to recover it turned out 499 00:29:30,070 --> 00:29:35,340 to be just a small adjustment at the boundary, so not a problem. 500 00:29:35,340 --> 00:29:35,880 Thanks. 501 00:29:35,880 --> 00:29:36,490 That's good. 502 00:29:36,490 --> 00:29:41,040 Yes, thanks. 503 00:29:41,040 --> 00:29:48,877 Sorry? 504 00:29:48,877 --> 00:29:49,960 When the boundary-- sorry. 505 00:29:49,960 --> 00:29:55,390 Two boundary conditions at the same point? 506 00:29:55,390 --> 00:29:57,070 That's a good question. 507 00:29:57,070 --> 00:29:59,510 So when would we have two? 508 00:29:59,510 --> 00:30:02,160 So instead of a boundary condition at zero 509 00:30:02,160 --> 00:30:05,120 and a boundary condition at one, you're putting them, 510 00:30:05,120 --> 00:30:06,200 is that what you mean? 511 00:30:06,200 --> 00:30:08,860 Put both boundary conditions at the end. 512 00:30:08,860 --> 00:30:10,000 Okay. 513 00:30:10,000 --> 00:30:13,830 So that would be, that would happen, 514 00:30:13,830 --> 00:30:15,440 I would think that would be more, 515 00:30:15,440 --> 00:30:18,370 it would be very typical in a, let 516 00:30:18,370 --> 00:30:22,430 me see if there's some space here, yeah. 517 00:30:22,430 --> 00:30:28,190 That would be very typical and we will do it, 518 00:30:28,190 --> 00:30:30,220 can I change x to t? 519 00:30:30,220 --> 00:30:38,060 Because that's what, if I have some. 520 00:30:38,060 --> 00:30:39,780 What does this problem look like? 521 00:30:39,780 --> 00:30:45,230 And u(0)=0 and u'(0)=0. 522 00:30:45,230 --> 00:30:52,260 Both at the same, at the start equals zero or whatever. 523 00:30:52,260 --> 00:30:56,400 So what kind of a problem is that? 524 00:30:56,400 --> 00:30:59,600 Now these are, I would say, initial values. 525 00:30:59,600 --> 00:31:02,270 Initial values instead of boundary 526 00:31:02,270 --> 00:31:05,550 values, I now have initial values. 527 00:31:05,550 --> 00:31:07,840 And can I solve it? 528 00:31:07,840 --> 00:31:10,020 Yes. 529 00:31:10,020 --> 00:31:11,890 So I'm starting at time zero. 530 00:31:11,890 --> 00:31:14,880 This is t=0. 531 00:31:14,880 --> 00:31:18,440 I'm starting at rest. 532 00:31:18,440 --> 00:31:23,180 No velocity and actually no displacement and just 533 00:31:23,180 --> 00:31:25,850 going forward in time. 534 00:31:25,850 --> 00:31:30,100 So I could solve that differential equation. 535 00:31:30,100 --> 00:31:32,620 I'd be interested in the corresponding difference 536 00:31:32,620 --> 00:31:36,170 equation. 537 00:31:36,170 --> 00:31:37,970 All fine. 538 00:31:37,970 --> 00:31:39,960 It's a different category of problem. 539 00:31:39,960 --> 00:31:49,670 This is an initial value problem. 540 00:31:49,670 --> 00:32:00,460 It's like tracking some mass that's, some satellite. 541 00:32:00,460 --> 00:32:04,790 So that's what you're doing in tracking a satellite 542 00:32:04,790 --> 00:32:09,360 or a planet or something. 543 00:32:09,360 --> 00:32:13,860 Yeah, tracking a planet or a satellite. 544 00:32:13,860 --> 00:32:16,630 You're solving equations like this. 545 00:32:16,630 --> 00:32:18,770 Forward in time. 546 00:32:18,770 --> 00:32:21,760 You know the initial position and you 547 00:32:21,760 --> 00:32:24,070 know the forces acting on it. 548 00:32:24,070 --> 00:32:26,650 Probably gravity. 549 00:32:26,650 --> 00:32:28,640 And you go forward in time. 550 00:32:28,640 --> 00:32:31,810 Yes. 551 00:32:31,810 --> 00:32:33,657 What would the matrix be? 552 00:32:33,657 --> 00:32:34,240 Good question. 553 00:32:34,240 --> 00:32:41,620 What would the matrix look like? 554 00:32:41,620 --> 00:32:46,810 So an electrical engineer would call a problem like this, 555 00:32:46,810 --> 00:32:50,080 and the kind of matrix that I'm going to write down, 556 00:32:50,080 --> 00:32:53,650 I think, would be called causal. 557 00:32:53,650 --> 00:32:57,710 That word just popped into my head, so let me mention it. 558 00:32:57,710 --> 00:33:01,210 You know, part of science and engineering, a big part of it 559 00:33:01,210 --> 00:33:04,440 is learning language, learning words. 560 00:33:04,440 --> 00:33:09,360 And you have to learn sort of the math language 561 00:33:09,360 --> 00:33:14,790 and the engineering language for whatever you're focusing on. 562 00:33:14,790 --> 00:33:19,410 But it's good to also to know a few other languages. 563 00:33:19,410 --> 00:33:21,950 Electrical engineering languages of filters 564 00:33:21,950 --> 00:33:28,030 and causal and other things that we'll see are important. 565 00:33:28,030 --> 00:33:32,120 What would the matrix look like? 566 00:33:32,120 --> 00:33:35,860 Here's what I think it would be. 567 00:33:35,860 --> 00:33:42,320 I've made this a plus there so I'll have to remember that. 568 00:33:42,320 --> 00:33:46,720 I think, so I'm looking at u_0, u_1-- No. 569 00:33:46,720 --> 00:33:48,646 Well, u_0 I actually know, so let 570 00:33:48,646 --> 00:33:50,020 me start with u_1, u_2, u_3, u_4. 571 00:33:50,020 --> 00:33:55,310 572 00:33:55,310 --> 00:33:58,671 What would a typical-- Equal sum, right side, f_1, f_2, 573 00:33:58,671 --> 00:33:59,170 f_3, f_4. 574 00:33:59,170 --> 00:34:02,330 575 00:34:02,330 --> 00:34:06,710 What do you think, what kind of a matrix am I going to get? 576 00:34:06,710 --> 00:34:09,490 Before I put it in there. 577 00:34:09,490 --> 00:34:11,620 This is a good question. 578 00:34:11,620 --> 00:34:14,690 What's the shape of this matrix? 579 00:34:14,690 --> 00:34:19,200 It's going to be triangular. 580 00:34:19,200 --> 00:34:22,530 Instead of being symmetric it's going to be triangular. 581 00:34:22,530 --> 00:34:25,540 I'm going to find, let's see, a typical value 582 00:34:25,540 --> 00:34:35,570 would be, say, u_3 because I've used a plus sign, oops! 583 00:34:35,570 --> 00:34:39,540 I can't make myself do it right. 584 00:34:39,540 --> 00:34:44,070 1, -2, 1. 585 00:34:44,070 --> 00:34:54,695 That would say u_3-2u_2+u_1 would be the new force. 586 00:34:54,695 --> 00:34:56,570 This is the kind of thing we're going to get. 587 00:34:56,570 --> 00:34:59,547 One, maybe one something. 588 00:34:59,547 --> 00:35:00,630 I don't know what this is. 589 00:35:00,630 --> 00:35:03,460 This is up in the boundary, in the initial values. 590 00:35:03,460 --> 00:35:10,060 But from now on it'll be below the diagonal. 591 00:35:10,060 --> 00:35:15,470 It'll be 1, -2, 1. 592 00:35:15,470 --> 00:35:17,390 Do you see? 593 00:35:17,390 --> 00:35:20,050 We're marching. 594 00:35:20,050 --> 00:35:25,140 We're marching forward. 595 00:35:25,140 --> 00:35:27,660 We start by knowing these and then 596 00:35:27,660 --> 00:35:29,690 the equation tells us the next one. 597 00:35:29,690 --> 00:35:31,730 Then the equation tells us the next one. 598 00:35:31,730 --> 00:35:34,560 That's what initial value problems do. 599 00:35:34,560 --> 00:35:38,610 You're told how you begin and you take a step, 600 00:35:38,610 --> 00:35:41,090 you take a step, take a step every time. 601 00:35:41,090 --> 00:35:47,140 And the new value just needs to know the older values. 602 00:35:47,140 --> 00:35:52,220 Do you see the big difference between that and our problems 603 00:35:52,220 --> 00:35:53,440 here? 604 00:35:53,440 --> 00:35:57,840 Our problem is looking left and right 605 00:35:57,840 --> 00:36:00,500 looking for back and forward. 606 00:36:00,500 --> 00:36:02,800 Back for one condition, forward for another. 607 00:36:02,800 --> 00:36:07,470 We start with one, but we're, it's more of a, 608 00:36:07,470 --> 00:36:10,340 it's like a hitting problem. 609 00:36:10,340 --> 00:36:15,150 We start forward, marching forward in our problems, 610 00:36:15,150 --> 00:36:19,240 but we have to hit the other end correctly. 611 00:36:19,240 --> 00:36:22,600 We don't know the slope, we don't know the starting slope, 612 00:36:22,600 --> 00:36:24,630 we know what we want to hit. 613 00:36:24,630 --> 00:36:32,340 Whereas these problems, we're told how we start and we just 614 00:36:32,340 --> 00:36:34,380 follow it in time. 615 00:36:34,380 --> 00:36:40,260 So that's the difference here. 616 00:36:40,260 --> 00:36:49,680 Yeah, sure, okay. 617 00:36:49,680 --> 00:36:51,710 That's true. 618 00:36:51,710 --> 00:36:54,600 So this'll be known. 619 00:36:54,600 --> 00:36:56,290 Yeah, that'll be known. 620 00:36:56,290 --> 00:36:58,550 Yeah. u_1 will also be known. 621 00:36:58,550 --> 00:37:02,360 Yeah, and really, maybe I should have got, 622 00:37:02,360 --> 00:37:06,060 let me put even the other known one. 623 00:37:06,060 --> 00:37:08,380 So we know this, we know this. 624 00:37:08,380 --> 00:37:12,000 So those are sort of not in our, yeah, 625 00:37:12,000 --> 00:37:23,500 that shouldn't be in our problem somehow. 626 00:37:23,500 --> 00:37:27,197 No, I think, what would we get in the end? 627 00:37:27,197 --> 00:37:28,530 You're always looking backwards. 628 00:37:28,530 --> 00:37:29,630 That's the point. 629 00:37:29,630 --> 00:37:32,580 Lower triangular matrices are always looking, 630 00:37:32,580 --> 00:37:35,020 they only look backwards for earlier values 631 00:37:35,020 --> 00:37:37,110 and then they give you the current value. 632 00:37:37,110 --> 00:37:41,620 So that's why lower triangular matrices are so easy to invert. 633 00:37:41,620 --> 00:37:42,330 No problem. 634 00:37:42,330 --> 00:37:46,840 If it's lower triangular, you just, like, march forward. 635 00:37:46,840 --> 00:37:50,740 And if it's upper triangular, which way do you march? 636 00:37:50,740 --> 00:37:53,490 So if you have an upper triangular problem, 637 00:37:53,490 --> 00:37:57,620 suppose I gave you the problem, let 638 00:37:57,620 --> 00:37:59,050 me make it upper triangular. 639 00:37:59,050 --> 00:37:59,570 So x+y+z=7. 640 00:37:59,570 --> 00:38:03,420 641 00:38:03,420 --> 00:38:08,760 2y+3z=12 and z=17. 642 00:38:08,760 --> 00:38:11,640 643 00:38:11,640 --> 00:38:13,980 So that's upper triangular. 644 00:38:13,980 --> 00:38:17,150 Where do we start in solving that one? 645 00:38:17,150 --> 00:38:18,490 From the bottom. 646 00:38:18,490 --> 00:38:20,930 From the right-hand end, the bottom. 647 00:38:20,930 --> 00:38:25,150 And we march backwards in time. 648 00:38:25,150 --> 00:38:29,490 And what I was saying about A, well L times U, 649 00:38:29,490 --> 00:38:31,910 yeah, this is worth seeing. 650 00:38:31,910 --> 00:38:38,840 What I was saying about A=LU, it was, you remember that? 651 00:38:38,840 --> 00:38:40,070 Those letters? 652 00:38:40,070 --> 00:38:45,030 What that was saying was that this matrix that's 653 00:38:45,030 --> 00:38:51,900 looking both ways can be written as a product of a matrix L 654 00:38:51,900 --> 00:38:59,030 that looks behind for old values and you can go forward with it. 655 00:38:59,030 --> 00:39:05,020 And a matrix U, like this one, this upper triangular, 1, 1, 1, 656 00:39:05,020 --> 00:39:10,920 zeroes below that diagonal, that you go backward with. 657 00:39:10,920 --> 00:39:12,750 Somehow that's appealing. 658 00:39:12,750 --> 00:39:18,590 That's like aesthetic to break up a two-way problem 659 00:39:18,590 --> 00:39:24,140 into a problem that marches one way and then the other. 660 00:39:24,140 --> 00:39:28,530 And of course, that's what elimination aims for, 661 00:39:28,530 --> 00:39:33,190 is this problem that it can solve by, the words would be, 662 00:39:33,190 --> 00:39:35,030 back substitution. 663 00:39:35,030 --> 00:39:37,390 When you've started with your original problem, got 664 00:39:37,390 --> 00:39:42,020 to this one, then you just have a back substitution, 665 00:39:42,020 --> 00:39:44,880 you go backwards. 666 00:39:44,880 --> 00:39:49,030 Oh, so much, I'll mention the Kalman filter. 667 00:39:49,030 --> 00:39:53,510 That's a similar process of going forward, 668 00:39:53,510 --> 00:39:55,120 that's called prediction. 669 00:39:55,120 --> 00:39:57,410 Going backward, that's called smoothing. 670 00:39:57,410 --> 00:40:00,750 And so, Kalman had the great idea 671 00:40:00,750 --> 00:40:05,640 that he could break these problems that 672 00:40:05,640 --> 00:40:14,030 were fundamental in space computations 673 00:40:14,030 --> 00:40:19,350 into prediction and smoothing. 674 00:40:19,350 --> 00:40:20,870 Once again, we've got off. 675 00:40:20,870 --> 00:40:25,630 Yes? 676 00:40:25,630 --> 00:40:28,420 Oh, the beam. 677 00:40:28,420 --> 00:40:33,520 Let me help you even more before the question. 678 00:40:33,520 --> 00:40:39,070 I said it's better to draw the beam this way. 679 00:40:39,070 --> 00:40:41,270 I like the beam better this way. 680 00:40:41,270 --> 00:40:46,310 Because the point of the beam problem is loads are acting -- 681 00:40:46,310 --> 00:40:48,360 and we'll see this, of course, later -- 682 00:40:48,360 --> 00:40:54,860 loads are acting perpendicular to the direction of the beam. 683 00:40:54,860 --> 00:40:57,010 That's why the beam bends. 684 00:40:57,010 --> 00:41:00,460 So it'll bend a little, right? 685 00:41:00,460 --> 00:41:04,250 And that is what leads us, its bending moments 686 00:41:04,250 --> 00:41:05,820 and other stuff. 687 00:41:05,820 --> 00:41:08,900 If you haven't met beams, well, it'll 688 00:41:08,900 --> 00:41:12,840 be great to just have a very, half a lecture about, 689 00:41:12,840 --> 00:41:15,800 or maybe a lecture about beams. 690 00:41:15,800 --> 00:41:19,860 That gives a fourth order equation 691 00:41:19,860 --> 00:41:21,560 that I'll write down again. 692 00:41:21,560 --> 00:41:24,970 Fourth derivative equal the load. 693 00:41:24,970 --> 00:41:39,630 Now, ready. 694 00:41:39,630 --> 00:41:42,990 Yeah, now here I don't have the negative sign. 695 00:41:42,990 --> 00:41:47,170 Because once I've got second derivatives twice, 696 00:41:47,170 --> 00:41:52,520 so the second derivative is, in some way, negative. 697 00:41:52,520 --> 00:41:55,090 I'll complete that sentence in a second. 698 00:41:55,090 --> 00:41:57,380 Somehow the second derivative, which 699 00:41:57,380 --> 00:42:02,540 is the guy that has the 1, -2, 1, somehow that's 700 00:42:02,540 --> 00:42:04,340 a negative thing. 701 00:42:04,340 --> 00:42:07,310 But fourth derivative is second derivative 702 00:42:07,310 --> 00:42:08,670 of the second derivative. 703 00:42:08,670 --> 00:42:12,560 Yeah, do you want to tell me what the numbers would be? 704 00:42:12,560 --> 00:42:17,640 As long as we're wildly looking forward to fourth derivatives, 705 00:42:17,640 --> 00:42:19,260 just, it helps. 706 00:42:19,260 --> 00:42:28,320 Do you want to guess what will a typical row of the matrix B 707 00:42:28,320 --> 00:42:32,830 when I go to finite differences, fourth differences? 708 00:42:32,830 --> 00:42:36,130 Probably you've never seen a fourth difference. 709 00:42:36,130 --> 00:42:38,710 You may not have seen second differences before. 710 00:42:38,710 --> 00:42:43,240 That was a big deal, then, to introduce second differences. 711 00:42:43,240 --> 00:42:46,050 Those 1, -2, 1's. 712 00:42:46,050 --> 00:42:48,600 That was second differences. 713 00:42:48,600 --> 00:42:50,520 Fourth? 714 00:42:50,520 --> 00:42:53,270 Yeah, 1, 4, 6, 4, 1 with minus sign. 715 00:42:53,270 --> 00:42:58,290 1, -4, 6, -4, and 1. 716 00:42:58,290 --> 00:43:06,190 In some way, I would get that by squaring this guy. 717 00:43:06,190 --> 00:43:17,710 So that would be a fourth difference. 718 00:43:17,710 --> 00:43:19,880 Oh, what's the deal with boundary conditions? 719 00:43:19,880 --> 00:43:25,070 What are you figuring, on beams, beam problems, 720 00:43:25,070 --> 00:43:28,630 for a fourth order equation and a matrix that's 721 00:43:28,630 --> 00:43:30,430 stretching out further. 722 00:43:30,430 --> 00:43:35,670 What's going to happen at the left-hand boundary? 723 00:43:35,670 --> 00:43:39,970 I guess my specific question is, how many boundary conditions 724 00:43:39,970 --> 00:43:41,580 do I now need? 725 00:43:41,580 --> 00:43:42,740 Four. 726 00:43:42,740 --> 00:43:44,880 And the typical is two at each end. 727 00:43:44,880 --> 00:43:46,400 That's the balanced way. 728 00:43:46,400 --> 00:43:50,630 That's the way that would make this matrix sort of symmetric. 729 00:43:50,630 --> 00:43:56,540 So maybe at this end I say it's held at zero 730 00:43:56,540 --> 00:44:01,250 and maybe it's just sitting on a log there. 731 00:44:01,250 --> 00:44:02,260 Right? 732 00:44:02,260 --> 00:44:06,900 That boundary condition I would call simply supported. 733 00:44:06,900 --> 00:44:11,420 That boundary condition says that u(0)=0. 734 00:44:11,420 --> 00:44:13,510 Because it's sitting there. 735 00:44:13,510 --> 00:44:17,150 And but the slope doesn't have to be zero. 736 00:44:17,150 --> 00:44:20,590 What does have to be zero there? 737 00:44:20,590 --> 00:44:23,560 Yeah, sort of the bending moment. 738 00:44:23,560 --> 00:44:27,780 Nobody's here twisting it, right? 739 00:44:27,780 --> 00:44:30,840 So the other condition in that picture 740 00:44:30,840 --> 00:44:36,140 would be second derivative equal zero. 741 00:44:36,140 --> 00:44:42,360 Maybe my point is that now you see what I said before, 742 00:44:42,360 --> 00:44:46,830 that getting the boundary conditions into the problem 743 00:44:46,830 --> 00:44:49,130 is often the hardest part. 744 00:44:49,130 --> 00:44:51,730 Because I have to replace u(0)=0, 745 00:44:51,730 --> 00:44:53,550 that shouldn't be too hard to do. 746 00:44:53,550 --> 00:44:57,430 But I have to use this other condition somehow, 747 00:44:57,430 --> 00:45:01,820 it's going to screw up the 1, -4, 6, -4, 1. 748 00:45:01,820 --> 00:45:06,140 I'll have two boundary rows at the top, 749 00:45:06,140 --> 00:45:08,940 two boundary rows at the bottom. 750 00:45:08,940 --> 00:45:11,860 I don't want to go further today. 751 00:45:11,860 --> 00:45:14,430 But I think maybe just mentioning 752 00:45:14,430 --> 00:45:19,910 this gives you the picture of sort 753 00:45:19,910 --> 00:45:23,490 of the how things fit together. 754 00:45:23,490 --> 00:45:28,150 We would still have some nice constant diagonals 755 00:45:28,150 --> 00:45:29,610 in the middle, but now we'll have 756 00:45:29,610 --> 00:45:33,040 two boundary rows at each end. 757 00:45:33,040 --> 00:45:36,750 So that's something to come. 758 00:45:36,750 --> 00:45:44,980 Yes, now back to reality, which is, any questions? 759 00:45:44,980 --> 00:45:52,540 Lower triangular guy, yeah. 760 00:45:52,540 --> 00:45:56,380 What do I mean by marching forward? 761 00:45:56,380 --> 00:46:01,240 So let's see. 762 00:46:01,240 --> 00:46:04,600 I'll replace this. 763 00:46:04,600 --> 00:46:06,310 Let's see it better. 764 00:46:06,310 --> 00:46:11,500 I would replace this by maybe u, I'll use a different letter, 765 00:46:11,500 --> 00:46:13,830 n+1. 766 00:46:13,830 --> 00:46:26,360 At u_(n+1)-2u_n+u_(n-1) is some right-hand side 767 00:46:26,360 --> 00:46:28,890 if there's a force acting on my thing. 768 00:46:28,890 --> 00:46:32,390 So f_n maybe. 769 00:46:32,390 --> 00:46:37,730 By marching forward, I just mean that this equation-- 770 00:46:37,730 --> 00:46:40,360 that I can go in order. 771 00:46:40,360 --> 00:46:43,290 I can start with u_0 and u_1. 772 00:46:43,290 --> 00:46:47,410 They come from the boundary conditions. 773 00:46:47,410 --> 00:46:51,880 Then this equation will tell me u_2. 774 00:46:51,880 --> 00:46:53,470 I use the equation. 775 00:46:53,470 --> 00:46:54,790 With n as one. 776 00:46:54,790 --> 00:47:01,490 This says u_2, some u_1's, some u_0's, f_1's, all that I know. 777 00:47:01,490 --> 00:47:07,450 In other words, once I get started, I'm on a roll. 778 00:47:07,450 --> 00:47:10,520 If I have two boundary conditions to get me started, 779 00:47:10,520 --> 00:47:14,120 then the equation tells me u_2. 780 00:47:14,120 --> 00:47:21,240 And then the next time, u_3-2u_2+u_1, I can find u_3. 781 00:47:21,240 --> 00:47:24,710 So I can get those, I can go forever. 782 00:47:24,710 --> 00:47:28,790 If you give me enough to start on, two things to start, 783 00:47:28,790 --> 00:47:32,630 then I march forward. 784 00:47:32,630 --> 00:47:37,120 Whereas in our problems, we've only got one thing to start on 785 00:47:37,120 --> 00:47:39,400 and we've got one goal to hit. 786 00:47:39,400 --> 00:47:44,090 And that's why we have to solve the whole system together. 787 00:47:44,090 --> 00:47:48,600 This is, we can solve it step-by-step. 788 00:47:48,600 --> 00:47:51,930 This is way faster of course. 789 00:47:51,930 --> 00:47:57,550 To be able to just go forward in time. 790 00:47:57,550 --> 00:48:01,550 I'll mention that the topic of initial value problems 791 00:48:01,550 --> 00:48:07,590 and finite differences for them, we can't get to that. 792 00:48:07,590 --> 00:48:10,370 So we're seeing a little bit here, 793 00:48:10,370 --> 00:48:16,740 but it's done properly in 18.086 in the second semester 794 00:48:16,740 --> 00:48:20,620 is the initial value problem start part. 795 00:48:20,620 --> 00:48:27,610 And that has it's own interesting questions. 796 00:48:27,610 --> 00:48:33,310 Somehow we've talked about fourth order equations, 797 00:48:33,310 --> 00:48:39,240 initial value problems. 798 00:48:39,240 --> 00:48:42,100 But no homework problems. 799 00:48:42,100 --> 00:48:46,280 So I'm ready for, or even related. 800 00:48:46,280 --> 00:48:51,790 But that's fine with me. 801 00:48:51,790 --> 00:48:53,730 Is there a question? 802 00:48:53,730 --> 00:48:54,400 Yeah, thanks. 803 00:48:54,400 --> 00:48:55,858 Or it doesn't have to be a homework 804 00:48:55,858 --> 00:49:05,350 question, another question. 805 00:49:05,350 --> 00:49:09,240 Oh, good question. 806 00:49:09,240 --> 00:49:11,210 You mean I should just send the homework out 807 00:49:11,210 --> 00:49:13,430 to Natick where MATLAB is. 808 00:49:13,430 --> 00:49:16,420 Do you know that MATLAB is just 15 miles away? 809 00:49:16,420 --> 00:49:22,410 I almost get there, I live 2/3 of the way there. 810 00:49:22,410 --> 00:49:25,370 Yeah, so we could just send the whole thing out there 811 00:49:25,370 --> 00:49:27,940 and get it back. 812 00:49:27,940 --> 00:49:36,150 That would save a lot of work. 813 00:49:36,150 --> 00:49:38,050 I suppose, I'm okay. 814 00:49:38,050 --> 00:49:40,400 Why should I say no? 815 00:49:40,400 --> 00:49:43,410 Anything MATLAB can do and you can make it do, 816 00:49:43,410 --> 00:49:45,060 I'm okay with that. 817 00:49:45,060 --> 00:49:48,270 I don't see that you have to do things by hand if you've 818 00:49:48,270 --> 00:49:50,180 got a better way. 819 00:49:50,180 --> 00:49:50,780 That's okay. 820 00:49:50,780 --> 00:49:53,560 And then probably the answer gets printed 821 00:49:53,560 --> 00:49:55,390 and you can graph it. 822 00:49:55,390 --> 00:49:57,960 So that's fine. 823 00:49:57,960 --> 00:50:03,020 So I mean, somehow a course like this has got two parts to it. 824 00:50:03,020 --> 00:50:05,010 Applied math has two parts to it. 825 00:50:05,010 --> 00:50:08,830 The modeling part, set up the equation, think, what is it 826 00:50:08,830 --> 00:50:10,260 you're supposed to do. 827 00:50:10,260 --> 00:50:13,080 And then, step two is do it. 828 00:50:13,080 --> 00:50:15,390 The numerical part, the computing part. 829 00:50:15,390 --> 00:50:19,620 And that's where MATLAB, Python, Fortran, whatever, 830 00:50:19,620 --> 00:50:25,220 is going to do a lot of the heavy lifting. 831 00:50:25,220 --> 00:50:27,390 Was there another question? 832 00:50:27,390 --> 00:50:34,330 So that first homework was certainly 833 00:50:34,330 --> 00:50:35,930 very general intentionally. 834 00:50:35,930 --> 00:50:40,610 Because I'm hoping you will read the book. 835 00:50:40,610 --> 00:50:46,820 The lecture, you'll be able to match the lectures 836 00:50:46,820 --> 00:50:51,100 with the book even later on when they separate 837 00:50:51,100 --> 00:50:54,004 a little or separate more. 838 00:50:54,004 --> 00:50:55,170 You'll see what we're doing. 839 00:50:55,170 --> 00:50:58,350 And those, the homework problems, 840 00:50:58,350 --> 00:51:01,380 you should look at some of the others just to see. 841 00:51:01,380 --> 00:51:02,860 Do I know how to do that? 842 00:51:02,860 --> 00:51:05,810 Right. 843 00:51:05,810 --> 00:51:08,010 Let's stop here for this first review. 844 00:51:08,010 --> 00:51:10,620 I'm sure we'll have more-- questions will build up 845 00:51:10,620 --> 00:51:12,870 for the second week.