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After defining the problem.

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Now we need to solve it in order to solve it.

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Well, we simply or not simply, we have to use the discretized form of the equation.

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And to calculate it a time, one time step at a time.

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Our solution will be what we call explicit, in which we will update the solution or we will find the

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solution.

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And for the next time, step.

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So we start from the initial condition and then go one time step at the time, one time step, and then

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see what is the solution.

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And then another time step and then another time step and another and another.

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This is what we call an explicit, explicit solution.

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And this is how we like discretized our equation.

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So let's see here what is happening.

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So we have do you over and of course equals K do the second derivative of U over.

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And now what we need to do we need to write this equation in terms of time plus DT.

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So every time we predict we will predict the value of the next time step.

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This is how it works.

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We have to predict the next time.

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Step what?

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What's going to happen?

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This goes, um, because this is a simulation.

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We need to always predict what what is going to happen.

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So here we start.

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Uh, well, first, calculating the time.

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Just give it some here just to move a little bit.

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Freely.

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And here we start for t in.

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Range.

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Starting from one.

30
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To the length.

31
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Of Victor.

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The victor.

33
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This one?

34
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Minus one.

35
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Y minus one because the the T actually in here, it will start from a in python.

36
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We start addressing the first element to zero.

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So the last element we will have one element lost.

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So this is why we always put it minus one.

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So it's a kind of python thing as it's, as it's known in range X also from one to the length of x.

40
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It's the same.

41
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And.

42
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A x vector, also minus one.

43
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And here we need to calculate.

44
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So what do we need to calculate?

45
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We need to calculate you.

46
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Act.

47
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Specific X at a specific time.

48
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But what is the time going to be?

49
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It's the time plus one.

50
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Y plus one, because in this simulation, in the explicit simulation, we're actually predicting what's

51
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going to happen in the future.

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So t plus one, this is what's going to happen in the future at this location.

53
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We every time step we will calculate what's happening in the future for every a the next time, step

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for every location.

55
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So this is basically what we are going to do.

56
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And this will equals what?

57
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This will equals.

58
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Let's see the equation we have.

59
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Well, we have.

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This.

61
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If we move this, you or we.

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We move this.

63
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We move it here.

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So K has to be multiplied by and divided by d x squared.

65
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So let's do that.

66
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So it's I think it's just a simple math of how we have to arrange these equation.

67
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So we will, we will do it.

68
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And so the first one is K.

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Multiplied.

70
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It will multiply.

71
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That is going to be divided by the square.

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And.

73
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This.

74
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Values because we again go to the discretized form.

75
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So we have k multiplied t divided by the square or dt divided by the x square multiplied k, and then

76
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all of this, these values has to be multiplied by all of this part.

77
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And later we have to plus u t dt so we may go and do it.

78
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So that.

79
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The term will be you.

80
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A little bit like this.

81
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U of T.

82
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X plus one.

83
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Minus two multiplied.

84
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Of course we have to go.

85
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And maybe it's better put it.

86
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Somewhere like this.

87
00:06:03,600 --> 00:06:03,950
Yeah.

88
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This way.

89
00:06:04,860 --> 00:06:05,580
So.

90
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You t x plus x, it means x plus one.

91
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The next point.

92
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This is our current point is x and next point will be x plus one.

93
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Multiply two which is this one two multiplied you at this time, which is going to be you.

94
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Time and x.

95
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This time the current location plus.

96
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Plus.

97
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This.

98
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Which is you?

99
00:06:40,660 --> 00:06:42,010
Current time, Of course.

100
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And.

101
00:06:43,300 --> 00:06:43,750
Oh, sorry.

102
00:06:43,750 --> 00:06:44,020
Sorry.

103
00:06:45,340 --> 00:06:46,150
And.

104
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X minus.

105
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One the last point.

106
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So all of this.

107
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Is written in here.

108
00:07:02,160 --> 00:07:07,770
So this all of this is in this part of the equation, this one.

109
00:07:09,100 --> 00:07:15,510
K This is going to be and then over the square is this one.

110
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So this is constant multiplied by this value that will depend on the all the time and the X.

111
00:07:25,080 --> 00:07:29,940
Now what is left is I would like to put a big.

112
00:07:31,040 --> 00:07:31,550
Parent.

113
00:07:31,580 --> 00:07:32,240
Oh, sorry.

114
00:07:32,360 --> 00:07:37,430
A big parentheses between here, although it might be not necessary.

115
00:07:37,430 --> 00:07:41,660
And this, which is We have all of this.

116
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We.

117
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We need to sum the value of you.

118
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And time and X at our specific.

119
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Time this one and specific X.

120
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So this is this equation.

121
00:08:04,880 --> 00:08:08,540
We'll just make it a little bit this way.

122
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We will calculate for every all the time.

123
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Starting from well, t equals one from the initial condition.

124
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And then we keep updating the initial condition.

125
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We will calculate for the whole space.

126
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And we will update for the time plus one for the next time step.

127
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And we keep doing this until we reach to the final time step.

128
00:08:39,390 --> 00:08:40,620
Which is this one?

129
00:08:41,420 --> 00:08:45,470
This, you move it here and this is going to be the you.

130
00:08:45,650 --> 00:08:47,980
And then we still have this bulky thing.

131
00:08:47,990 --> 00:08:50,690
This one will be here.

132
00:08:51,680 --> 00:08:53,510
The x squared is here.

133
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D is here because we moved it here.

134
00:08:56,690 --> 00:08:59,060
The x is this one and then k.

135
00:08:59,940 --> 00:09:01,260
This constant.

136
00:09:01,680 --> 00:09:06,550
This values will be summed with this value.

137
00:09:06,570 --> 00:09:08,100
So what is left?

138
00:09:08,130 --> 00:09:08,760
Nothing.

139
00:09:08,760 --> 00:09:10,440
We just run it.

140
00:09:12,980 --> 00:09:13,730
It.

141
00:09:17,890 --> 00:09:18,790
You.

142
00:09:19,670 --> 00:09:22,010
Minus two x.

143
00:09:23,350 --> 00:09:24,670
A 100.

144
00:09:26,700 --> 00:09:26,960
It's.

145
00:09:34,660 --> 00:09:35,260
Uh, sorry.

146
00:09:35,290 --> 00:09:36,310
T vector.

147
00:09:37,120 --> 00:09:37,420
Um.

148
00:09:38,350 --> 00:09:39,490
I made a mistake.

149
00:09:39,520 --> 00:09:40,870
This is x vector.

150
00:09:40,900 --> 00:09:42,840
I put it X here, but I forgot.

151
00:09:42,850 --> 00:09:43,350
Okay.

152
00:09:43,360 --> 00:09:44,560
Now it should work.

153
00:09:44,650 --> 00:09:53,890
So now it's computing, and we can see how after this finish, we can see how you become.

154
00:09:54,630 --> 00:09:59,390
So it's now computing, Now it's finished and we can see you.

155
00:09:59,430 --> 00:10:04,410
We print it out and we can see the values got updated.

156
00:10:04,440 --> 00:10:13,110
Of course, these values, you can see this is the 200 is the far right, and then it will start diffusing

157
00:10:13,110 --> 00:10:18,120
little by little here and then here, basically all the rods.

158
00:10:18,120 --> 00:10:25,860
And then this is the values, of course, starting from zero and step by step, updating these values.

159
00:10:27,070 --> 00:10:30,780
Okay, so this is how we solved our equation.

160
00:10:30,790 --> 00:10:33,490
Next time we will just check the solution.