Then - Islamic University of Gaza
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1 . PDE
Transcript of Then - Islamic University of Gaza
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PDE
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Then
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T equation is
And
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𝑢𝑢𝑡𝑡𝑡𝑡 − 𝑐𝑐2𝑢𝑢𝑥𝑥𝑥𝑥 = 𝐻𝐻(𝑥𝑥), 0 < 𝑥𝑥 < 𝐿𝐿, 𝑡𝑡 > 0.
𝑢𝑢(0, 𝑡𝑡) = 0, 𝑢𝑢(𝐿𝐿, 𝑡𝑡) = 0, (3)
𝑢𝑢𝑡𝑡(𝑥𝑥, 0) = 0, 𝑢𝑢(𝑥𝑥, 0) = 0.
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The new equation is 𝑣𝑣𝑡𝑡𝑡𝑡 = 𝑐𝑐2𝑣𝑣𝑥𝑥𝑥𝑥 is homogeneous .
Applying the conditions 𝑢𝑢(0, 𝑡𝑡) = 𝑢𝑢(𝐿𝐿, 𝑡𝑡) we have
Now the problem is 𝑣𝑣𝑡𝑡𝑡𝑡 = 𝑐𝑐2𝑣𝑣𝑥𝑥𝑥𝑥 , 0 < 𝑥𝑥 < 𝐿𝐿, 𝑡𝑡 > 0.
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Then 𝜓𝜓(𝑥𝑥) = 𝑇𝑇1−𝑇𝑇0𝐿𝐿
𝑥𝑥 + 𝑇𝑇0 .
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𝑋𝑋′′ − 𝜆𝜆𝑋𝑋 = 0, 𝑌𝑌′′ + 𝜆𝜆𝑌𝑌 = 0.
and continue to find X and Y.
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Excercises:
Ex 1.
Ex2.
Ex 3.
Ex 4.
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Ex 5.
Exercise 7.9 page 265 Q(1,2,3,4,11,12,13)