9. Ut = 4uxx, (xo, to) = (5,4)

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Hand written..partial differential..solve only Ques. 9
9. Utt = 4uxx, (xo, to) = (5,4)
10. Utt = 3Uzz, (xo, to) = (0,5)
11. For the problem in Example 2, use only characteristics to find the fol-
lowing: u(0,2), u(0,4), u(5,5), u(10, 6), u(-5, 3). See Figure 5.10. By
the way, again, notice that, in those cases where ut(x, 0) = 0, the value
of u(xo, to) depends only on the values of u along the two characteristics
containing (zo, to). This means that a disturbance at some point on the
string arrives at, and leaves from, the point x = xo instantaneously. In
this particular example, a square wave hits the point x =
through it and leaves as sharply as it arrived we say that it has sharp
leading and trailing edges. When all disturbances propagate in this
manner, the system is said to satisfy Huygens's Principle.
xo, moves
12. We wish to show that Huygens's Principle is not satisfied when u, (x, 0) is
not the zero-function. Specifically, consider the problem in Example 2,
but with f(r) and g(x) interchanged. Show that the solution u is as
given in Figure 5.12, and then describe what the graph of u looks like
for various times t.
Transcribed Image Text:9. Utt = 4uxx, (xo, to) = (5,4) 10. Utt = 3Uzz, (xo, to) = (0,5) 11. For the problem in Example 2, use only characteristics to find the fol- lowing: u(0,2), u(0,4), u(5,5), u(10, 6), u(-5, 3). See Figure 5.10. By the way, again, notice that, in those cases where ut(x, 0) = 0, the value of u(xo, to) depends only on the values of u along the two characteristics containing (zo, to). This means that a disturbance at some point on the string arrives at, and leaves from, the point x = xo instantaneously. In this particular example, a square wave hits the point x = through it and leaves as sharply as it arrived we say that it has sharp leading and trailing edges. When all disturbances propagate in this manner, the system is said to satisfy Huygens's Principle. xo, moves 12. We wish to show that Huygens's Principle is not satisfied when u, (x, 0) is not the zero-function. Specifically, consider the problem in Example 2, but with f(r) and g(x) interchanged. Show that the solution u is as given in Figure 5.12, and then describe what the graph of u looks like for various times t.
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