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

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

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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