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pset_1 - MIT 18.385j/2.036j First Problem Set Suggested...

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First Problem Set ________________________________________________________________________ Suggested Readings (textbook): Chapters 1-2-3. Suggested Problems (textbook): Ch. 2: 2.2.9 2.2.12 2.2.13 2.3.3 2.4.9 2.6.1 2.8.3 2.8.5 Ch. 3: 3.3.1 3.4.5 3.4.7 3.4.8 3.4.9 3.4.10 ________________________________________________________________________ Problems to hand in for grading (textbook): Ch. 2: 2.2.8 2.2.10 2.3.2 2.5.4 2.5.5 2.5.6 Ch. 3: 3.2.6 3.2.7 3.3.2 3.4.6 ________________________________________________________________________ PROBLEM TO HAND IN FOR GRADING (not in textbook): PDE_Blow_Up In the lectures we considered the PDE problem initial value problem: u_t + u*u_x = 0; u(x, 0) = F(x). Notation: 1) u_t and u_x are the partial derivatives, with respect to t and x (resp.). 2) t is time and x is space. 3) * is the multiplication operator. 4) ^ denotes taking a power [u^2 is the square of u]. 4) u = u(x, t) is a function of x and t. We showed that the solution to this problem ceased to exist at a finite time (the derivatives of u become infinite and, beyond that, u becomes multiple valued) whenever dF/dx was negative anywhere. This was shown "graphically". It can be shown analytically as follows: --- A. Consider the CHARACTERISTIC CURVES dx/dt = u(x, t), as instroduced in the lecture.
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