homework12-solutions

homework12-solutions - Homework 12: Extra Problems –...

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Unformatted text preview: Homework 12: Extra Problems – Solutions 1 The partial differential equation u t = au x is known as the transport equation (or sometimes the advection equation ). In this problem we’ll find solutions u ( x,t ) to this equation. In particular, we’ll include an initial condition and solve the initial value problem (IVP) ( * ) ∂u ∂t = a ∂u ∂x u ( x, 0) = f ( x ) . (a) Here is a graph of u = f ( x ) (an example initial condition for the IVP): ................................................................................................................................................................................................................................................................................................................................ . . . . . . . . . . . . . x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . u ................................................................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................................................................. • • • A B C This graph is of u ( x, 0) = f ( x ). Is the value of u x ( x, 0) positive, negative, or zero at the points x = A , x = B , and x = C ? Solution: This is simply asking: for the function u = f ( x ) graphed above, is f ( x ) positive, negative, or zero at the points x = A , x = B , and x = C ? From single-variable calculus, we know that f ( A ) > 0 while f ( B ) < 0 and f ( C ) < 0. Thus u x ( A, 0) > while u x ( B, 0) < 0 and u x ( C, 0) < 0. (b) If u ( x,t ) is a solution of the IVP labeled ( * ) for a > 0, what is the sign of u t ( x, 0) at the points x = A , x = B , and x = C ?...
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homework12-solutions - Homework 12: Extra Problems –...

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