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# Solved: Humans gather most of their information on the outside world through sight and

ISBN: 9781284105902 342

## Solution for problem 33 Chapter 15.2

Advanced Engineering Mathematics | 6th Edition

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Advanced Engineering Mathematics | 6th Edition

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Problem 33

Humans gather most of their information on the outside world through sight and sound. But many creatures use chemical signals as their primary means of communication; for example, honeybees, when alarmed, emit a substance and fan their wings feverishly to relay the warning signal to the bees that attend to the queen. These molecular messages between members of the same species are called pheromones. The signals may be carried by moving air or water or by a diffusion process in which the random movement of gas molecules transports the chemical away from its source. FIGURE 15.2.4 shows an ant emitting an alarm chemical into the still air of a tunnel. If c(x, t) denotes the concentration of the chemical x centimeters from the source at time t, then c(x, t) satisfies k 02 c 0x 2 0c 0t , x . 0, t . 0, and k is a positive constant. The emission of pheromones as a discrete pulse gives rise to a boundary condition of the form 0u 0x 2 x0 Ad(t), where d(t) is the Dirac delta function. (a) Solve the boundary-value problem if it is further known that c(x, 0) 0, x 0, and limxS q c(x, t) 0, t 0. (b) Use a CAS to plot the graph of the solution in part (a) for x 0 at the fixed times t 0.1, t 0.5, t 1, t 2, t 5. (c) For a fixed time t, show that eq 0 c(x, t) dx Ak. Thus Ak represents the total amount of chemical discharged.

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