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CYBR 150 – Computer Science II Object-Oriented Programming

1. Heat is flowing steadily in a metal plate whose shape is an infinite rectangle occupying the −a < x < a，y > 0 of the (x, y) plane. The temperature at the point (x, y) is denoted by u(x, y). The sides x = ±a are insulated, the temperature approaches zero as y → ∞, while the side y = 0 is maintained at a fifixed temperature −T for −a < x < 0 and T for 0 < x < a. It is known that u(x, y) satisfifies the Laplace equation

∂2u /∂x2+∂2u /∂y2 = 0

1. Sketch by hand the configuration of the metal plate and specify all boundary conditions corresponding to each side of the metal plate

1. Use the method of separation to obtain the solution in the form:

u(x, y) =Aπ ∞X n=01B eC D

Analyse all three cases of a separation constants (λ <, =, > 0). Coefficients A, B, C and D (D is a trigonometric expression) have to be calculated and highlighted in your assignment. Full marks are awarded for a complete step by step proof.

iii. Take T the temperature from part a) to be equal to the last two figures of your student Monash ID number (if ID XXXXXX31, take T=31; ID XXXXXX09, take T=9; ID XXXXX1100, take T=10).

And take a = 1. In MATLAB, on the same graph plot the partial sum up to the 50th harmonic of u(x, y) for 10 relevant values y = 0, 0.01, 0.02, …… and continuing with any y of your own choice.

Label and ADD a legend to the graph and publish the graph of your solution, and attach it to the assignment.

iii. For what value y does the temperature drop to 10% of the initial temperature for 0 < x < a?

TOTAL=4+18+10+1+2[neat]=35 marks