这是一篇来自澳洲的关于解决一些偏微分程的Assignment 4 Matlab代写
- Heat is flflowing steadily in a metal plate whose shape is an infifinite 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
- Sketch by hand the confifiguration of the metal plate and specify all boundary conditions corresponding to each side of the metal plate
- 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). Coeffiffifficients 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 fifigures 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?
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