# Math Assignment | Custom Assignment Help

Question 1 [10 marks] The following partial differential equation (PDE) describes the temperature u(x,t) in a metal rod of length 1. The rod is insulated at both ends, but heat escapes throughout the length of the rod at a rate proportional to the local temperature:

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∂u ∂t

= D∂2u ∂x2 −hu, ∂u ∂x

(0,t) = 0, ∂u ∂x

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(1,t) = 0, u(x,0) = x(1−x). In this equation, D and h are positive constants.

(a) (4 marks) Assuming a solution of the form un(x,t) = Xn(x)Tn(t), Apply separation of variables to the problem to ﬁnd the ordinary differential equations for Xn(x) and Tn(t). Explain why the relevant eigenfunction solutions Xn(x) are Xn(x) = cos(nπx), n = 0,1,2,….

(b) (4 marks) Solve the differential equation for Tn(t), and hence form the general solutiontothepartialdifferentialequation. Applytheinitialconditiontodetermine the arbitrary constants in the equation. You may use the following Fourier cosine series expansion  Get mathematics  assignment homework help today

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