Question

Solve the equation ut = uxx with the boundary conditions u(x,0) = 3 sin (nπx) and u(0,t)=0=u(1,t) where 0<x<1 and t>0.

(a) \(3∑_{n=1}^∞ \) e^-n^2 π^2 t  cos⁡(nπx)

(b) \(∑_{n=1}^∞ \) e^-n^2 π^2 t  sin⁡(nπx)

(c) \(3∑_{n=1}^∞ \) e^-n^2 π^2 t  sin⁡(nπx)

(d) \(∑_{n=1}^∞ \) e^-n^2 π^2 t  cos(nπx)

I got this question in an internship interview.

This interesting question is from Solution of 1D Heat Equation in portion Applications of Partial Differential Equations of Engineering Mathematics

Answer 1

The correct option is (c) \(3∑_{n=1}^∞ \) e^-n^2 π^2 t  sin⁡(nπx)

The explanation: u(x,t) = (c cospx + c’ sinpx) (c’’  e^-p^2t)

When x=0, c=0 and when x=1, p=nπ.

When t=0, 3 sin (nπx) = \(∑_{n=1}^∞ \) bn e^-n^2 π^2 t  sin⁡(nπx)

Therefore bn =3 for all n

Hence the solution is \(3∑_{n=1}^∞ \) e^-n^2 π^2 t  sin⁡(nπx).