Consider the initial boundary value problem for the heat equation: [begin{array}{cc} u_{t}=2 u_{x x}, & 0 0 . end{array}] Use the finite transform method to

Consider the initial boundary value problem for the heat equation:

\[\begin{array}{cc} u_{t}=2 u_{x x}, & 00, \\ u(1, t)=0, & t>0 . \end{array}\]

Use the finite transform method to solve this problem. Namely, assume that the solution takes the form \(u(x, t)=\sum_{n=1}^{\infty} b_{n}(t) \sin n \pi x\) and obtain an ordinary differential equation for \(b_{n}\) and solve for the \(b_{n}{ }^{\prime}\) 's for each \(n\).