A temperature field T(x,t) in a one-dimensional structure (x = [0, L]) is determined by 1 ar 8T the following governing equation Based on previous experience, one a t proposes that the temperature field T(x,t) can be constructed as a form of expansion of cosine function cos(x): T(x,1)= cos(2x)T(2,1). (1) K-0 where An = (2n+1)z/(2L) (n=0, 1, 2, ...) are known. T(,1) can be viewed as the corresponding coefficients of the expansion in terms of cos(x). Using the orthogonality property of cos(x) in domain x = [0, L], one obtains [T(x,1) cos(2x)dx T(2,1)= (2) cos (2,x)dx Shown in Eq. 1, the only thing unknown at this moment is the coefficient inside the expansion. One needs to solve for T(21) to finally determine the temperature field T(x, t). Please follow the steps described below: dT(2,1) dt inside the integration in Eq. 2. (B) Then based on the governing equation (Eq. 1), replace inside the integration (A) Derive the expression for ar from Eq. 2, you should end up with a ar t 0T in Eq. 2 with a You should end up with an integration with inside. (Please note that t 2T du (C) You then rearrange it using integration by parts twice to the derived integration in (B). In procedures of integration by parts, you will see an integration is modified as fudx = uvla-fv- dx, leave the first term without further derivation, continue to work on the 2nd term via performing integration by parts again. dx (D) Eventually, you should end up with an ordinary differential equation for T(2,1). If not, continue to check your derivations. at (x,t) ox may not be equal to - A sin(x)7(2)!!! Just follow the guideline mentioned, even if you may not understand completely why I ask you to do