Fourier Serier and boundary Value Problems

[SoLuTioN]

จงใช้ method of variation of parameters เพื่อ solve หา $u(x,t)$ จาก Initial Boundary Value Problem

(1.1)$\qquad$PDE$\qquad$ $u_t(x,t)\,=\,ku_{xx}(x,t)\,+\,g(t)\quad,\quad 0 <x<1 \quad , t>0$

(1.2)$\qquad$B.C.$\qquad$ $t>0 , \, \cases{u_x(0 ,t)\,=\,0\cr u(1,t)\,=\,0}$

(1.3)$\qquad$I.C.$\qquad$ $u(x,0)\,=\,0 \quad,\quad 0<x<1$