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We are asked to solve the initial-boundary value problem (IBVP) [ u_t = 2 u_{xx}, \quad 0 < x < 1, \ t>0, ] [ u(0,t) = 1, \quad u(1,t) = 1, \quad t>0, ] [ u(x,0) = 1 + \sin(\pi x), \quad 0 < x < 1, ] using the Laplace transform method. --- Step 1: Laplace transform in (t) Let (\bar{u}(x,s) = \mathcal{L}{u(x,t)} = \int_0^\infty e^{-st} u(x,t), dt). The Laplace transform of (u_t) gives [ \mathcal{L}{u_t} = s \bar{u}(x,s) - u(x,0) = s \bar{u}(x,s) - (1 + \sin(\pi x)). ] The PDE transforms to [ s \bar{u} - (1 + _____ __________ _____ _______ ______ __________ ____ _________ ________ _____.
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