How to solve this heat conduction equation $u_t = u_{xx} -2u_x +x +2t, 0 \lt x \lt1, t \gt 0, u \left( 0,t \right )=u \left(1,t \right) = t, u \left( x,0 \right) = e^x\sin \pi x$ ? I tried to reduce this problem on equation with homogeneous boundary conditions, but I can't express Fourier coefficients.
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