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Let's reconstruct this from first principles: Defining the ODE system In the method-of-lines discretization you solve an ODE system $\dot U=F(U)$, $U=(U_0,U_1,...,U_{M+1})$, $U_k(t)=u(x_k,t)$, and similarly $F=(F_0,F_1...,F_{M+1})$. Because of the boundary conditions $$u(x, 0) = 40 · x^2 · (1 - x) / 3 \\ u(0, t) = u(1, t) = 0$$ $U_0=U_{M+1}=0$ and ...