Cubic spline technique for solution of Burgers' equation with a semi-linear boundary condition

By using the splitting-up technique and cubic spline method, numerical solutions of Burgers' equation ut = uxx + ϵuux, 0 0 and u(x, 0) known, are obtained constructively by taking either of the two pairs of the semi-linear boundary conditions: (i) u(0, t) = 0, ux(1, t) = aup(1, t), t > 0; (ii) u(1, t) = 0, ux(0, t), = - aup(0, t), where 0 < p < ∞. The stability analysis of the proposed method is discussed.