The Spline-Collocation and the Spline-Galerkin Methods for Orr-Sommerfeld Problem

Recently the projectional mesh methods of solving the boundary value problems for ordinary differential equa­tions are intensively developing. The theoretical and prac­tical aspects of the spline-collocation method have been examined by R.D.Russell and L.F.Shampine [15] . C. de Boor and B.Swartz [5] have studied a question of choosing the points in the spline-collocation method. They have shown that the greatest rate will be achieved by the choice of the Gaussian points as the collocation points. B.P.Kolobov and A.G.Sleptsov [11,12] have made use of the following method for solving the problem 2$$ Ly = f,\left( {{B_o}y} \right)\left( a \right) = 0,\left( {{B_1}y} \right)\left( b \right) = 0 $$ where L is linear mth order differential operator. Let x₁=a < x₂ <… < x_N+1=b, n > 0 is some interger. For each k=1, … , N−m the approximate solution will be a polynomial of the degree m+n−1 on the segment [x_k,x_k+m] . m coefficients of this polynomial are determined by the conditions u_k(x_i)=v_i, i=k, … , k+m−1, the rest of the coefficients are determined by the collocation equations.The equalities v_k+m=u_k(x_k+m), k=1, … , N−m, give N−m linear equations to determine the N unknowns v_i. The boundary conditions yield m more equations.