V. K. SAUL'EV

ON THE NUMERICAL SOLUTION OF A BOUNDARY-VALUE PROBLEM FOR A SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS

(Presented by Academician S. L. Sobolev on 24 IX 1956)

Let the problem be given

\[ a_i \frac{d^2 u_i}{dx^2}+\sum_{j=1}^{n} b_{i,j}u_j+\lambda \sum_{j=1}^{n} c_{i,j}u_j=0,\quad x\in(0,R),\quad x\ne r_k; \tag{1} \]

\[ u_i(0)=u_i(R)=0; \tag{2} \]

\[ u_i(r_k-0)=u_i(r_k+0); \tag{3} \]

\[ a_i(r_k+0)\frac{du_i(r_k+0)}{dx} - a_i(r_k-0)\frac{du_i(r_k-0)}{dx} = \frac{a_i(r_k+0)-a_i(r_k-0)}{r_k}\,u_i(r_k), \tag{4} \]

\[ 0=r_0<r_1<\cdots<r_k<\cdots<r_{s-1}<r_s=R, \]

\[ k=1,2,\ldots,s-1, \]

\[ i=1,2,\ldots,n, \]

where the unknowns are the vector-functions \(u\{u_1,u_2,\ldots,u_n\}\) and the scalar \(\lambda\); the real coefficients \(a_i, b_{i,j}\), and \(c_{i,j}\) \((i,j=1,2,\ldots,n)\) may have discontinuities of the first kind at the points \(r_k\) \((k=1,2,\ldots,s-1)\).

Divide \([r_{k-1},r_k]\) into \(m_k\) parts; denoting the resulting step by \(h_k\) \(\bigl(h_k=(r_k-r_{k-1})/m_k,\ k=1,2,\ldots,s\bigr)\), we approximate the original problem (1)—(4) by the following algebraic one:

\[ a_{i,k}\frac{\Delta^2 u_{i,k,j}^{(h)}}{h_k^2} + \sum_{p=1}^{n} b_{i,p,k,j}u_{p,k,j}^{(h)} + \lambda^{(h)}\sum_{p=1}^{n} c_{i,p,k,j}u_{p,k,j}^{(h)} =0, \tag{5} \]

\[ j=1,2,\ldots,m_k-1,\quad k=1,2,\ldots,s,\quad i=1,2,\ldots,n; \]

\[ u_{i,1,0}^{(h)}=u_{i,s,m_s}^{(h)}=0,\quad i=1,2,\ldots,n; \tag{6} \]

\[ a_{i,k+1}\frac{\Delta u_{i,k,m_k}^{(h)}}{h_{k+1}} - a_{i,k}\frac{\Delta u_{i,k,m_k-1}^{(h)}}{h_k} = \frac{a_{i,k+1}-a_{i,k}}{r_k}\,u_{i,k,m_k}^{(h)}, \tag{7} \]

\[ k=1,2,\ldots,s-1,\quad i=1,2,\ldots,n, \]

where

\[ \Delta\varphi_j=\varphi_{j+1}-\varphi_j;\qquad \Delta^2\varphi_j=\varphi_{j-1}-2\varphi_j+\varphi_{j+1}; \]

\[ a_{i,k}=a_i(r_{k-1}+jh_k);\qquad \varphi_{k,j}=\varphi(r_{k-1}+jh_k). \]

If we assume that it is always possible to choose a partition of the intervals \([r_{k-1},r_k]\) into \(m_k\) parts such that

\[ h_k=h_{k+1}=h \quad (k=1,2,\ldots,s), \tag{8} \]

then equations (5), under conditions (6) and (7), form a system of linear homogeneous equations of order \(n\sum_{k=1}^{s}(m_k-1)\) with a symmetric matrix.

Let \(\lambda\) and \(\lambda^{(h)}\) be the first, i.e., the smallest in absolute value, eigenvalues of problems (1)—(4) and (5)—(7), respectively.

Theorem *. If, for all \(i, j, k\) satisfying the inequalities \(1\leq i,j\leq n,\ 1\leq k\leq s\), the following conditions hold: 1) \(a_i\) are constant on \([r_{k-1},r_k]\); 2) \(b_{i,j}, c_{i,j}\) are twice continuously differentiable on \([r_{k-1},r_k]\); 3) \(a_i>0\); 4) \(b_{i,j}=b_{j,i},\ c_{i,j}=c_{j,i}\) for all \(x\) from \((r_{k-1},r_k)\); 5) for any numbers \(\xi_1,\xi_2,\ldots,\xi_n\) and all \(x\) from \((r_{k-1},r_k)\) one has

\[ \sum_{i,j=1}^{n} c_{i,j}\xi_i\xi_j \geq \alpha \sum_{i=1}^{n}\xi_i^2,\qquad \alpha=\operatorname{const}>0, \]

and also conditions (8), then as \(h\to 0\)

\[ |\lambda-\lambda^{(h)}|<ch^2, \tag{9} \]

where \(c\) is a constant independent of \(h\).

To prove the theorem, first of all the inequalities are established

\[ \lambda \leq \frac{D[u,u]}{I[u,u]},\qquad \lambda^{(h)} \leq \frac{D_h[u^{(h)},u^{(h)}]}{I_h[u^{(h)},u^{(h)}]} \tag{10} \]

for arbitrary functions \(u\) and \(u^{(h)}\) satisfying conditions (2)—(4) and (6)—(7), respectively. Here we have put

\[ D[u,v]= \sum_{i=1}^{n}\sum_{k=1}^{s}\int_{r_{k-1}}^{r_k} a_i\frac{du_i}{dx}\frac{dv_i}{dx}\,dx -\sum_{i,l=1}^{n}\sum_{k=1}^{s}\int_{r_{k-1}}^{r_k} b_{i,l}u_i v_l\,dx+ \]

\[ +\sum_{i=1}^{n}\sum_{k=1}^{s-1} \frac{a_i(r_k+0)-a_i(r_k-0)}{r_k}\,u_i(r_k)v_i(r_k), \]

\[ D_h[u^{(h)},v^{(h)}]= \sum_{i=1}^{n}\sum_{k=1}^{s} a_{i,k}h \sum_{j=0}^{m_k-1} \frac{\Delta u_{i,h,j}^{(h)}}{h}\, \frac{\Delta v_{i,h,j}^{(h)}}{h} \]

\[ -\sum_{i,l=1}^{n}\sum_{k=1}^{s}\sum_{j=0}^{m_k-1} b_{i,l,k,j}u_{i,k,j}^{(h)}v_{l,k,j}^{(h)} +\sum_{i=1}^{n}\sum_{k=1}^{s-1} \frac{a_{i,k+1}-a_{i,k}}{r_k}\, u_{i,h,m_k}^{(h)}v_{i,h,m_k}^{(h)}, \]

\[ I[u,v]= \sum_{i,l=1}^{n}\sum_{k=1}^{s}\int_{r_{k-1}}^{r_k} c_{i,l}u_i v_l\,dx; \]

\[ I_h[u^{(h)},v^{(h)}]= \sum_{i,l=1}^{n}\sum_{k=1}^{s} h\sum_{j=0}^{m_k-1} c_{i,l,k,j}u_{i,h,j}^{(h)}v_{l,h,j}^{(h)}. \]

Equality in (10) occurs only in the cases when \(u\) and \(u^{(h)}\) are the first eigenfunctions of problems (1)—(4) and (5)—(7), respectively.

Further, starting from the first eigenfunction \(u\) of problem (1)—(4), which is defined, in particular, at the nodes \(r_{k-1}+jh\) \((k=1,2,\ldots,s;\)

* This theorem is analogous to the corresponding theorem of Collatz (1) for the case of a single equation \((n=1)\) and absence of matching \((s=1)\).

\(j=0,1,\ldots,m_k-1\)), the difference is estimated as

\[ \lambda^{(h)}-\lambda \leq \frac{D_h[u,u]}{I_h[u,u]}-\frac{D[u,u]}{I[u,u]}+O(h^2)<c_1h^2. \tag{11} \]

Similarly, proceeding from the first difference eigenfunction \(u^{(h)}\) of problem (5)—(7), extended linearly on \((r_{k-1}+jh,\ r_{k-1}+(j+1)h)\) \((j=0,1,\ldots,m_k-1;\ k=1,2,\ldots,s;\ i=1,2,\ldots,n)\):

\[ \bar u_i(x)=u^{(h)}_{i,k,j}+ \frac{\Delta u^{(h)}_{i,k,j}}{h}\,[x-(r_{k-1}+jh)], \]

we establish the estimate

\[ \lambda-\lambda^{(h)} \leq \frac{D[\bar u,\bar u]}{I[\bar u,\bar u]}- \frac{D_h[u^{(h)},u^{(h)}]}{I_h[u^{(h)},u^{(h)}]}<c_2h^2. \tag{12} \]

Since \(c_1\) and \(c_2\) do not depend on \(h\) (to prove uniform boundedness in \(h\) of \(c_2\), it was necessary, in particular, to use Courant’s inequality \((^2)\), § 12), the theorem follows immediately from (11) and (12).

We note that, although the first derivatives in the matching conditions \((4)\) are approximated in the crudest way, namely with accuracy \(O(h)\), the final error (see (9)) in the solution is majorized by a quantity of order \(O(h^2)\). This circumstance does not occur, for example, in the Neumann problem: when the Laplace equation is approximated at interior nodes with accuracy \(O(h^2)\), and the normal derivative at boundary nodes with accuracy \(O(h)\), the error in the solution is determined by the cruder approximation and is \(O(h)\) \((^3)\).

Received
26 VI 1956

CITED LITERATURE

\(^1\) L. Collatz, Deutsch. Math., 2, 189 (1937).
\(^2\) R. Courant, Acta Math., 49, 1 (1926).
\(^3\) E. Batschelet, Zs. angew. Math. u. Phys., 3, 165 (1952).