Solution of two-point boundary value problems involving Kronecker products. (English) Zbl 0765.65084
The authors consider a two-point boundary value problem of the form \((P\otimes Q)y'+(R\otimes S)y=f(t,y(t))\), \((M\otimes N)y(a)+(U\otimes V)y(b)=a\), where \(P\), \(R\), \(M\), \(U\) are \(m\times n\) matrices, \(Q\), \(S\), \(V\), \(N\) are \(p\times q\), \(y\) is a column matrix of order \(nq\times 1\), \(a\) is \(mp\times 1\), and \(f: [a,b]\times \mathbb{R}^{nq} \to \mathbb{R}^{nq}\) is continuous.
The solution of the Kronecker product nonhomogeneous boundary value problem is first obtained in the form of an integral transform involving a Green matrix. The unique solution of the two-point boundary value problem is constructed by using Banach’s fixed-point theorem. A solution algorithm using the modified Gram-Schmidt process of J. R. Rice [Math. Comput. 20, 325-328 (1966; Zbl 0228.65034)] is presented.
The solution of the Kronecker product nonhomogeneous boundary value problem is first obtained in the form of an integral transform involving a Green matrix. The unique solution of the two-point boundary value problem is constructed by using Banach’s fixed-point theorem. A solution algorithm using the modified Gram-Schmidt process of J. R. Rice [Math. Comput. 20, 325-328 (1966; Zbl 0228.65034)] is presented.
Reviewer: P.Chocholatý (Bratislava)
MSC:
65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
34B15 | Nonlinear boundary value problems for ordinary differential equations |
Keywords:
two-point boundary value problem; Kronecker product; integral transform; Green’s matrix; Banach’s fixed-point theorem; algorithm; Gram-Schmidt processCitations:
Zbl 0228.65034References:
[1] | Cole, R. H., Theory of Ordinary Differential Equations (1968), Appleton-Century-Crofts · Zbl 0206.37402 |
[2] | Brewer, J. W., Kronecker products and matrix calculus in system theory, (IEEE Trans. Circ. Sys., CAS-25 (1978)), 772-781 · Zbl 0397.93009 |
[3] | Golub, G. H.; Van Loan, C. F., Matrix Computations (1985), Johns Hopkins University Press: Johns Hopkins University Press Baltimore, Maryland · Zbl 0592.65011 |
[4] | Rice, J. R., Experiments on Gram-Schmidt orthogonalization, Math. Comp., 20, 325-328 (1966) · Zbl 0228.65034 |
[5] | Bjorck, A., Solving linear least-squares problems by Gram-Schmidt orthogonalization, Bit, 7, 1-21 (1967) · Zbl 0183.17802 |
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