Jódar, L.; Navarro, E.; Martin, J. A. Exact and analytic-numerical solutions of strongly coupled mixed diffusion problems. (English) Zbl 0949.35033 Proc. Edinb. Math. Soc., II. Ser. 43, No. 2, 269-293 (2000). Coupled partial differential systems with coupled boundary conditions are frequent in quantum mechanical scattering problems, chemical physics, thermoelastoplastic modelling, coupled diffusion problems and other fields. In this paper systems of the type \[ \begin{alignedat}{2} u_t(x,t)-A u_{xx}(x,t)&=0, &\quad &0<x<1,\;t>0,\\ A_1 u(0,t) + B_1 u_x(0,t)& =0, &\quad & t>0,\\ A_2 u(1,t) + B_2 u_x(1,t)&=0, &\quad &t>0,\\ u(x,0)&=f(x), &\quad &0\leq x \leq 1,\end{alignedat} \] are considered, where the unknown \(u=(u_1,u_2,\dots,u_m)^T\) and \(f=(f_1,f_2,\dots,f_m)^T\) are \(m\)-dimensional vectors, \(A_i, B_i, i=1,2\) are \(m \times m\) complex matrices, where no simultaneous diagonalizable hypothesis is assumed, and \(A\) is a positive stable matrix, such that \( \text{Re}(z)>0\) for all eigenvalues \(z\) of \(A\). The construction of exact and analytical-numerical solutions with apriori bounds is shown. Given an admissible error \(\varepsilon\) and a bounded subdomain \(D\), an approximate solution whose error with respect to an exact series solution is less than \(\varepsilon\) uniformly in \(D\) is constructed. Reviewer: Angela Handlovičová (Bratislava) Cited in 1 ReviewCited in 13 Documents MSC: 35C10 Series solutions to PDEs 15A24 Matrix equations and identities 35M10 PDEs of mixed type Keywords:coupled diffusion problem; coupled boundary conditions; vector boundary differential system; analytic-numerical solution; Moore-Penrose pseudoinverse PDFBibTeX XMLCite \textit{L. Jódar} et al., Proc. Edinb. Math. Soc., II. Ser. 43, No. 2, 269--293 (2000; Zbl 0949.35033) Full Text: DOI References: [1] DOI: 10.1063/1.444610 [2] DOI: 10.1016/0898-1221(95)00071-6 · Zbl 0839.65105 [3] Mikhailov, Unifield analysis and solutions of heat and mass diffusion (1984) [4] Dunford, Linear operators (1957) [5] Crank, The mathematics of diffusion (1995) [6] Campbell, Generalized inverses of linear transformations (1979) · Zbl 0417.15002 [7] Axelsson, Iterative solution methods (1994) · Zbl 0795.65014 [8] Atkinson, Discrete and continuous boundary value problems (1964) · Zbl 0117.05806 [9] Apostol, Mathematical analysis (1977) [10] DOI: 10.1063/1.452154 [11] DOI: 10.1016/0021-9991(84)90115-3 · Zbl 0531.65079 [12] DOI: 10.1063/1.445338 [13] DOI: 10.1063/1.1673214 [14] DOI: 10.1016/0898-1221(95)00030-3 · Zbl 0831.65102 [15] Inge, Ordinary differential equations (1927) [16] Coddington, Theory of ordinary differential equations (1967) · Zbl 0179.46903 [17] Hueckel, Numerical methods in transient and coupled problems pp 213– (1987) [18] Golub, Matrix computation (1989) [19] Folland, Fourier analysis and its applications (1992) · Zbl 0786.42001 [20] Stakgold, Green’s functions and boundary value problems (1979) · Zbl 0421.34027 [21] DOI: 10.1063/1.438450 [22] Reid, Ordinary differential equations (1971) · Zbl 0212.10901 [23] Rao, Generalized inverse of matrices and its applications (1971) · Zbl 0236.15004 [24] DOI: 10.1016/0377-0427(92)90222-J · Zbl 0747.65070 [25] Pryce, Numerical solution of Sturm–Liouville problems (1993) · Zbl 0795.65053 [26] DOI: 10.1137/1020098 · Zbl 0395.65012 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.