TY - JOUR
T1 - Matrix difference equations in applied mathematics
AU - Zabarankin, Michael
AU - Grechuk, Bogdan
N1 - Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics.
PY - 2020
Y1 - 2020
N2 - In various fields of applied mathematics, e.g., electrostatics, heat conduction, fluid mechanics, elastostatics, etc., boundary-value problems involving regions described in spheroidal, toroidal, and bispherical coordinate systems reduce to a system of second-order difference equations, whose solution, { x/n} ∞ n=0 with x/n in Rm, should vanish asymptotically, i.e., limn→ ∞ xn = 0. There are several methods for constructing such { x/n} ∞ n=0. However, in general, those methods do not guarantee limn→ ∞ xn = 0. Moreover, in actual computations, they yield an approximate solution { ˆ x/n} Nn =0 different from the truncated true solution { x/n} Nn =0 and coinciding with the solution of the system being truncated at N with xN+1 set to 0. This work establishes sufficient conditions for the existence of an asymptotically vanishing solution to the system and provides the rate of convergence of the solution to the truncated system. Those results are used to analyze systems of second-order difference equations arising in the boundary-value problems in electrostatics, heat conduction, fluid mechanics, and elastostatics when a medium contains an inhomogeneity having the shape of either a torus or two unequal spheres.
AB - In various fields of applied mathematics, e.g., electrostatics, heat conduction, fluid mechanics, elastostatics, etc., boundary-value problems involving regions described in spheroidal, toroidal, and bispherical coordinate systems reduce to a system of second-order difference equations, whose solution, { x/n} ∞ n=0 with x/n in Rm, should vanish asymptotically, i.e., limn→ ∞ xn = 0. There are several methods for constructing such { x/n} ∞ n=0. However, in general, those methods do not guarantee limn→ ∞ xn = 0. Moreover, in actual computations, they yield an approximate solution { ˆ x/n} Nn =0 different from the truncated true solution { x/n} Nn =0 and coinciding with the solution of the system being truncated at N with xN+1 set to 0. This work establishes sufficient conditions for the existence of an asymptotically vanishing solution to the system and provides the rate of convergence of the solution to the truncated system. Those results are used to analyze systems of second-order difference equations arising in the boundary-value problems in electrostatics, heat conduction, fluid mechanics, and elastostatics when a medium contains an inhomogeneity having the shape of either a torus or two unequal spheres.
KW - Biharmonic equation
KW - Bispherical coordinates
KW - Harmonic equation
KW - Second-order difference equation
KW - Toroidal coordinates
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U2 - 10.1137/19M1256671
DO - 10.1137/19M1256671
M3 - Article
AN - SCOPUS:85084506173
SN - 0036-1399
VL - 80
SP - 753
EP - 771
JO - SIAM Journal on Applied Mathematics
JF - SIAM Journal on Applied Mathematics
IS - 2
ER -