Matrix difference equations in applied mathematics

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.