Analysis of implicit hyperbolic multivariable systems Wieslaw Marszalek zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Department of Mathematics, North Carolina State University, Raleigh, NC, USA Zdzislaw W. Trzaska Department of Electrical Engineering, W arsaw University of Technology, W arsaw , Poland This paper deals with the coupled zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA distributed parameter systems described by the system of partial difSerentia1 equations (PDEs) with singular matrix coeficients. A basic classification of such systems is provided, and some properties of the systems are analyzed using the spectral theory of matrix polynomial pencils. This theory is used next for analysis of the transient behavior of superconducting energy storage coils. Several illustrative examples are included. and some numerical results showing the distribution of voltages and electric field stresses are presented. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Keywords: implicit systems, poles, finite differences 1. Introduction Recently an interest has been shown in extending some results for singular lumped parameter systems to distributed parameter systems with singular matrix coefficients.1’3 Such systems of coupled second-order partial differential equations appear in a number of practical applications. 4*5 In this paper, we are interested in a study of the distribution of electromagnetic signals along a superconducting energy storage coil, which is a major part of any superconducting magnetic energy storage (SMES) plant. The SMES is inherently very efficient and has siting requirements that are somewhat different from other technologies, such as, for example, compressed air, underground pumped hydro, or batteries. It promises to be an attractive technology for both defense and utility use. One of the important problems appearing in the design of a superconducting coil structure is a good knowledge of its electromagnetic characteristics, especially the strength of the coil insulation on the surge waves excited by energizing processes in the SMES systems. Studies of such phenomena are based on solutions of respective coupled partial differential equations (PDEs) with singular coefficients. Address reprint requests to Dr. W. Marszalek at the Deuartment of Mathemat&, North Carolina State University, Box 8205, Raleigh, NC 27695-8205, USA. Received 11 January 1994; revised 17 December 1994; accepted 16 January 1995 Appl. Math. Modelling 1995, Vol. 19, July 0 1995 by Elsevier Science Inc. 655 Avenue of the Americas, New York, NY 10010 Unlike for the regular PDEs (coefficient matrices are nonsingular6s8), the understanding of the singular PDEs is much more complex and no good theory for classification of such systems exists.‘.” Another problem is to analyze the eigenstructure of the systems using the spectral theory of matrix polynomial pencils. Next, it may be interesting to see if the usual discretization techniques (e.g., finite differences) can be used to solve the singular PDEs numerically. This paper tries to identify and solve some of these problems using the Fourier technique of separation of the solution and properties of the resulting matrix polynomial pencils. We discuss the problems based on the two kinds of singular PDEs, which were already considered in Refs. 3 and 11. An application of the theory for calculation of the voltage and electric field stresses distribution within a superconducting coil due to quenches is presented. 2. Coupled systems of PDEs We shall be concerned here with the analysis of the solution 4x9 t) = G(t)H(x), G(t) E R” ’ “, H(x) E R” (1) of the initial-boundary problem for *A?!!&! zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP ax2 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC at (2) or *‘r=,!?? ax2 at2 (3) 0307-904x/95/$10.00 SSDI 0307-904X(95300013-A