By J.M. Ball
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Additional resources for Systems of Nonlinear Partial Differential Equations (NATO Science Series C: (closed))
For the polynomial-matrix determinants, the reason for the property of being ill-posed is quite clear: this property arises wherever, with nominal values of coefficients, the terms with the highest degree of λ cancel. It is clear that, even with arbitrarily small values of coefficients, we have no such cancellation; as a result, under small parameter variations, the degree of the polynomial of λ in the determinant undergoes changes; as a result, another polynomial root emerges. Nevertheless, the fact that calculation of determinants of some polynomial matrices presents an ill-posed problem means that some of even more important and commonly encountered problems, problems on solving systems of ordinary differential equations, are also ill-posed problems.
Yet, for high-order systems wherein the total number of coefficients is large, this simple method becomes too labor-consuming. All the aforesaid suggest a simple criterion that permits easy identification of suspicious systems that may be ill-conditioned: the highest term Chapter 1. Simplest ill-posed problems 29 of the characteristic polynomial of such systems (or the highest term and several terms of lesser degree) is substantially smaller in value than other terms. ) indicative of the fact that the higher term of the characteristic polynomial has emerged as a small difference between large terms in the initial system, and small variations of these terms may give rise to large (relative) changes in the highest term or may make the term to change its sign for the opposite.
This statement concerns the vicinity of the system, contending that other systems close (but not identical) to the initial system in the space of parameters has solutions that all are stable. When we say that the solution of the equation x˙ + x = 0 is parametrically stable, and the problem of stability prediction for this equation is a well-posed problem, this statement is equivalent to the statement that all solutions of the family of equations (1 + ε1 )x˙ + (1 + ε2 )x = 0, which, with small ε1 and ε2 , represent a vicinity of the equation x˙ + x = 0, are stable.
Systems of Nonlinear Partial Differential Equations (NATO Science Series C: (closed)) by J.M. Ball