Finding the Zeros of a High-Degree Polynomial Sequence

Abstract

A 1-parameter initial-boundary value problem for a linear spatially 1-dimensional homogeneous degenerate wave equation, posed in a space-time rectangle, in case of strong degeneracy, was reduced to a linear integro-differential equation of convolution type (JODEA, 29(1) (2021), pp. 1–31). The former was then solved by applying the Laplace transformation, and the solution formula was inverted in the form of the Neumann series. The current study deals with an other approach to the inversion of the solution formula, based on invoking the Bromwich integral and the Cauchy residue theorem for the integrand. The denominator of the integrand being an infinite series with respect to rational functions of the complex variable, converges quite rapidly and can be approximated with finite series of m terms. Therefore finding the zeros of the approximated denominator reduces to finding the zeroes of a polynomial of degree 2m. For the resulting polynomial sequence some numerical approaches have been applied.

Keywords

• degenerate wave equation
• linear integro-differential equation of convolution type
• laplace transformation