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.
Highlights
Introduction and the problem formulationThe current study supplements our previous publication [1] dealing with the following 1-parameter simplified initial boundary value problem (IBVP) for the degenerate wave equation in the space-time rectangle [0, T ] × [−1, +1] ∂2u(t, x; α) ∂ ∂u(t, x; α) ∂t2 = a(x; α)∂x u(t, −1; α) = h2(t; α), u(t, +1; α) = h1(t; α) ∂u(0, x; α) = ∗u∗(x; α)∂t u(0, x; α) u∗
There are known a lot of techniques to find zeros of analytic functions, in particular, polynomials, in a region Ω, for example [5, 8, 9]
We think the reason for this is that the polynomials grow rapidly with distance from the point τ = 0, whereas the Gibbs phenomenon is negligibly small compared to this growth to produce false zeros and avoid the convergence of the zeros
Summary
The current study supplements our previous publication [1] dealing with the following 1-parameter simplified initial boundary value problem (IBVP) for the degenerate wave equation in the space-time rectangle [0, T ] × [−1, +1]. When implementing the continuity condition to the above families of solutions, we succeeded in deriving a linear integro-differential equation of convolution type with respect to the difference h(t; α) ≡ h3(t; α) − h4(t; α) of the required functions. In the current study we will try to satisfy all the conditions for the application of the Bromwich integral to (1.4) First of all, this means finding the singularities of the right-hand side of (1.4). From our study [1], we know that the above singularities are nothing but the zeros of the denominator of the right-hand side of (1.4) From this there stems our concern in finding the zeros of the function 1 + Q1(τ ; α).
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