Abstract:We develop a unified analytical and computational framework for the generalized Abel ordinary differential equation $y^{\prime }(x)=a_n(x)\bigl(% y^n+\lambda_{n-1}(x)y^{n-1}+\dots+\lambda_0(x)\bigr)$ of arbitrary degree $% n\ge1$ on the unbounded interval $[x_0,\infty)$. Under mild structural hypotheses on the coefficients and on the existence of a stable moving equilibrium branch $E(x)$, we prove a new \emph{Asymptotic Plateau Theorem} establishing that the solution issued from $y(x_0)=0$ is globally defined, strictly monotone, trapped between zero and $E(x)$, and converges to a finite positive limit $L=\lim_{x\to\infty}E(x)$. We further obtain an explicit, computable rate of convergence and a degree-reduction principle that generalizes the classical Liouville substitution. The theory is complemented by a high-order Radau IIA implementation whose output reproduces the predicted plateaus to nine significant digits. A detailed application to a generalized Merton structural credit-risk model derives an Abel-type equation for the long-maturity state profile of the credit spread and illustrates the economic relevance of the framework.
From: Dragos-Patru Covei D.P. [view email]
[v1]
Mon, 4 May 2026 17:07:25 UTC (110 KB)
[v2]
Mon, 18 May 2026 06:30:30 UTC (113 KB)
[v3]
Tue, 23 Jun 2026 18:44:04 UTC (83 KB)
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