[Paper Review] Convergence of solutions of fractional differential equations to power-type functions

Mohammed D. Kassim, Nasser‐eddine Tatar|arXiv (Cornell University)|Nov 18, 2020

Nonlinear Differential Equations Analysis4 citations

TL;DR

This paper investigates the asymptotic behavior of solutions to nonlinear fractional differential equations with Caputo derivatives, proving that solutions converge to power-type functions under specific growth conditions on the nonlinearity. Using fractional integral transforms, weighted norm estimates, and a novel Caputo fractional version of L’Hôpital’s rule, the authors establish convergence to $ \tau^{\beta} $-type functions for $ 0 < \beta < \alpha < 1 $, generalizing classical results for ordinary differential equations.

ABSTRACT

In this article we study the asymptotic behavior of solutions of some fractional differential equations. We prove convergence to power type functions under some assumptions on the nonlinearities. Our results extend and generalize some existing well-known results on solutions of ordinary differential equations. Appropriate estimations and lemmas such as a fractional version of L'Hopital's rule are used.

Motivation & Objective

• To analyze the long-term behavior of solutions to fractional differential equations with Caputo derivatives.
• To extend classical asymptotic results from ordinary differential equations to the fractional setting.
• To establish sufficient conditions under which solutions converge to power-type functions $ \tau^{\beta} $.
• To develop and apply a fractional version of L’Hôpital’s rule for asymptotic analysis.
• To generalize boundedness and convergence results under nonlinear growth constraints on the right-hand side.

Proposed method

• Transform the fractional initial value problems into equivalent integral equations using the Caputo fractional integral.
• Apply weighted $ L^q $-norm estimates and Hölder’s inequality with conjugate exponents $ p, q $ to control nonlinear terms.
• Use a newly derived fractional version of L’Hôpital’s rule to analyze the asymptotic behavior of solutions.
• Employ comparison theorems and Gronwall-type inequalities in weighted function spaces to bound solutions and their fractional derivatives.
• Introduce auxiliary functions $ z(\tau) $ to dominate $ |x(\tau)| $ and $ |{}^{C}\mathfrak{D}_{0}^{\beta}x(\tau)| $, enabling the use of integral inequalities.
• Verify integrability conditions on the function $ h(\tau) $ and the growth functions $ \varphi_1, \varphi_2 $ to ensure boundedness of solutions.

Experimental results

Research questions

• RQ1 Under what conditions do solutions of fractional differential equations with Caputo derivatives converge to power-type functions?
• RQ2 How can a fractional version of L’Hôpital’s rule be constructed and applied to analyze asymptotic behavior?
• RQ3 What growth restrictions on the nonlinearity ensure boundedness and convergence of solutions?
• RQ4 How do the orders $ \alpha $ and $ \beta $ of the fractional derivatives influence the asymptotic profile of solutions?
• RQ5 Can classical asymptotic results for ODEs be generalized to the fractional case with Caputo derivatives?

Key findings

• Solutions of the initial value problem $ {}^{C}\mathfrak{D}_{0}^{\alpha}x(\tau) = f(\tau, x(\tau), {}^{C}\mathfrak{D}_{0}^{\beta}x(\tau)) $ converge to $ c\tau^{\beta} $ as $ \tau \to \infty $, for $ 0 < \beta < \alpha < 1 $, under appropriate growth conditions.
• For the problem $ {}^{C}\mathfrak{D}_{0}^{\alpha}x(\tau) = f(\tau, x(\tau), {}^{C}\mathfrak{D}_{0}^{\beta}x(\tau)) $, solutions are bounded and satisfy $ |x(\tau)| \leq C $, $ |{}^{C}\mathfrak{D}_{0}^{\beta}x(\tau)| < C $ for some constant $ C $, provided $ \int_{\xi_0}^{\infty} \frac{ds}{\varphi_1^q(s^{1/q})\varphi_2^q(s^{1/q})} = \infty $.
• A new fractional version of L’Hôpital’s rule is established and used to analyze the asymptotic behavior of ratios of functions involving fractional derivatives.
• The method relies on transforming the differential equation into an integral equation and applying Hölder’s inequality with carefully chosen weights to control nonlinearities.
• The results generalize known asymptotic behaviors from ordinary differential equations (e.g., $ x(\tau) \sim c\tau + b $) to fractional-order systems with power-type convergence.
• An example is provided where $ \alpha = 2/3 $, $ \beta = 1/3 $, and the nonlinearity involves exponential decay and power functions, confirming boundedness and convergence via the main theorems.