[Paper Review] Global Solution to a Nonlinear Fractional Differential Equation for the Caputo-Fabrizio Derivative

Sabrina Roscani, Domingo A. Tarzia+1 more|Conicet|Sep 6, 2018

Fractional Differential Equations Solutions1 references3 citations

TL;DR

This paper establishes the existence and uniqueness of a global solution to a nonlinear fractional differential equation using the Caputo-Fabrizio derivative, which features a non-singular exponential kernel. By extending a local solution via iterative application of a contraction mapping principle under a Lipschitz condition with $ L < \frac{1}{1-\alpha} $, the authors prove the solution exists for all time, with convergence to the classical first derivative as $ \alpha \nearrow 1 $. The analysis includes explicit computations of the derivative for power, trigonometric, and exponential functions using Mittag-Leffler functions.

ABSTRACT

This paper deals with the fractional Caputo--Fabrizio derivative and some basic properties related. A computation of this fractional derivative to power functions is given in terms of Mittag--Lefler functions. The inverse operator named the fractional Integral of Caputo--Fabrizio is also analyzed. The main result consists in the proof of existence and uniqueness of a global solution to a nonlinear fractional differential equation, which has been solved previously for short times by Lozada and Nieto (Progr. Fract. Differ. Appl., 1(2):87--92, 2015). The effects of memory as well as the convergence of the obtained results when $\al earrow 1$ (and the classical first derivative is recovered) are analyzed throughout the paper.

Motivation & Objective

• To establish the existence and uniqueness of a global solution to a nonlinear fractional differential equation involving the Caputo-Fabrizio derivative.
• To analyze the behavior of the Caputo-Fabrizio derivative in the limit as $ \alpha \nearrow 1 $, showing convergence to the classical first derivative.
• To provide explicit formulas for the Caputo-Fabrizio derivative of power functions, sine, cosine, and exponential functions in terms of Mittag-Leffler functions.
• To investigate the memory effects inherent in the Caputo-Fabrizio derivative, particularly the non-local dependence on the entire history of the function.
• To extend a previously known local solution to a global solution by iteratively solving sub-problems on successive time intervals.

Proposed method

• The authors use the Caputo-Fabrizio derivative with a non-singular exponential kernel: $ {}^{CF}D^\alpha f(t) = \frac{1}{1-\alpha} \int_0^t f'(\tau) e^{-\frac{\alpha(t-\tau)}{1-\alpha}} d\tau $.
• They derive explicit expressions for the derivative of $ t^n $, $ \sin(at) $, $ \cos(at) $, and $ e^{bt} $ in terms of Mittag-Leffler functions.
• The inverse operator, the fractional integral of Caputo-Fabrizio, is analyzed to support the solution framework.
• A contraction mapping principle is applied iteratively on successive time intervals $[T_k, T_{k+1}]$ to extend a local solution globally.
• The solution is constructed by solving sub-problems of the form $ {}^{CF}D^\alpha f(t) = \Phi_k(t, f(t)) $ with modified source terms that account for memory effects.
• The global solution is established under the condition $ L < \frac{1}{1-\alpha} $, ensuring the contraction mapping applies at each step.

Experimental results

Research questions

• RQ1 Does a global solution exist for a nonlinear fractional differential equation involving the Caputo-Fabrizio derivative, given a local solution exists?
• RQ2 How does the Caputo-Fabrizio derivative behave as $ \alpha \nearrow 1 $, and does it converge to the classical first derivative?
• RQ3 Can the Caputo-Fabrizio derivative of elementary functions like $ t^n $, $ \sin(at) $, $ \cos(at) $, and $ e^{bt} $ be expressed in terms of Mittag-Leffler functions?
• RQ4 What is the role of memory in the Caputo-Fabrizio derivative, and how does it affect the initial conditions required for solving the equation?
• RQ5 Under what conditions can a local solution be extended to a global solution over $ \mathbb{R}^+ $?

Key findings

• The global solution to the nonlinear fractional differential equation $ {}^{CF}D^\alpha f(t) = \varphi(t, f(t)) $ exists and is unique when the Lipschitz constant $ L $ of $ \varphi $ satisfies $ L < \frac{1}{1-\alpha} $.
• The solution is constructed iteratively by solving sub-problems on intervals $[T_k, T_{k+1}]$ with $ T_{k+1} - T_k < \frac{1 - (1-\alpha)L}{\alpha L} $, ensuring convergence at each step.
• As $ \alpha \nearrow 1 $, the Caputo-Fabrizio derivative converges pointwise to the classical first derivative, and the solution converges to the classical solution of the ODE.
• The derivative of $ t^n $ under the Caputo-Fabrizio operator is expressed as a combination of Mittag-Leffler functions, explicitly computed in the paper.
• The derivative of $ \sin(at) $, $ \cos(at) $, and $ e^{bt} $ is also computed in terms of Mittag-Leffler functions, showing the operator's tractability for elementary functions.
• The memory effect is evident in the solution process: the initial condition for each sub-problem depends on the full history up to the previous time point, not just the value at the boundary.