A Simple Solution for the General Fractional Ambartsumian Equation
Abstract
:1. Introduction
2. On the Fractionalisation Problem
- not distinguishing between constant and Heaviside functions;
- not giving the derivative of the ;
- the derivative of a sinusoid is not a sinusoid;
- (Riemann–)Liouville derivative [30]The usual Riemann–Liouville derivative is a particular case of this one, obtained for functions null outside any interval .
- Liouville–Caputo derivative [25]The Caputo derivative is a particular case of this one as happens with the RL derivative.
- Regularized Liouville derivative [40]
- Linearity;
- Additivity and Commutativity of the orders;
- Neutral and inverse elementsFrom (10) we conclude that there is always an inverse element; that is, for every there is always the order that we called above the anti-derivative.
- Backward compatibility ()If , then:We obtain this expression repeating the first order derivative.
- The generalized Leibniz ruleThis rule gives the FD of the product of two functions and assumes the format [30]
3. Problem Formulation and a First Solution
4. Solution Using Mittag–Leffler Functions
5. Numerical Aspects
- Use the bilinear transformation [8]
- Set , to get a discrete-time Fourier transform of [39]. To avoid the effect of the branchcut points at , we found it better to move them slightly into the unit circle. This is accomplished by settingWe can use a simple procedure that consists of the following steps:
- (a)
- Set , where represents the discrete Fourier transform that is implemented by the fast Fourier transform algorithm;
- (b)
- Similarly, set ;
- (c)
- Compute ;
- (d)
- Obtain the approximation to
- Sample in a uniform grid, making , where N is in high enough agreement with T and the interval where we want to compute the function. In our simulations, we used and .
- We could use the fast Fourier transform to compute the inverse of , . However, this is not very good from numerical aspects, due to the singularity at the origin. We found it was better to separate the computation in three steps:
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Ortigueira, M.D.; Bengochea, G. A Simple Solution for the General Fractional Ambartsumian Equation. Appl. Sci. 2023, 13, 871. https://doi.org/10.3390/app13020871
Ortigueira MD, Bengochea G. A Simple Solution for the General Fractional Ambartsumian Equation. Applied Sciences. 2023; 13(2):871. https://doi.org/10.3390/app13020871
Chicago/Turabian StyleOrtigueira, Manuel Duarte, and Gabriel Bengochea. 2023. "A Simple Solution for the General Fractional Ambartsumian Equation" Applied Sciences 13, no. 2: 871. https://doi.org/10.3390/app13020871
APA StyleOrtigueira, M. D., & Bengochea, G. (2023). A Simple Solution for the General Fractional Ambartsumian Equation. Applied Sciences, 13(2), 871. https://doi.org/10.3390/app13020871