Consider elementary integer-order calculus. Below is an integration and differentiation using the example function :
Now, on exchanging the order of composition:
Where C is the constant of integration. Even if it was not obvious, the initialized condition ƒ'(0) = C, ƒ''(0) = D, etc. could be used. If we neglected those initialization terms, the last equation would show the composition of integration, and differentiation (and vice versa) would not hold.
Working with a properly initialized differ integral is the subject of initialized fractional calculus. If the differ integral is initialized properly, then the hoped-for composition law holds. The problem is that in differentiation, information is lost, as with C in the first equation.
However, in fractional calculus, given that the operator has been fractionalized and is thus continuous, an entire complementary function is needed. This is called complementary function .
The Fractional Calculus of Sets (FCS), first introduced in the article titled "Sets of Fractional Operators and Numerical Estimation of the Order of Convergence of a Family of Fractional Fixed-Point Methods" [1], is a methodology derived from fractional calculus[2]. The primary concept behind FCS is the characterization of fractional calculus elements using sets due to the plethora of fractional operators available.[3][4][5] This methodology originated from the development of the Fractional Newton-Raphson method[6] and subsequent related works [7][8][9].
Illustration of some lines generated by the fractional Newton–Raphson method for the same initial condition but with different orders of the implemented fractional operator. Source: Applied Mathematics and Computation
Fractional calculus, a branch of mathematics dealing with derivatives of non-integer order, emerged nearly simultaneously with traditional calculus. This emergence was partly due to Leibniz's notation for derivatives of integer order: . Thanks to this notation, L'Hopital was able to inquire in a letter to Leibniz about the interpretation of taking in a derivative. At that moment, Leibniz couldn't provide a physical or geometric interpretation for this question, so he simply replied to L'Hopital in a letter that "... is an apparent paradox from which, one day, useful consequences will be drawn".
The name "fractional calculus" originates from a historical question, as this branch of mathematical analysis studies derivatives and integrals of a certain order . Currently, fractional calculus lacks a unified definition of what constitutes a fractional derivative. Consequently, when the explicit form of a fractional derivative is unnecessary, it is typically denoted as follows:
Fractional operators have various representations, but one of their fundamental properties is that they recover the results of traditional calculus as . Considering a scalar function and the canonical basis of denoted by , the following fractional operator of order is defined using Einstein notation[10]:
Denoting as the partial derivative of order with respect to the -th component of the vector , the following set of fractional operators is defined:
Considering a function and the following set of fractional operators:
Then, taking a ball , it is possible to define the following set of fractional operators:
which allows generalizing the expansion in Taylor series of a vector-valued function in multi-index notation. As a consequence, the following result can be obtained:
Let be a function with a point such that . Then, for some and a fractional operator , it is possible to define a type of linear approximation of the function around as follows:
which can be expressed more compactly as:
where denotes a square matrix. On the other hand, as and given that , the following is inferred:
As a consequence, defining the matrix:
the following fractional iterative method can be defined:
Illustration of some lines generated by the fractional Newton–Raphson method for the same initial condition but with different orders of the fractional operator implemented. The fractional Newton–Raphson method usually generates lines that are not tangent to the function whose zeros are sought, unlike the classical Newton–Raphson method. Source: MDPI
The use of fractional operators in fixed-point methods has been widely studied and cited in various academic sources. Examples of this can be found in several articles published in renowned journals, such as those appearing in ScienceDirect,[14][15]Springer,[16]World Scientific,[17] and MDPI.[18][19][20][21][22][23][24][25] Studies from Taylor & Francis (Tandfonline),[26]Cubo,[27]Revista Mexicana de Ciencias Agrícolas,[28]Journal of Research and Creativity,[29]MQR,[30] and Актуальные вопросы науки и техники.[31] These works highlight the relevance and applicability of fractional operators in problem-solving.
^Torres-Hernandez, A.; Brambila-Paz, F.; Ramirez-Melendez, R. (2022). "Sets of Fractional Operators and Some of Their Applications". Operator Theory - Recent Advances, New Perspectives and Applications.
^Shams, M.; Kausar, N.; Agarwal, P.; Jain, S. (2024). "Fuzzy fractional Caputo-type numerical scheme for solving fuzzy nonlinear equations". Fractional Differential Equations. pp. 167–175. doi:10.1016/B978-0-44-315423-2.00016-3. ISBN978-0-443-15423-2.
^Shams, M.; Kausar, N.; Agarwal, P.; Edalatpanah, S.A. (2024). "Fractional Caputo-type simultaneous scheme for finding all polynomial roots". Recent Trends in Fractional Calculus and Its Applications. pp. 261–272. doi:10.1016/B978-0-44-318505-2.00021-0. ISBN978-0-443-18505-2.
^Al-Nadhari, A.M.; Abderrahmani, S.; Hamadi, D.; Legouirah, M. (2024). "The efficient geometrical nonlinear analysis method for civil engineering structures". Asian Journal of Civil Engineering. 25 (4): 3565–3573. doi:10.1007/s42107-024-00996-z.
^Shams, M.; Kausar, N.; Samaniego, C.; Agarwal, P.; Ahmed, S.F.; Momani, S. (2023). "On efficient fractional Caputo-type simultaneous scheme for finding all roots of polynomial equations with biomedical engineering applications". Fractals. 31 (4): 2340075–2340085. Bibcode:2023Fract..3140075S. doi:10.1142/S0218348X23400753.
^Shams, M.; Kausar, N.; Agarwal, P.; Jain, S.; Salman, M.A.; Shah, M.A. (2023). "On family of the Caputo-type fractional numerical scheme for solving polynomial equations". Applied Mathematics in Science and Engineering. 31 (1): 2181959. doi:10.1080/27690911.2023.2181959.
^Nayak, S.K.; Parida, P.K. (2024). "Global convergence analysis of Caputo fractional Whittaker method with real world applications". Cubo (Temuco). 26 (1): 167–190. doi:10.56754/0719-0646.2601.167.
^Rebollar-Rebollar, S.; Martínez-Damián, M.Á.; Hernández-Martínez, J.; Hernández-Aguirre, P. (2021). "Óptimo económico en una función Cobb-Douglas bivariada: una aplicación a ganadería de carne semi extensiva". Revista mexicana de ciencias agrícolas. 12 (8): 1517–1523. doi:10.29312/remexca.v12i8.2915.
^Mogro, M.F.; Jácome, F.A.; Cruz, G.M.; Zurita, J.R. (2024). "Sorting Line Assisted by A Robotic Manipulator and Artificial Vision with Active Safety". Journal of Robotics and Control (JRC). 5 (2): 388–396. doi:10.18196/jrc.v5i2.20327 (inactive 2024-09-03).{{cite journal}}: CS1 maint: DOI inactive as of September 2024 (link)
^Luna-Fox, S.B.; Uvidia-Armijo, J.H.; Uvidia-Armijo, L.A.; Romero-Medina, W.Y. (2024). "Exploración comparativa de los métodos numéricos de Newton-Raphson y bisección para la resolución de ecuaciones no lineales". MQRInvestigar. 8 (2): 642–655. doi:10.56048/MQR20225.8.2.2024.642-655.
^Tvyordyj, D.A.; Parovik, R.I. (2022). "Mathematical modeling in MATLAB of solar activity cycles according to the growth-decline of the Wolf number". Vestnik KRAUNC. Fiziko-Matematicheskie Nauki. 41 (4): 47–64. doi:10.26117/2079-6641-2022-41-4-47-65 (inactive 2024-09-03).{{cite journal}}: CS1 maint: DOI inactive as of September 2024 (link)