Initialized fractional calculus
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In mathematical analysis, initialization of the differintegrals is a topic in fractional calculus.
Composition rule of Differintegrals
[edit]The composition law of the differintegral operator states that although:
wherein D−q is the left inverse of Dq, the converse is not necessarily true:
Example
[edit]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.
Description of initialization
[edit]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 .
Set of Fractional Operators
[edit]The Fractional Calculus of Sets (FCS), first mentioned 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 main concept behind FCS is the characterization of the elements of fractional calculus using sets due to the large number 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]
![](http://upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Applied_mathematics_and_computation-fig.png/500px-Applied_mathematics_and_computation-fig.png)
Fractional calculus, a branch of mathematics dealing with derivatives of non-integer order, emerged almost simultaneously with traditional calculus. This emergence was partly due to Leibniz's notation for integer-order derivatives: . Thanks to this notation, L’Hopital was able to ask Leibniz in a letter about the interpretation of taking in a derivative. At the time, Leibniz could not provide a physical or geometrical interpretation for this question, so he simply responded to L’Hopital in a letter that "... it 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 it is not necessary to explicitly specify the form of a fractional derivative, it is typically denoted as follows:
Fractional operators have several 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 [11] [12]:
whose complement is:
As a consequence, the following set is defined:
Extension to Vector Functions
[edit]For a function , the set is defined as:
where denotes the -th component of the function .
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 [13], [14], Springer [15], World Scientific [16], and MDPI [17], [18], [19], [20], [21], [22], [23], [24]. Studies from Taylor & Francis (Tandfonline) [25], Cubo [26], Revista Mexicana de Ciencias Agrícolas [27], Journal of Research and Creativity [28], MQR [29], and Актуальные вопросы науки и техники [30]. These works highlight the relevance and applicability of fractional operators in problem-solving.
See also
[edit]References
[edit]- Lorenzo, Carl F.; Hartley, Tom T. (2000), Initialized Fractional Calculus (PDF), NASA (technical report).
- ^ Sets of Fractional Operators and Numerical Estimation of the Order of Convergence of a Family of Fractional Fixed-Point Methods
- ^ Applications of fractional calculus in physics
- ^ A review of definitions for fractional derivatives and integral
- ^ A review of definitions of fractional derivatives and other operators
- ^ How many fractional derivatives are there?
- ^ Fractional Newton-Raphson Method
- ^ Acceleration of the order of convergence of a family of fractional fixed-point methods and its implementation in the solution of a nonlinear algebraic system related to hybrid solar receivers
- ^ Code of a multidimensional fractional quasi-Newton method with an order of convergence at least quadratic using recursive programming
- ^ Sets of Fractional Operators and Some of Their Applications
- ^ Einstein summation for multidimensional arrays
- ^ Torres-Hernandez, A.; Brambila-Paz, F. (December 29, 2021). "Sets of Fractional Operators and Numerical Estimation of the Order of Convergence of a Family of Fractional Fixed-Point Methods". Fractal and Fractional. 5 (4): 240. doi:10.3390/fractalfract5040240.
- ^ Acceleration of the order of convergence of a family of fractional fixed-point methods and its implementation in the solution of a nonlinear algebraic system related to hybrid solar receivers
- ^ 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. ISBN 978-0-443-15423-2.
- ^ 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. ISBN 978-0-443-18505-2.
- ^ 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.
- ^ Wang, X.; Jin, Y.; Zhao, Y. (2021). "Derivative-free iterative methods with some Kurchatov-type accelerating parameters for solving nonlinear systems". Symmetry. 13 (6): 943. Bibcode:2021Symm...13..943W. doi:10.3390/sym13060943.
- ^ Tverdyi, D.; Parovik, R. (2021). "Investigation of Finite-Difference Schemes for the Numerical Solution of a Fractional Nonlinear Equation". Fractal and Fractional. 6 (1): 23. doi:10.3390/fractalfract6010023.
- ^ Tverdyi, D.; Parovik, R. (2022). "Application of the fractional Riccati equation for mathematical modeling of dynamic processes with saturation and memory effect". Fractal and Fractional. 6 (3): 163. doi:10.3390/fractalfract6030163.
- ^ Srivastava, H.M. (2023). "Editorial for the Special Issue "Operators of Fractional Calculus and Their Multidisciplinary Applications"". Fractal and Fractional. 7 (5): 415. doi:10.3390/fractalfract7050415.
- ^ Shams, M.; Carpentieri, B. (2023). "Efficient inverse fractional neural network-based simultaneous schemes for nonlinear engineering applications". Fractal and Fractional. 7 (12): 849. doi:10.3390/fractalfract7120849.
- ^ Candelario, G.; Cordero, A.; Torregrosa, J.R.; Vassileva, M.P. (2023). "Solving Nonlinear Transcendental Equations by Iterative Methods with Conformable Derivatives: A General Approach". Mathematics. 11 (11): 2568. doi:10.3390/math11112568.
- ^ Shams, M.; Carpentieri, B. (2023). "On highly efficient fractional numerical method for solving nonlinear engineering models". Mathematics. 11 (24): 4914. doi:10.3390/math11244914.
- ^ Martínez, F.; Kaabar, M.K.A.; Martínez, I. (2024). "Novel Results on Legendre Polynomials in the Sense of a Generalized Fractional Derivative". Mathematical and Computational Applications. 29 (4): 54. doi:10.3390/mca29040054.
- ^ 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).
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: 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).
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: CS1 maint: DOI inactive as of September 2024 (link)