Abstract
This chapter aims to familiarize the reader with the new field of mathematics known as ‘fractional calculus’, as well as the operators, techniques, and additional mathematical definitions that will be useful in the work that will be covered in subsequent chapters. A significant component of contemporary biological research is its dependence on computers. In order to understand the multilayered complexity of biological systems, mathematical models are essential. The purpose of the dynamic mathematical model is to support biological research. The mechanism of modeling of any biological phenomenon is given in this chapter along with the diagrammatical representation. The methods used for solving the fractional differential equations used in the next chapters are also elaborated here.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Agarwal RP, Humbert P (1953) On the mittag-leffler function and some of its generalizations. Bull Sci Math 77(2):180–185
Alber HD (1998) Materials with memory. Initial boundary value problems for constitutive equations with internal variables. Springer, Berlin, Germany, p 170
Anderson DR, Ulness DJ (2015) Newly defined conformable derivatives. Adv Dyn Syst Appl 10(2):109–137
Assadi I, Charef A, Copot D, De Keyser R, Bensouici T, Ionescu C (2017) Evaluation of respiratory properties by means of fractional order models. Biomed Signal Proc Control 34:206–213
Atangana A (2017) Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system. Chaos, Solitons & Fractals 102:396–406
Bagley RL, Torvik PJ (1983) A theoretical basis for the application of fractional calculus to viscoelasticity. J Rheol 27:201–210
Baleanu D, Fernandez A, Akgül A (2020) On a fractional operator combining proportional and classical differintegrals. Mathematics 8(3):360
Bildik N (2017) General convergence analysis for the perturbation iteration technique. Turkish J Math Comput Sci 6:1–9
Blair GS (1944) Analytical and integrative aspects of the stress-strain-time problem. J Sci Instrum 21(5):80
Boltzmann L (1874) Theory of elastic aftereffect [zur theorie der elastischen nachwirkung]. Wien Akad Sitzungsber 70:275–306
Boltzmann L (1876) Theory of elastic aftereffect [zur theorie der elastischen nachwirkung]. Annalen der Physik und Chemie: Erganzungsband 7:624–654
Boltzmann L (2012a) Theory of elastic aftereffect [zur theorie der elastischen nachwirkung]. In: Hasenohrl F (ed) Wissenschaftliche Abhandlungen, vol 1, pp 616–644. Cambridge University Press, Cambridge, UK
Boltzmann L (2012b) Theory of elastic aftereffect [zur theorie der elastischen nachwirkung]. In: Hasenohrl F (ed) Wissenschaftliche Abhandlungen, vol 2, pp 318–320. Cambridge University Press, Cambridge, UK
Bozler E (1954) Relaxation in extracted muscle fibers. J Gen Physiol 38:149–159
Caputo M (1967) Linear models of dissipation whose q is almost frequency independent-ii. Geophys J Inter 13(5):529–539
Caputo M, Fabrizio M (2015) A new definition of fractional derivative without singular kernel. Progr Fract Differ Appl 1(2):1–13
Chen W (2006) Time–space fabric underlying anomalous diffusion. Chaos, Solitons & Fractals 28(4)(4):923–929
Chen W, Sun H, Zhang X, Korošak D (2010) Anomalous diffusion modeling by fractal and fractional derivatives. Comput & Math Appl 59(5):1754–1758
Crank J, Nicolson P (1947) A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Math Proc Camb Philos Soc 43(1):50–67. Cambridge University Press
Davis TN (1992) What’s new with calcium? Cell 71(4):557–564
Erdelyi A, Magnus W, Oberhettinger F, Tricomi FG (1955) Higher transcendental functions, vol 3. McGraw - Hill, New York, Toronto and London
Feliu-Talegon D, San-Millan A, Feliu-Batlle V (2016) Fractional-order integral resonant control of collocated smart structures. Control Eng Pract
Granger CWJ (1964) The typical spectral shape of an economic variable. Technical Report No. 11; Department of Statistics, Stanford University, Stanford, CA, USA, p 21
Granger CWJ (1966) The typical spectral shape of an economic variable. Econometrica 34:150–161
Granger CWJ, Joyeux R (1980) An introduction to long memory time series models and fractional differencing. J Time Ser Anal 1:15–39
Greenenko AA, Chechkin AV, Shul’Ga NF (2004) Anomalous diffusion and lévy flights in channeling. Phys Lett A 324(1):82–85
Grunwald AK (1867) Uber begrenzte derivationen and deren anwendung. Z Angew Math Und Phys 12:441–480
Hadmard J (1892) Essai sur l’etude des functions donnees par leur developpement de taylor. J Math Pures et Appl Ser 4:101–186
Hardy GH, Littlewood JE (1925) Some properties of fractional integrals. Proc London Math Soc Ser 2(24):37–41
Herrmann R (2011) Fractional calculus: an introduction for physicist. World Scientific, New Jersey
Hilfer R (2000) Applications of fractional calculus in physics. World scientific, Germany
Holmgren HJ (1865) Om differenlialkalkylen med indices of hoad nature som helst. Kongl Svenska Vetenskaps Akad Handl Stockholm 5(1):1–83
Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of epidemics. Proc R Soc Lond Ser A Contain Papers Math Phys Character 115(772):700–721
Kilbas A, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations, vol 204. Elsevier, Amsterdam
Kiryakova V (1994) Generalized fractional calculus and applications. Longman and J. Wiley, New York, USA
Kriyakova VS (1986) On operators of fractional integration invovling meijer’s g-function. C R Acad Bulg Sci 39(10):25–28
Kurg A (1890) Theorie der derivationen. Akad Wiss Wien, Denkenschriften Math-Natur Kl 57:151–226
Laplace PS (1820) Analytic theory of probabilities. Courcier, Paris
Laurent H (1884) On the calculation of the d ’e riv é es ‘a arbitrary indices. Nouvelles Annales de Math e Matiques: J Cand Polytech Norm Sch 3:240–252
Letnikov AV (1868) Theory of differentiation with an arbtraly indicator. Matem Sbornik 3:1–68
Li Y, Yu SL (2006) Fractional order difference filters and edge detection. Opto-Electr Eng 33(19):71–74
Liao S (1997) Homotopy analysis method: a new analytical technique for nonlinear problems. Commun Nonlinear Sci Numer Simul 2(2):95–100
Liao S (2004) On the homotopy analysis method for nonlinear problems. Appl Math Comput 147(2):499–513
Liouville J (1832) Memoire sur le calcul des differentielles a indices quelcon- ques. Ibid 71–162
Liu W, Hillen T, Freedman H (2007) A mathematical model for m-phase specific chemotherapy including the \( g_0 \)-phase and immunoresponse. Math Biosci & Eng 4(2):239
Logan JD (2013) Applied mathematics. Wiley, New Jersey
Logan JD (2015) A first course in differential equations, 3rd edn. Springer, Switzerland
Losada J, Nieto JJ (2015) Properties of a new fractional derivative without singular kernel. Progr Fract Differ Appl 1(2):87–92
Mainardi F (1997) Fractional calculus. Fractals and fractional calculus in continuum mechanics. Springer, Vienna, pp 291–348
Mainardi F (2012) An historical perspective on fractional calculus in linear viscoelasticity. Fract Calc Appl Anal 15:712–717
Marchaud A (1927) Sur les derivees et sur les differences des functions de variables reelles. J Math Pures et Appl 6(4):337–425
Marinangeli L, Alijani F, Hossein Nia SH (2018) Fractional-order positive position feedback compensator for active vibration control of a smart composite plate. J Sound Vib 412:1–16
Meerschaert MM, Tadjeran C (2004) Finite difference approximations for fractional advection-dispersion flow equations. J Comput Appl Math 172(1):65–77
Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley-Interscience, New York
Mittag-Leffler GM (1903) On the new function \(e_{a}(x)\). CR Acad Sci Paris 137(2):554–558
Mohamed MS, Al-Malki F, Al-Humyani M (2014) Homotopy analysis transform method for timespace fractional gas dynamics equation. Gen Math Notes 24(1):1–16
Oldham K, Spanier J (1974) The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier
Pakdemirli M, Aksoy Y, Boyacı H (2011) A new perturbation-iteration approach for first order differential equations. Math Comput Appl 16(4):890–899
Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol 198. Elsevier, San Diego
Pu YF (2007) Fractional differential analysis for texture of digital image. J Alg Comput Technol 1(03):357–380
Riesz M (1949) L’integrale de riemann-liouville et le probleme de cauchy. Acta Math 81:1–223
Rubin BS (1972) On the spaces of fractional integrals on straight line contour. Izr Akad Nauk Armyan SSR Ser Mat 7(5):373–386
Samko SG (1987) Fractional integrals and derivatives, theory and applications. Gordon and Breach Science Publishers, Switzerland, USA
Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives theory and applications. Gordon and Breach, New York, NY, USA
Tomovski Z (2011) Generalised cauchy type problems for nonlinear fractional differential equation with composite fractional derivative operator. Nonlinear Anal- Theor 75:3364–3384
Volterra V (1928) On the mathematical theory of hereditary phenomena. J Mathematiques Pures et Appliquees 7:249–298
Volterra V (1930) Theory of functionals and of integral and integro-differential equations. Blackie and Son Ltd., London, UK; Glasgow, Scotland, p 226
Weyl H (1917) Bemerkungen zum begriff des differential quotienten gebrochener. ordung. Vir Natur Ges Zurich 62:296–302
Wiesner TF, Berk BC, Nerem RM (1996) A mathematical model of cytosolic calcium dynamics in human umbilical vein endothelial cells. Amer J Physiol-Cell Physiol 270(5):C1556–C1569
Wiman A (1905) Über den fundamentalsatz in der theorie der funktionen ea (z). Acta Math 29(1):191–201
Zhang J, Wei Z (2011) A class of fractional-order multiscale variational models and alternating projection algorithm for image denoising. Appl Math Model 35:2516–2528
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Agarwal, R., Purohit, S.D., Kritika (2024). Introduction to Fractional Calculus and Modelling. In: Modeling Calcium Signaling. SpringerBriefs in Biochemistry and Molecular Biology. Springer, Singapore. https://doi.org/10.1007/978-981-97-1651-7_1
Download citation
DOI: https://doi.org/10.1007/978-981-97-1651-7_1
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-97-1650-0
Online ISBN: 978-981-97-1651-7
eBook Packages: Chemistry and Materials ScienceChemistry and Material Science (R0)