A New Approach for Modeling with Discrete Fractional Equations

• Autorzy: Atıcı F. M. , Atıcı M. , Belcher M. , Marshall D.

Identyfikatory

DOI

10.3233/FI-2017-1494

• Języki publikacji: EN

Abstrakty

EN

In this paper, we introduce a new class of nonlinear discrete fractional equations to model tumor growth rates in mice. For the data fitting purpose, we develop a new method which can be considered as an improved version of the partial sum method for parameter estimations. We demonstrate the goodness of fit by comparing the models with three statistical measures.

Słowa kluczowe

EN

discrete fractional calculus; parameter estimations; data fitting; method of partial sums

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2017
• Tom: Vol. 151, nr 1/4
• Strony: 313--324
• Opis fizyczny: Bibliogr. 21 poz., tab., wykr.

Twórcy

autor

Atıcı F. M.

• Department of Mathematics, Western Kentucky University, Bowling Green, Kentucky 42101-3576 USA

autor

Atıcı M.

• Department of Computer Science, Western Kentucky University, Bowling Green, Kentucky 42101-3576 USA

autor

Belcher M.

• Gatton Academy of Science and Mathematics, Bowling Green, Kentucky 42101 USA

autor

Marshall D.

• Department of Pathology, Anatomy and Cell Biology, Meharry Medical College, Nashville, Tennessee 37208 USA

Bibliografia

• [1] Annadurai G, Rajesh Babu S, and Srinivasamoorthy VR. Development of mathematical models (Logistic, Gompertz and Richards Models) describing the growth pattern of Pseudomonas Putida (NCM 2174), Bioprocess Engineering, 2000; (23): 607-612. doi: 10.1007/s004490000209.
• [2] Atıcı FM, and Eloe PW. Two-point boundary value problems for finite fractional difference equations, J. Difference Equations and Applications, 2011; 17 (4): 445-456. doi: 10.1080/10236190903029241.
• [3] Atıcı FM, and Eloe PW. Initial value problems in discrete fractional calculus, Proceedings of the American Mathematical Society, 2009; 137 (3): 981-989. URL http://www.jstor.org/stable/20535823.
• [4] Atici FM, and Eloe PW. A transform method in discrete fractional calculus, International Journal of Difference Equations, 2007; 2 (2): 165-176. URL http://www.ripublication.com/ijde.htm.
• [5] Atici FM, Atici M, Hrushesky WM, and Nguyen N. Modeling tumor growth with basic functions of fractional calculus, Progress in Fractional Differentiation and Applications, 2015; 1 (4): 1-13. URL http://dx.doi.org/10.18576/pfda/010401.
• [6] Atici FM and Şengül S. Modeling with fractional difference equations, Journal of Mathematical Analysis and Applications, 2010; 369 (1): 1-9. doi: 10.1016/j.jmaa.2010.02.009.
• [7] Bassukas ID, Schultze BM. The recursion formula of the Gompertz function: A simple method for the estimation and comparison of tumor growth curves, Growth Dev. Aging, 1988; (52): 113-122. ISSN: 1041-1232.
• [8] Bassukas ID. Comparative Gompertzian analysis of alterations of tumor growth patterns, Cancer Research. 1994; (54): 4385-4392. URL http://cancerres.aacrjournals.org/content/54/16/4385.
• [9] Benzekry S, Lamont C, Beheshti A, Tracz A, Ebos JML, Hlatky L, and Hahnfeldt P. Classical mathematical models for description and prediction of experimental tumor growth, PLOS Computational Biology. 2014; (10): 1-19. URL http://dx.doi.org/10.1371/journal.pcbi.1003800.
• [10] Dokoumetzidis A, Magin R, and Macheras P. Fractional kinetics in multi-compartmental systems, J. Pharmacokinet. Pharmacodyn., 2010; (37): 507-524. doi: 10.1007/s10928-010-9170-4.
• [11] Farebrother RW. Non-linear curve fitting and the true method of least squares, Journal of the Royal Statistical Society Series D (The Statiscian), 1998; 47 (1): 137-147. doi: 10.1111/1467-9884.00119.
• [12] Ferreira RAC. A discrete fractional Gronwall inequality, Proc. Amer. Math. Soc. 2012; 140 (5): 1605-1612. URL http://www.ams.org/journals/proc/2012-140-05/S0002-9939-2012-11533-3/.
• [13] Ferreira RAC, and Goodrich CS. Positive solution for a discrete fractional periodic boundary value problem, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 2012; 19 (5): 545-557. ISSN: 1201-3390.
• [14] Goodrich CS. On discrete fractional boundary value problems with nonlocal, nonlinear boundary conditions, Commun. Appl. Anal., 2012; 16 (3): 433-445. ISSN: 1083-2564.
• [15] Goodrich CS. On a fractional boundary value problem with fractional boundary conditions, Appl. Math Lett., 2012; 25 (8): 1101-1105. URL http://dx.doi.org/10.1016/j.am1.2011.11.028.
• [16] Gompertz B. On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies, Philos. Trans. R. Soc. Lond., 1825; 115: 513-585. URL http://www.jstor.org/stable/107756.
• [17] Goodrich C, and Peterson AC. Discrete Fractional Calculus, Springer, 2016, doi: 10.1007/978-3-319-25562-0.
• [18] Holm MT. Sum and difference compoistions and applications in discrete fractional calculus, Cubo. 2011; 13: 153-184. URL http://dx.doi.org/10.4067/S0719-06462011000300009.
• [19] Nobile AG, Ricciardi LM, and Sacerdote L. On Gompertz growth model and related difference equations, Biol. Cybern. 1982; 42: 221-229. doi: 10.1007/BF00340079.
• [20] Donatelli M, Mazza M, and Serra-Capizzano S. Spectral analysis and structure preserving preconditioners for fractional diffusion equations. J. Comput. Phys. 2016; 307: 262-279. URL http://dx.doi.org/10.1016/j.jcp.2015.11.061.
• [21] Meerschaert MM. and Tadjeran C. Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 2004; 172 (1): 65-77. URL http://dx.doi.org/10.1016/j.cam.2004.01.033.