Abstract
In this paper, we study a singular nonlocal fractional dynamic system arising in the abstract model for bioprocess. Conditions for the exact iterative solution to the problem are established, followed by development of an iterative technique for generating approximate solution to the problem. The iterative technique has been proved to give sequences converging uniformly to the exact solution, and formulate for estimation of the approximation error and the convergence rate have been derived. An example is also given in the paper to demonstrate the application of our theoretical results.
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Jesus, I., Machado, J., Cunha, J.: Fractional electrical impedances in botanical elements. J. Vib. Control 14, 1389–1402 (2008)
Jesus, I., Machado, J., Cunha, J.: Fractional order electrical impedance of fruits and vegetables. In: Proceedings of the 25th IASTED International Conference Modeling, Identification, and Control, February 6–8, Lanzarote, Canary Islands, Spain (2006)
Petrovic, L.M., Spasic, D.T., Atanackovic, T.M.: On a mathematical model of a human root dentin. Demonstr. Math. 21, 125–128 (2005)
Cole, K.: Electric conductance of biological systems. In: Proceedings of Cold Spring Harbor Symposium on Quantitative Biology, pp. 107-116. Cold Spring Harbor, New York (1993)
Djordjević, V., Jarić, J., Fabry, B., Fredberg, J., Stamenović, D.: Fractional derivatives embody essential features of cell rheological behavior. Ann. Biomed. Eng. 31, 692–699 (2003)
Ahmed, E., El-Saka, H.A.: On fractional order models for Hepatitis C. Nonlinear Biomed. Phys. 4, 1 (2010)
Perelson, A.S.: Modeling the interaction of the immune system with HIV. In: Castillo-Chavez, C. (ed.) Mathematical and Statistical Approaches to AIDS Epidemiology. Lecture Notes in Biomathematics, vol. 83, pp. 350. Springer, New York (1989)
Perelson, A.S., Kirschner, D.E., Boer, R.D.: Dynamics of HIV infection of CD4+ T cells. Math. Biosci. 114, 81–125 (1993)
Arafa, A.A.M., Rida, S.Z., Khalil, M.: Fractional modeling dynamics of HIV and CD4+ T-cells during primary infection. Nonlinear Biomed. Phys. 6, 1 (2012)
Zhang, X., Liu, L., Wu, Y., Lu, Y.: The iterative solutions of nonlinear fractional differential equations. Appl. Math. Comput. 219, 4680–4691 (2013)
Goodrich, C.: Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions. Comput. Math. Appl. 61, 191–202 (2011)
Goodrich, C.: Positive solutions to boundary value problems with nonlinear boundary conditions. Nonlinear Anal. 75, 417–432 (2012)
Rehman, M., Khan, R.: Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations. Appl. Math. Lett. 23, 1038–1044 (2010)
Zhang, X., Liu, L., Wu, Y.: The uniqueness of positive solution for a singular fractional differential system involving derivatives. Commun. Nonlinear Sci. Numer. Simul. 18, 1400–1409 (2013)
Zhang, X., Liu, L., Wiwatanapataphee, B., Wu, Y.: The eigenvalue for a class of singular \(p\)-Laplacian fractional differential equations involving the Riemann–Stieltjes integral boundary condition. Appl. Math. Comput. 235, 412–422 (2014)
Zhang, X., Liu, L., Wu, Y.: The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium. Appl. Math. Lett. 37, 26–33 (2014)
Zhang, X., Liu, L.: Positive solutions of fourth-order four-point boundary value problems with \(p\)-Laplacian operator. J. Math. Anal. Appl. 336, 1414–1423 (2007)
Zhang, X., Liu, L., Wiwatanapataphee, B., Wu, Y.: The spectral analysis for a singular fractional differential equation with a signed measure. Appl. Math. Comput. 257, 252–263 (2015)
Zhang, X., Liu, L.: Multiple positive solutions of a singular fractional differential equation with negatively perturbed term. Math. Comput. Model. 55, 1263–1274 (2012)
Webb, J.R.L., Infante, G.: Positive solutions of nonlocal boundary value problems involving integral conditions. Nonlinear Differ. Equ. Appl. 15, 45–67 (2008)
Webb, J.R.L., Infante, G.: Positive solutions of nonlocal boundary value problems: a unified approach. J. Lond. Math. Soc. 74, 673–693 (2006)
Webb, J.R.L.: Nonlocal conjugate type boundary value problems of higher order. Nonlinear Anal. 71, 1933–1940 (2009)
Podlubny, I.: Fractional Differential Equations. Mathematics in Science and Engineering. Academic Press, New York (1999)
Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. Elsevier B.V, Netherlands (2006)
Yuan, C.: Multiple positive solutions for \((n-1,n)\)-type semipositone conjugate boundary value problems of nonlinear fractional differential equations. Electron. J. Qual. Theory Differ. Equ. 36, 12 pp (2010)
Borberg, K.B.: Cracks and Fracture. Academic Press, San Diego (1999)
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The authors were supported financially by the National Natural Science Foundation of China (11371221, 11571296) and the Natural Science Foundation of Shandong Province of China (ZR2014AM009).
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Zhang, X., Mao, C., Liu, L. et al. Exact Iterative Solution for an Abstract Fractional Dynamic System Model for Bioprocess. Qual. Theory Dyn. Syst. 16, 205–222 (2017). https://doi.org/10.1007/s12346-015-0162-z
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DOI: https://doi.org/10.1007/s12346-015-0162-z