Volume 8, no. 2Pages 55 - 68

Existence of Lower and Upper Solutions in Reverse Order with Respect to a Variable in a Model of Acidogenesis to Anaerobic Digestion

M.M. Higuera, A.V. Sinitsyn
We prove existence of upper and lower solutions in reverse order with respect a part of the variables in a system of nonlinear ordinary differential equations modelling acidogenesis in anaerobic digestion. The corresponding existence theorems are established. The upper and lower solutions are constructed analytically, by defining semi-trivial solutions for each of the variables in the model. We introduce the concept of indicator semi-trivial solutions. Finally, we numerically solve the system supported by the Matlab software and matching the graphs of the numerical solutions with analytical solutions is found.
Full text
upper-lower solutions; inverse order; system of nonlinear differential equations; anaerobic digestion.
1. Jewell W. Anaerobic Sewage Treatment. Environmental Science and Technology, 1987, vol. 21, no. 1, pp. 9-21. DOI: 10.1021/es00155a002
2. Bernard O., Sadok Z.H., Dochain D., Genovesi A., Steyer J.-P. Dynamical Model Development and Parameter Identification for an Anaerobic Wastewater Treatment Process. Biotechnology and Bioengineering, 2001, vol. 75, no. 4, pp. 424-438. DOI: 10.1002/bit.10036
3. Alcaraz V., Genovesi A., Harmand J., Gonz'alez V., Rapaport A., Steyer J.P. Robust Exponetial Nonlinear Interval Observers for a Class of Lumped Models Useful in Chemical and Biochemical Engineering. Application to a Wastewater Treatment Process. International Workshop on Application of Interval Analysis to Systems and Control, MISC'99, Girona, Spain, 1999, pp. 24-26.
4. Pao C.-V. Nonlinear Parabolic and Elliptic Equations. N.Y., Plenum Press, 1992.
5. Delgado M., Su'arez A. Existence of Solutions for Elliptic Systems with Holder Continuous Nonlinearities. Diferential and Integral Equations, 2000, vol. 13, no. 4, 6, pp. 453-477.
6. Franco D., Nieto J.J., O'Regan D. Upper and Lower Solutions for First Order Problems with Nonlinear Boundary Conditions. Estracta Mathematicae, 2003, vol. 18, no. 2, pp. 153-160
7. Benyahia B., Sari T., Cherki B., Harmand J. Bifurcation and Stability Analysis of a Two Step Model for Monitoring Anaerobic Digestion Processes. Journal of Process Control, 2012, vol. 22, no. 6, pp. 1008-1019. DOI: 10.1016/j.jprocont.2012.04.012
8. Sidorov N., Loginov B., Sinitsyn A., Falaleev M. Lyapunov - Schmidt Methods in Nonlinear Analysis and Applications. Dordrecht, Boston, London, Kluwer Academic Publisher, 2002. 568 p. DOI: 10.1007/978-94-017-2122-6
9. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, Koln, Tokyo, VSP, 2003. 216 p. DOI: 10.1515/9783110915501
10. Lorenzo Y., Obaya M.C. La digesti'on anaerobia. aspectos te'oricos. parte I. ICIDCA. Sobre los Derivados de la Cana de Azucar, 2005, vol. 39, no. 1, pp. 35-48.
11. McKenna P.-J., Walter W. On the Dirichlet Problem for Elliptic Systems. Applicable Analysis, 1986, no. 21, pp. 207-224. DOI: 10.1080/00036818608839592
12. Salinas E., Munoz R., Sosa J.-C., L'opez B. Analysis to the Solutions of Abel's Differential Equations of the First Kind under Transformation y = u(x)z(x)+v(x). Applied Mathematical Sciences, 2013, vol. 7, no. 42, pp. 2075-2092.