Solutions of Impulsive Diffusion and Von-Foerster-Makendrick Models Using the B-Transform

Benjamin Oyediran Oyelami, Samson Olatunji Ale

National Mathematical Centre, Abuja, Nigeria.

DOI: 10.4236/am.2013.412223                

Abstract

In this paper we explore the possibility of using the scientific computing method to obtain the inverse B-Transform of Oyelami and Ale [1]. Using some suitable conditions and the symbolic programming method in Maple 15 we obtained the asymptotic expansion for the inverse B-transform then used the residue theorem to obtain solutions of Impulsive Diffusion and Von-Foerster-Makendrick models. The results obtained suggest that drugs that are needed for prophylactic or chemotherapeutic purposing the concentration must not be allowed to oscillate about the steady state. Drugs that are to be used for immunization should not oscillate at steady state in order to have long residue effect in the blood. From Von-Foerster-Makendrick model, we obtained the conditions for population of the specie to attain super saturation level through the “dying effect” phenomenon ([2-4]). We used this phenomenon to establish that the environment cannot accommodate the population of the specie anymore which mean that a catastrophic stage t^* is reached that only the fittest can survive beyond this regime (i.e. t > t^*) and that there would be sharp competition for food, shelter and waste disposal etc.

Keywords

B-Transform; Impulsive Diffusion; Von-Foerster-Makendrick Models; Residue Theorem; Maple Symbolic Programme and Asymptotic Expansion

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	B. O. Oyelami and S. O. Ale, “B-transform Method and its Applications, in obtaining Solutions of some Impulsive Models,” International Journal of Mathematical Education in Science and Technology, Vol. 31, No. 4, 2000, pp. 525-538. http://dx.doi.org/10.1080/002073900412633
[2]	S. O. Ale and B. O. Oyelami, “Impulsive System and Applications,” International Journal of Mathematical Education in Science and Technology, Vol. 31, No. 4, 2000, pp. 539-544. http://dx.doi.org/10.1080/002073900412642
[3]	D. D. Bainov, V. Lakshikantham and P. S. Simeonov, “Theory of Impulsive Differential Equations,” World Scientific Publication, Singapore, 1989.
[4]	P. S. Simeonov and D. D. Bainov, “Theory of Impulsive Differential Equations: Periodic Solutions and Applications,” Essex, Longman, 1993.
[5]	E. Beltrami, “Mathematics for Dynamic Modeling,” Academy Press, London, 1987.
[6]	B. O. Oyelami, “On Military Model for Impulsive Reinforcement Functions using Exclusion and Marginalization Techniques,” Nonlinear Analysis, Vol. 35, No. 8, 1999, pp. 947-958. http://dx.doi.org/10.1016/S0362-546X(98)00114-X
[7]	B. O. Oyelami, S. O. Ale, P. Onumanyi and J. A. Ogidi, “B-Transform Method and Application to Sickle Cell Aneamia,” 2008, pp. 202-220. http://sirius-c.ncat.edu/asn/ajp/allissue/ajp-ISOTPAND/index.html
[8]	B. O. Oyelami and S. O. Ale, “On Existence of Solution, Oscillation and Non-Oscillation Properties of Delay Equations Containing ‘Maximum’,” Acta Applicandae Mathematicae Journal, Vol. 109, No. 3, 2010, pp. 683701. http://dx.doi.org/10.1007/s10440-008-9340-1
[9]	P. S. Simeonov and D. D. Bainov, “Impulsive Differential Equation: Asymptotic Properties of the Solutions,” World Scientific Publication, Singapore, 1989.
[10]	S. G. Pandit and H. Deo Sudashiv, “Differential Systems Involving Impulsive. Lecture Notes in Maths,” SpringerVerlag, Berlin-Heidelberg-New York, 1982.
[11]	B. O. Oyelami and S. O. Ale, “B-Transform and Its Applications to a Fish-Hyacinth Model,” International Journal of Mathematical Education in Science and Technology, Vol. 33, No. 4, 2002, pp. 565-573. http://dx.doi.org/10.1080/00207390210131353
[12]	B. O. Oyelami, S. O. Ale, P. Onumanyi and J. A. Ogidi, “Impulsive HIV-1 Model in the Presence of Antiretroviral Drugs Using B-Transform Method,” Proceedings of African Mathematical Union, Vol. 1, No. 1, 2003, pp. 6276.
[13]	B. O. Oyelami, S. O. Ale and P. Onumanyi, “Impulsive HIV Model Using B-Transform,” The Proceedings of National Mathematical Centre on Conference on Computational Mathematics, Vol. 2, No. 1, 2005, pp. 50-64. http://nmcabuja.org/nmc_proceedings.html
[14]	B. Davies, “Brian Davies Integral Transforms and Their Applications,” Springer Publisher, Berlin-HeidelbergNew York, 2002.
[15]	M. B. V. Robert, “Biology; a Functional Approval,” Nelson Butler and Tanner Ltd., Rome and London, 1971.
[16]	V. Lakshimikantham and Z. Dric, “Positivity and Boundedness of Solutions of Impulsive Reaction-Diffusion Equations,” Journal of Computational and Applied Mathematics, Vol. 88, No. 1, 1988, pp. 175-184. http://dx.doi.org/10.1016/S0377-0427(97)00210-0
[17]	L. H. Erbe, H. I. Freeman, X. Z. Lin and H. J. Wu, “Comparison Principles for Impulsive Parabolic Species Growth,” The Journal of the Australian Mathematical Society, Series B, Vol. 32, No. 4, 1991, pp. 382-400.
[18]	S. O. Ale and B. O. Oyelami, “On Chemotherapy of Impulsive Models Involving Malignant Cancer Cells,” Abacus, Journal of Mathematical Association of Nigeria, Vol. 24, No. 2, 1996, pp. 1-10.
[19]	B. O. Oyelami and S. O. Ale, “Impulsive Differential Equations and Applications to Some Models: Theory and Applications. A Monograph,” Lambert Academic Publisher, Saarbrücken, 2012.