×

Nonstandard finite difference schemes for Michaelis–Menten type reaction-diffusion equations. (English) Zbl 1255.65141

Summary: We compare and investigate the performance of the exact scheme of the Michaelis–Menten (M–M) ordinary differential equation with several new nonstandard finite difference (NSFD) schemes that we construct using Mickens’ rules. Furthermore, the exact scheme of the M–M equation is used to design several dynamically consistent NSFD schemes for related reaction-diffusion equations, advection-reaction equations, and advection-reaction-diffusion equations. Numerical simulations that support the theory and demonstrate computationally the power of NSFD schemes are presented.

MSC:

65L12 Finite difference and finite volume methods for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] O’Malley, Singular perturbation methods for ordinary differential equations, Applied mathematical science series (1991) · doi:10.1007/978-1-4612-0977-5
[2] Segel, The quasi-steady-state assumption: a case study in perturbation, SIAM Rev 31 pp 446– (1989) · Zbl 0679.34066 · doi:10.1137/1031091
[3] Dimitrov, Positive and elementary stable nonstandard numerical methods with applications to predator-prey models, J Comput Appl Math 189 pp 98– (2006) · Zbl 1087.65068 · doi:10.1016/j.cam.2005.04.003
[4] D. T. Dimitrov H. V. Kojouharov Nonstandard numerical methods for a class of predator-prey models with predator interference 2007 · Zbl 1130.37410
[5] Mickens, Nonstandard finite difference models of differential equations (1994) · Zbl 0810.65083
[6] Applications of nonstandard finite difference schemes (2000)
[7] Advances in the applications of nonstandard finite difference schemes (2005) · Zbl 1079.65005
[8] Mickens, An exact discretization of Michaelis-Menten type population equations, J Biol Dyn 5 pp 391– (2011) · Zbl 1225.92048 · doi:10.1080/17513758.2010.515690
[9] Corless, On the Lambert W function, Adv Comput Math 5 pp 329– (1996) · Zbl 0863.65008 · doi:10.1007/BF02124750
[10] Anguelov, Topological dynamic consistency of nonstandard finite difference schemes for dynamical systems, J Difference Equation Appl 17 pp 1769– (2011) · Zbl 1230.65089 · doi:10.1080/10236198.2010.488226
[11] Schnell, Enzyme kinetics at high enzyme concentration, Bull Math Biol 62 pp 483– (2000) · Zbl 1323.92099 · doi:10.1006/bulm.1999.0163
[12] Handbook of mathematical functions with formulas, graphs and mathematical tables (1972) · Zbl 0543.33001
[13] CRC standard mathematical tables (1987)
[14] Murray, Mathematical biology I: an introduction (2002) · Zbl 1006.92001
[15] Miller, Nonlinear Volterra integral equations, Mathematics lecture note series (1971) · Zbl 0448.45004
[16] Walter, Differential and integral inequalities (1970) · doi:10.1007/978-3-642-86405-6
[17] Anguelov, Contributions to the mathematics of the nonstandard finite difference method and applications, Numer Methods Partial Differential Equations 17 pp 518– (2001) · Zbl 0988.65055 · doi:10.1002/num.1025
[18] Anguelov, On non-standard finite difference models of reaction-diffusion equations, J Comput Appl Math 175 pp 11– (2005) · Zbl 1070.65071 · doi:10.1016/j.cam.2004.06.002
[19] Dumont, Non-standard finite difference methods for vibro-impact problems, Proc R Soc Lond Ser A: Math Phys Eng Sci 461A pp 1927– (2005) · doi:10.1098/rspa.2004.1425
[20] Logan, Nonlinear differential equations (1994) · Zbl 0834.35001
[21] Smoller, Shock waves and reaction diffusion equations (1983) · Zbl 0508.35002 · doi:10.1007/978-1-4684-0152-3
[22] Mickens, Relation between the time and space step-sizes in nonstandard finite-difference schemes for the Fisher equation, Numer Methods Partial Differential Equations 13 pp 51– (1997) · Zbl 0872.65080 · doi:10.1002/(SICI)1098-2426(199701)13:1<51::AID-NUM4>3.0.CO;2-L
[23] Morton, Numerical solutions of partial differential equations (1994)
[24] Varga, Matrix iterative analysis (1962)
[25] Berman, Nonnegative matrices in the mathematical sciences (1979) · Zbl 0484.15016
[26] Mickens, Nonstandard finite difference scheme for scalar advection-reaction partial differential equations, Neural, Parallel Sci Comput 12 pp 163– (2004) · Zbl 1071.65123
[27] H. V. Kojouharov B. M. Chen Nonstandard methods for advection-diffusion-reaction equations, Applications of nonstandard finite difference schemes R. E. Mickens World Scientific Singapore 2000 55 108
[28] Lubuma, Solving singularly perturbed advection reaction equation via non-standard finite difference methods, Math Methods Appl Sci 30 pp 1627– (2007) · Zbl 1132.35008 · doi:10.1002/mma.858
[29] Garba, Dynamically-consistent nonstandard finite difference method for an epidemic model, Math Comput Model 53 pp 131– (2011) · Zbl 1211.65102 · doi:10.1016/j.mcm.2010.07.026
[30] Chapwanya, From enzyme kinetics to epidemiological Models with Michaelis-Menten contact rate: design of nonstandard finite difference schemes, Computers and Mathematics with Applications, doi:10.1016/j.camwa.2011.12.058 · Zbl 1252.65131
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.