Abstract
An algorithm for the reaction–diffusion system including model problems Brusselator, Schnakenberg and Gray–Scott is introduced. The integration of the system is managed by combining the Crank–Nicolson method in time and the cubic trigonometric B-spline collocation method in space. Our aim here is to provide a new code to understand and implement reaction–diffusion-type events. Some problems chosen from the literature to illustrate the efficiency of the algorithm are studied for each model problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abbas M, Majid AA, Ismail AIM, Rashid A (2014a) The application of the cubic trigonometric B-spline to the numerical solution of the hyperbolic problems. Appl Math Comput 239:74–88
Abbas M, Majid AA, Ismail AIM, Rashid A (2014b) Numerical method using cubic trigonometric B-spline technique for nonclassical diffusion problems. Abstr Appl Anal 849682:11
Arora G, Joshi V (2016) Comparison of numerical solution of 1D hyperbolic telegraph equation using B-Spline and trigonometric B-Spline by differential quadrature method. Indian J Sci Technol 9(45). https://doi.org/10.17485/ijst/2016/v9i45/106356
Ay B, Dag I, Gorgulu MZ (2015) Trigonometric quadratic B-spline subdomain Galerkin algorithm for the Burgers’ equation. Open Phys 13:400–406
Chou CS, Zhang Y, Zhao R, Nie Q (2007) Numerical methods for stiff reaction–diffusion systems. Discr Contin Dyn Syst Ser B 7:515–525
Craster RV, Sassi R (2006) Spectral algorithms for reaction–diffussion equations. Technical Report. Note del Polo No. 99. https://air.unimi.it/handle/2434/24276?mode=simple.316#.W56v4s4zYdU
Ersoy O, Dag I (2015) Numerical solutions of the reaction–diffusion system by using exponential cubic B-spline collocation algorithms. Open Phys 13:414–427
Fairweather G, Meade D (1989) A survey of spline collocation methods for the numerical solution of differential equations. In: Diaz JC (ed) Mathematics for large scale computing, vol 120. Lecture notes in pure and applied mathematics. Marcel Dekker, New York, pp 297–341
Garcia-Lopez CM, Ramos JI (1996) Linearized O-methods part II: reaction–diffusion equations. Comput Methods Appl Mech Eng 137:357–378
Gray P, Scott SK (1984) Autocatalytic reactions in the isothermal, continuous stirred tank reactor: oscillations and instabilities in the system A+2B 3B. B C Chem Eng Sci 39:1087–1097
Gupta Y, Kumar M (2011) A computer based numerical method for singular boundary value problems. Int J Comput Appl 1:0975–8887
Hamid NNA, Majid AA, Ismail AIM (2010) Cubic trigonometric B-spline applied to linear two-point boundary value problems of order two. World Acad Sci Eng Technol 47:478–803
Madzvamuse A, Wathen AJ, Maini PK (2003) A moving grid finite element method applied to a biological pattern generator. J Comput Phys 190:478–500
Mittal RC, Rohila R (2016) Numerical simulation of reaction–diffusion systems by modified cubic B-spline differential quadrature method. Chaos Solitons Fractals 92(1):1339–1351
Nazir T, Abbas M, Ismail AIM, Majid AA, Rashid A (2016) The numerical solution of advection-diffusion problems using new cubic trigonometric B-splines approach. Appl Math Model 40(7–8):4586–4611
Nikolis A (2004) Numerical solutions of ordinary differential equations with quadratic trigonometric splines. Appl Math E Notes 4:142–149
Nikolis A, Seimenis I (2005) Solving dynamical systems with cubic trigonometric splines. Appl Math E Notes 5:116–123
Prigogine I, Lefever R (1968) Symmetry breaking instabilities in dissipative systems. J Chem Phys 48:1695–1700
Rubin SG, Graves RA (1975) A cubic spline approximation for problems in fluid mechanics. Nasa TR R-436, Washington
Ruuth SJ (1995) Implicit–explicit methods for reaction–diffusion problems in pattern formation. J Math Biol 34:148–176
Sahin A (2009) Numerical solutions of the reaction–diffusion equations with B-spline finite element method Ph. D. dissertation. Department of Mathematics. Eskişehir Osmangazi University, Eskiş ehir
Schoenberg IJ (1964) On trigonometric spline interpolation. J Math Mech 13:795–826
Schnakenberg J (1979) Simple chemical reaction systems with limit cycle behavior. J Theor Biol 81:389–400
Thomas D (1975) Artificial enzyme membranes, transport, memory and oscillatory phenomena. Anal Control immobilized Enzyme Syst 8:115–150
Wang G, Chen Q, Zhou M (2004) NUAT B-spline curves. Comput Aided Geom Design 21:193–205
Zegeling PA, Kok HP (2004) Adaptive moving mesh computations for reaction–diffusion systems. J Comput Appl Math 168(1–2):519–528
Zin SM, Majid AA, Ismail AIM, Abbas M (2014a) Cubic trigonometric B-spline approach to numerical solution of wave equation. Int J Math Comput Phys and Quan Eng 8(10):1212–1216
Zin SM, Abbas M, Majid AA, Ismail AIM (2014b) A new trigonometric spline approach to numerical solution of generalized nonlinear Klien–Gordon equation. PLoS One 9(5):e95774
Acknowledgements
Portion of the paper was presented in International Conference on Quantum Science and Applications (ICQSA-2016) in Eskisehir/Turkey. The authors would like to thank the anonymous reviewers for their comments and suggestions for improving the article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jose Alberto Cuminato.
Rights and permissions
About this article
Cite this article
Onarcan, A.T., Adar, N. & Dag, I. Trigonometric cubic B-spline collocation algorithm for numerical solutions of reaction–diffusion equation systems. Comp. Appl. Math. 37, 6848–6869 (2018). https://doi.org/10.1007/s40314-018-0713-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40314-018-0713-4