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 JAMP  Vol.6 No.4 , April 2018
Finite Difference Implicit Schemes to Coupled Two-Dimension Reaction Diffusion System
Abstract: In this research article, two finite difference implicit numerical schemes are described to approximate the numerical solution of the two-dimension modified reaction diffusion Fishers system which exists in coupled form. Finite difference implicit schemes show unconditionally stable and second-order accurate nature of computational algorithm also the validation and comparison of analytical solution, are done through the examples having known analytical solution. It is found that the numerical schemes are in excellent agreement with the analytical solution. We found, second-implicit scheme is much faster than the first with good rate of convergence also we used NVIDA devices to accelerate the computations and efficiency of the algorithm. Numerical results show our proposed schemes with use of HPC (High performance computing) are very efficient and reliable.
Cite this paper: Hasnain, S. , Saqib, M. and Al-Harbi, N. (2018) Finite Difference Implicit Schemes to Coupled Two-Dimension Reaction Diffusion System. Journal of Applied Mathematics and Physics, 6, 737-753. doi: 10.4236/jamp.2018.64066.
References

[1]   Fisher, R.A. (1937) The Wave of Advance of Advantageous Genes. Annals of Eugenics, 7, 355-369.
https://doi.org/10.1111/j.1469-1809.1937.tb02153.x

[2]   Canosa, J.C. (1973) On a Nonlinear Diffusion Equation Describing Population Growth. IBM Journal of Research Development, 17, 307-313.
https://doi.org/10.1147/rd.174.0307

[3]   Arnold, A.R., Showalter, K.S. and Tyson, J.J. (1987) Propagation of Chemical Reactions in Space. Journal of Chemical Education, 64, 740-743.
https://doi.org/10.1021/ed064p740

[4]   Tuckwell, H.C. (1988) Introduction to Theoretical Neurobiology. Cambridge University Press, Cambridge, UK.

[5]   Gazdag, G.J. and Canosa, J.C. (1974) Numerical Solutions of Fisher’s Equation. Journal of Applied Probability, 11, 445-457.
https://doi.org/10.2307/3212689

[6]   Abdullaev, U.G. (1994) Stability of Symmetric Traveling Waves in the Cauchy Problem for the KPP Equation. Journal of Differential Equations, 30, 377-386.

[7]   Logan, D.J. (1994) An Introduction to Nonlinear Partial Differential Equations. Wiley, New York.

[8]   Evans, D.J. and Sahimi, M.S. (1989) The Alternating Group Explicit (AGE) Iterative Method to Solve Parabolic and Hyperbolic Partial Differential Equations. Annals of Numerical Fluid Mechanics and Heat Transfer, 2, 283-389.

[9]   Segel, L.A. (1984) Chapter 8: Modelling Dynamic Phenomena in Molecular and Cellular Biology. Cambridge University Press, Cambridge.

[10]   Pao, C.V. (1981) Asymptotic Stability of Reaction-Diffusion Systems in Chemical Reactor and Combustion Theory. Journal of Mathematical Analysis and Applications, 82, 503-526.
https://doi.org/10.1016/0022-247X(81)90213-4

[11]   Schiesser, W.E. and Griffiths, G.W. (2009) A Compendium of Partial Differential Equation Models. Cambridge University Press, Cambridge.
https://doi.org/10.1017/CBO9780511576270

[12]   Argyris, J.A., Haase, M.H. and Heinrich, J.C. (1991) Finite Approximation to Two-Dimensional Sine Gordon Equations. Computer Methods in Applied Mechanics and Engineering, 86, 1-26.
https://doi.org/10.1016/0045-7825(91)90136-T

[13]   Aronson, D.G. and Weinberger, H.F. (1978) Multidimensional Non-Linear Diffusion Arising in Population Genetics. Advance in Mathematics, 30, 33-76.
https://doi.org/10.1016/0001-8708(78)90130-5

[14]   Hilborn, R.H. (1975) The Effect of Spatial Heterogeneity on the Persistence of Predator-Prey Interactions. Theoretical Population Biology, 8, 346-355.
https://doi.org/10.1016/0040-5809(75)90051-9

[15]   Hans, L.P. (1999) Computational Partial Differential Equations. Springer Verlag, Berlin.

[16]   Hayhoe, M.N. (1978) Numerical Study of Quasi-Analytic and Finite Difference Solutions of the Soil-Water Transfer Function. Soil Science, 125, 68-79.

[17]   Linker, P.L. (1990) Numerical Methods for Solving the Reactive Diffusion Equation in Complex Geometries. Technische Universitat Darmstadt, Darmstadt, Hessen.

[18]   Tang, T.S. and Weber, R.O. (1991) Numerical Study of Fisher’s Equations by a Petrov-Galerkin Finite Element Method. The ANZIAM Journal, 33, 27-38.
https://doi.org/10.1017/S0334270000008602

[19]   Khaled, K.A. (2001) Numerical Study of Fisher’s Diffusion-Reaction Equation by the Sinc Collocation Method. Journal of Computational and Applied Mathematics, 137, 245-255.
https://doi.org/10.1016/S0377-0427(01)00356-9

[20]   Ames, W.F. (1965) Nonlinear Partial Differential Equations in Engineering. Academic Press, New York.

[21]   Ames, W.F. (1969) Numerical Methods for Partial Differential Equations. Barnes and Noble, Inc., New York.

[22]   Noye, J.N. (1989) Finite Difference Methods for Partial Differential Equations. North-Holland Publishing Comp. Conference in Queen’s College, University of Melbourne, Australia, 23-87.

[23]   Mittal, R.C. and Kumar, S.K. (2009) Numerical Study of Fisher’s Equation by Wavelet Galerkin Method. International Journal of Computer Mathematics, 83, 287-298.

[24]   Dhawam, S.D., Kapoor, S.K. and Kumar, S.K. (2012) Numerical Method for Advection Diffusion Equation Using FEM and B-Splines. Journal of Computer Science, 3, 429-437.
https://doi.org/10.1016/j.jocs.2012.06.006

[25]   Tamseer, M.T., Srivastava, V.K. and Mishra, P.D. (2016) Numerical Simulation of Three Dimensional Advection-Diffusion Equations by Using Modified Cubic B-Spline Differential Quadrature Method. Asia Pacific Journal of Engineering Science and Technology, 2, 1-13.

[26]   Fletcher, C.A. (2016) Generating Exact Solutions of the Two-Dimensional Burgers Equations. International Journal for Numerical Methods in Fluids, 3, 213-216.
https://doi.org/10.1002/fld.1650030302

[27]   Hill, M.D. and Michael, R.M. (2008) Amdahl’s Law in the Multicore Era. Computer, 41, 33-38.
https://doi.org/10.1109/MC.2008.209

[28]   Heath, M.T. (1997) Scientific Computing, an Introductory Survey. University of Illinois at Urbana-Champaign, Urbana and Champaign, IL.

[29]   Ashraf, M.U., Fouz, F.F. and Fathy, A.E. (2016) Empirical Analysis of HPC Using Different Programming Models. International Journal of Modern Education and Computer Science, 6, 27-34.

[30]   Busenberg, S.N. and Travis, C.C. (1983) Epidemic Models with Spatial Spread Due to Population Migration. Journal of Mathematical Biology, 16, 181-198.
https://doi.org/10.1007/BF00276056

[31]   Crank, J.C. and Nicolson, P.N. (1947) A Practical Method for the Numerical Evaluation of Solutions of Partial Differential Equations of the Heat-Conduction Type. Mathematical Proceedings of the Cambridge Philosophical Society, 43, 50-67.
https://doi.org/10.1017/S0305004100023197

[32]   Lakoba, T.L. (2015) The Heat Equation in 2 and 3 Spatial Dimensions. In: MATH 337, University of Vermont.

[33]   Srivastava, V.K. and Tamsir, M.T. (2012) Crank-Nicolson Semi-Implicit Approach for Numerical Solutions of Two-Dimensional Coupled Nonlinear Burgers’ Equations. International Journal of Applied Mechanics and Engineering, 17, 571-581.

[34]   Smith, G.D. (1986) Numerical Solution of Partial Differential Equations: Finite Difference Methods. 3rd Edition, Oxford University Press, Oxford.

[35]   Saber, E.N. (1999) An Introduction to Differential Equations. 2nd Edition, Springer Verlag, New York.

[36]   Shalf, J.S., Sundip, D.S. and John, M.J. (2010) Exascale Computing Technology Challenges. In: Palma, J.M.L.M., Daydé, M., Marques, O. and Lopes, J.C., Eds., High Performance Computing for Computational Science—VECPAR 2010, Lecture Notes in Computer Science, Vol. 6449, Springer, Berlin, Heidelberg, 1-25.

[37]   Shekhar, B.S. (2007) Thousand Core Chips: A Technology Perspective. Proceedings of the 44th annual Design Automation Conference, San Diego, CA, 4-8 June 2007, 746-749.

[38]   Bajellan, A.A.A.F. (2015) Computation of the Convection-Diffusion Equation by the Fourth-Order Compact Finite Difference Method. Izmir Institute of Technology, January 2015.

[39]   Ge, Y., Tian, Z.F. and Zhang, J. (2013) An Exponential HO Compact ADI Method for 3D Unsteady Convection Diffusion Problems. Numerical Methods for Partial Differential Equations, 29, 186-205.
https://doi.org/10.1002/num.21705

 
 
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