The Differential Quadrature Solution of Reaction-Diffusion Equation Using Explicit and Implicit Numerical Schemes

Affiliation(s)

Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig, Egypt.

Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig, Egypt.

ABSTRACT

In this paper, two different numerical schemes, namely the Runge-Kutta fourth order method and the implicit Euler method with perturbation method of the second degree, are applied to solve the nonlinear thermal wave in one and two dimensions using the differential quadrature method. The aim of this paper is to make comparison between previous numerical schemes and detect which is more efficient and more accurate by comparing the obtained results with the available analytical ones and computing the computational time.

Cite this paper

M. Salah, R. Amer and M. Matbuly, "The Differential Quadrature Solution of Reaction-Diffusion Equation Using Explicit and Implicit Numerical Schemes,"*Applied Mathematics*, Vol. 5 No. 3, 2014, pp. 327-336. doi: 10.4236/am.2014.53033.

M. Salah, R. Amer and M. Matbuly, "The Differential Quadrature Solution of Reaction-Diffusion Equation Using Explicit and Implicit Numerical Schemes,"

References

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http://dx.doi.org/10.1364/AO.22.003169

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http://dx.doi.org/10.1016/S0960-0779(02)00435-6

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[8] R. G. Marcus, “Finite-Difference Schemes for Reaction-Diffusion Equations Modeling Predator-Prey Interactions in MATLAB,” Bulletin of Mathematical Biology, Vol. 69, No. 3, 2007, pp. 931-956. http://dx.doi.org/10.1007/s11538-006-9062-3

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[11] G. Meral and M. S. Tezer, “Solution of Nonlinear Reaction-Diffusion Equation by Using Dual Reciprocity Boundary Element Method with Finite Difference or Least Squares Method,” Advances in Boundary Element Techniques, Vol. 3, 2008, pp. 317322.

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http://dx.doi.org/10.1007/978-1-4471-0407-0

[13] T. Y. Wu and G. R. Liu, “A Differential Quadrature as a Numerical Method to Solve Differential Equations,” Computational Mechanics, Vol. 24, No. 3, 1999, pp. 197-205.

http://dx.doi.org/10.1007/s004660050452

[14] V. Kajal, “Numerical Solutions of Some Reaction-Diffusion Equations by Differential Quadrature Method,” International Journal of Applied Mathematics and Mechanics, Vol. 6, No. 14, 2010, pp. 68-80.

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[16] W. Y. Yang, W. Cao, T. Chung, J. Morris, et al., “Applied Numerical Methods Using Matlab,” John Wiley & Sons, Hoboken, 2005. http://dx.doi.org/10.1002/0471705195

[17] J. S. Nadjafi and A. Ghorbani, “He’s Homotopy Perturbation Method: An Effective Tool for Solving Nonlinear Integral and Integro-Differential Equations,” Computers and Mathematics with Applications, Vol. 58, No. 11-12, 2009, pp. 2379-2390.

http://dx.doi.org/10.1016/j.camwa.2009.03.032

[18] H. S. Prasad and Y. N. Reddy, “Numerical Solution of Singularly Perturbed Differential-Difference Equations with Small Shifts of Mixed Type by Differential Quadrature Method,” American Journal of Computational and Applied Mathematics, Vol. 2, No. 1, 2012, pp. 46-52.

http://dx.doi.org/10.5923/j.ajcam.20120201.09

[19] J. P. Hambleton and S. W. Sloan, “A Perturbation Method for Optimization of Rigid Block Mechanisms in the Kinematic Method of Limit Analysis,” Computers and Geotechnics, Vol. 48, 2013, pp. 260-271.

http://dx.doi.org/10.1016/j.compgeo.2012.07.012

[20] M. Bastani and D. K. Salkuyeh, “A Highly Accurate Method to Solve Fisher’s Equation,” Indian Academy of Sciences, Vol. 78, No. 3, 2012, pp. 335-346.

[21] E. Kreyszig, “Advanced Engineering Mathematics,” John Wiley & Sons, Columbus, 2006.

[1] J. Opsal, A. Rosencwaig and D. L. Willenborg, “Thermal-Wave Detection and Thin-Film Thickness Measurements with Laser Beam Deflection,” Applied Optics, Vol. 22, No. 20, 1983, pp. 3169-3176.

http://dx.doi.org/10.1364/AO.22.003169

[2] Z. Y. Yan, “Generalized Method and Its Application in the Higher-Order Nonlinear Schrodinger Equation in Nonlinear Optical Fibres,” Chaos Solitons and Fractals, Vol. 16, No. 5, 2003, pp. 759-766.

http://dx.doi.org/10.1016/S0960-0779(02)00435-6

[3] J. Mei, H. Zhang and D. Jiang, “New Exact Solutions for a Reaction-Diffusion Equation and a Quasi-Camassa Holm Equation,” Applied Mathematics E-Notes, Vol. 4, 2004, pp. 85-91.

[4] E. S. Fahmy and H. A. Abdusalam, “Exact Solutions for Some Reaction Diffusion Systems with Nonlinear Reaction Polynomial Terms,” Applied Mathematical Sciences, Vol. 3, No. 11, 2009, pp. 533-540.

[5] M. S. H. Chowdhury and I. Hashim, “Analytical Solution for Cauchy Reaction-Diffusion Problems by Homotopy Perturbation Method,” Sains Malaysiana, Vol. 39, No. 3, 2010, pp. 495-504.

[6] S. Puri and K. Wiese, “Perturbative Linearization of Reaction-Diffusion Equations,” Journal of Physics A: Mathematical and General, Vol. 36, No. 8, 2003, pp. 2043-2054. http://dx.doi.org/10.1088/0305-4470/36/8/303

[7] L. D. Ropp, N. J. Shadid and C. C. Ober, “Studies of the Accuracy of Time Integration Methods for Reaction-Diffusion Equations,” Journal of Computational Physics, Vol. 194, No. 2, 2004, pp. 544-574. http://dx.doi.org/10.1016/j.jcp.2003.08.033

[8] R. G. Marcus, “Finite-Difference Schemes for Reaction-Diffusion Equations Modeling Predator-Prey Interactions in MATLAB,” Bulletin of Mathematical Biology, Vol. 69, No. 3, 2007, pp. 931-956. http://dx.doi.org/10.1007/s11538-006-9062-3

[9] B. Liu, M. B. Allen, H. Kojouharov, B. Chen, et al., “Finite-Element Solution of Reaction-Diffusion Equations with Advection,” Computational Mechanics, 1996, pp. 3-12.

[10] X. Christos and L. Oberbroeckling, “On the Finite Element Approximation of Systems of Reaction-Diffusion Equations by p/hp Methods,” Global Science, Vol. 28, No. 3, 2010, pp. 386-400.

[11] G. Meral and M. S. Tezer, “Solution of Nonlinear Reaction-Diffusion Equation by Using Dual Reciprocity Boundary Element Method with Finite Difference or Least Squares Method,” Advances in Boundary Element Techniques, Vol. 3, 2008, pp. 317322.

[12] C. Shu, “Differential Quadrature and Its Application in Engineering,” Springer Verlag, London, 2000.

http://dx.doi.org/10.1007/978-1-4471-0407-0

[13] T. Y. Wu and G. R. Liu, “A Differential Quadrature as a Numerical Method to Solve Differential Equations,” Computational Mechanics, Vol. 24, No. 3, 1999, pp. 197-205.

http://dx.doi.org/10.1007/s004660050452

[14] V. Kajal, “Numerical Solutions of Some Reaction-Diffusion Equations by Differential Quadrature Method,” International Journal of Applied Mathematics and Mechanics, Vol. 6, No. 14, 2010, pp. 68-80.

[15] G. Meral, “Solution of Density Dependent Nonlinear Reaction-Diffusion Equation Using Differential Quadrature Method,” World Academy of Science, Engineering and Technology, Vol. 41, 2010, pp. 1178-1183.

[16] W. Y. Yang, W. Cao, T. Chung, J. Morris, et al., “Applied Numerical Methods Using Matlab,” John Wiley & Sons, Hoboken, 2005. http://dx.doi.org/10.1002/0471705195

[17] J. S. Nadjafi and A. Ghorbani, “He’s Homotopy Perturbation Method: An Effective Tool for Solving Nonlinear Integral and Integro-Differential Equations,” Computers and Mathematics with Applications, Vol. 58, No. 11-12, 2009, pp. 2379-2390.

http://dx.doi.org/10.1016/j.camwa.2009.03.032

[18] H. S. Prasad and Y. N. Reddy, “Numerical Solution of Singularly Perturbed Differential-Difference Equations with Small Shifts of Mixed Type by Differential Quadrature Method,” American Journal of Computational and Applied Mathematics, Vol. 2, No. 1, 2012, pp. 46-52.

http://dx.doi.org/10.5923/j.ajcam.20120201.09

[19] J. P. Hambleton and S. W. Sloan, “A Perturbation Method for Optimization of Rigid Block Mechanisms in the Kinematic Method of Limit Analysis,” Computers and Geotechnics, Vol. 48, 2013, pp. 260-271.

http://dx.doi.org/10.1016/j.compgeo.2012.07.012

[20] M. Bastani and D. K. Salkuyeh, “A Highly Accurate Method to Solve Fisher’s Equation,” Indian Academy of Sciences, Vol. 78, No. 3, 2012, pp. 335-346.

[21] E. Kreyszig, “Advanced Engineering Mathematics,” John Wiley & Sons, Columbus, 2006.