Numerical simulations for coupled pair of diffusion equations by MLPG method

CMES - Computer Modeling in Engineering and Sciences

Volumn 71, Issue 1, 2011, Pages 15-37

  Abbasbandy, S ^a   Sladek, V ^b   Shirzadi, A ^a   Sladek, J ^b  

^a IMAM KHOMEINI INTERNATIONAL UNIVERSITY   (Iran)

^b INSTITUTE OF CONSTRUCTION AND ARCHITECTURE   (Slovakia)

Author keywords

Boundary conditions; MLPG; Moving least squares approximation; Numerical integration procedures; System of parabolic partial differential equations

Indexed keywords

DIFFUSION EQUATIONS; EXACT SOLUTION; FIELD VARIABLES; ILLUSTRATIVE EXAMPLES; INITIAL-BOUNDARY VALUE PROBLEMS; LINEAR INTERPOLATION; LOCAL INTEGRAL EQUATIONS; MLPG; MLPG METHOD; MOVING LEAST SQUARES; MOVING LEAST SQUARES APPROXIMATION; NONEQUILIBRIUM SYSTEM; NONLINEAR DIFFUSION EQUATIONS; NUMERICAL INTEGRATIONS; NUMERICAL SIMULATION; SELF-ORGANIZATIONS; SPATIAL VARIATIONS; SYSTEM OF PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS; TIME DEPENDENCE;

BOUNDARY CONDITIONS; DIFFERENTIATION (CALCULUS); FINITE DIFFERENCE METHOD; INITIAL VALUE PROBLEMS; INTEGRAL EQUATIONS; NONLINEAR EQUATIONS; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS;

LEAST SQUARES APPROXIMATIONS;

EID: 79955487465     PISSN: 15261492     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (37)

1. Abbasbandy, S.; Shirzadi, A. (2010): A meshless method for two-dimensional diffusion equation with an integral condition. Eng. Anal. Bound. Elem., vol. 34, pp. 1031-1037.
2. Abbasbandy, S.; Shirzadi, A. (2011): MLPG method for two-dimensional diffusion equation with Neumann's and non-classical boundary conditions. Appl. Numer. Math., vol. 61, pp. 170-180.
3. Atluri, S. N. (2004): The Meshless Method, (MLPG) For Domain & BIE Discretizations. Tech. Science Press, Forsyth.
4. Atluri, S. N.; Han, Z.; Shen, S. (2003): Meshless Local Patrov-Galerkin (MLPG) approaches for weaklysingular traction & displacement boundary integral equations. CMES: Computer Modeling in Engineering & Sciences, vol. 4, no. 5, pp. 507-517.
5. Atluri,S. N.; Shen, S. (2002): The Meshless Local Petrov-Galerkin (MLPG) Method: A Simple & Less-costly Alternative to the Finite Element and Boundary Element Methods. CMES, vol. 3, pp. 11-51. (Pubitemid 135695145)
6. Brunner, H.; Ling, L.; Yamamoto, M. (2010): Numerical simulations of 2D fractional subdiffusion problems. J. Comput. Phys., vol. 229, pp. 6613-6622.
7. Castets, V.; Dulos, E.; Boissonade, J.; De Kepper, P. (1990): Experimental evidence of a sustained Turing-type equilibrium chemical pattern. Phys. Rev. Lett., vol. 64, pp 2953-2956.
8. Ching, H.-K.; Batra, R. C. (2001): Determination of crack tip fields in linear elastostatics by the meshless local Petrov-Galerkin (MLPG) method. CMES: Computer Modeling in Engineering & Sciences, vol. 2, pp. 273-289. (Pubitemid 33626575)
9. Ebeling,W. (1976): Strukturbildung bei Irreversiblen Prozessen. Eine Einfürung in die Theorie dissipativer Strukturen. BSB B.G. Teubner Verlagsgesellschaft, Leipzig.
10. Haken, H. (1975): Cooperative phenomena in systems far from thermal equilibrium and in nonphysical systems. Rev. Mod. Phys., vol. 47, pp. 67-121.
11. Haken, H. (1978): Synergetics - An Introduction, Nonequilibrium Phase Transitions and Self-Organization in Physics, Chemistry and Biology. Springer-Verlag, Berlin.
12. Kondo, S.; Asai, R. (1995): A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus. Nature, vol. 376, pp. 765-768.
13. Lancaster, P.; Salkauskas, K. (1981): Surfaces generated by moving least square methods. Math. Comput., vol. 37, pp. 141-158.
14. Libre, N. A.; Emdadi, A.; Kansa, E. J.; Shekarchi, M.; Rahimian, M. (2009): A multiresolution prewavelet-based adaptive refinement scheme for RBF approximations of nearly singular problems. Eng. Anal. Bound. Elem., vol. 33, pp. 901-914.
15. Ling, L.; Kansa, E. J. (2004): Preconditioning for radial basis functions with domain decomposition methods. Math. Compt. Modelling, vol. 40, no. 13, pp. 1413-1427. (Pubitemid 40607887)
16. Ling, L.; Hon, Y. C. (2005): Improved numerical solver for Kansa's method based on affine space decomposition. Eng. Anal. Bound. Elem., vol. 29, no. 12, pp. 1077-1085. (Pubitemid 41706173)
17. Liu, G. R. (2003): Mesh Free Methods, Moving Beyond the Finite Element Method. CRC Press, Boca Raton.
18. Lotka, A. J. (1956): Elements of Mathematical Biology. Dover, New York.
19. Meinhardt, H. (2009): The Algorithmic Beauty of Sea Shells. 4th ed., Springer-Verlag, Berlin.
20. Miura, T.; Shiota, K.; Morriss-Kay, G.; Maini, P. (2006): Mixed mode pattern in Doublefoot mutant mouse limb-Turing reaction-diffusion model on a growing domain limb development. J. Theor. Biol., vol. 240, pp. 562-573. (Pubitemid 43765938)
21. Moore, P. K.; Flaherty, J. E. (1992): Adaptive local overlapping grid methods for parabolic systems in two space dimensions. J. Comput. Phys., vol. 98, no. 1, pp. 54-63.
22. Nicolis, G.; Prigogine, I. (1977): Self-Organization in Nonequilibrium Systems. John Wiley & Sons, New York.
23. Prigogine, I.; Nicolis, G. J. (1967): On symmetry-breaking instabilities in dissipative systems. J. Chem. Phys., vol. 46, pp. 3542-3550.
24. Prigogine, I.; Lefever, R. (1968): Symmetry breaking instabilities in dissipative systems II. J. Chem. Phys., vol. 48, pp. 1695-1700.
25. Ouyang, Q.; Swinney, H. (1991): Transition from a uniform state to hexagonal and striped Turing patterns. Nature. vol. 352, pp. 610-612.
26. Schlögl, F. (1971a): On stability of steady states. Z. Physik, vol. 243, pp. 303-310.
27. Schlögl, F. (1971b): On thermodynamics near a steady state. Z. Physik, vol. 248, pp. 446-458.
28. Schlögl, F. (1972): Chemical reaction models for non-equilibrium phase transitions. Z. Physik, vol. 253, pp. 147-161.
29. Segal, L.; Jackson, J. (1972): Dissipative structure: an explanation and an ecological example. J. Theor. Biol., vol. 37, pp. 545-559.
30. Sladek, J.; Sladek, V.; Atluri, S. N. (2004): Meshless local Petrov-Galerkin method for heat conduction problem in an anisotropic medium. CMES: Computer Modeling in Engineering & Sciences, vol. 6, pp. 309-318.
31. Sladek, V.; Sladek, J.; Tanaka, M.; Zhang, C.-H. (2005): Transient heat conduction in anisotropic and functionally graded media by local integral equations. Eng. Anal. Bound. Elem., vol. 29, pp. 1047-1065. (Pubitemid 41586131)
32. Sladek, J.; Sladek, V.; Tan, C. L.; Atluri, S. N. (2008): Analysis of transient heat conduction in 3D anisotropic functionally graded solids by the MLPG method. CMES: Computer Modeling in Engineering & Sciences, vol. 32, pp. 161-174.
33. Smith, G. D. (1978): Numerical Solution of Partial Differential Equations: Finite Diffrence Methods. Clarendon Press, Oxford.
34. Turing, A. M. (1952): The chemical basis of morphogenesis. Phil. Trans. Roy. Soc., B237, pp. 37-72.
35. Volterra, V. (1931): Lesons sur la theorie mathematique de la lutte pur la vie. Gauthier-Villars, Paris.
36. Zhabotinskiy, A. M. (1964): Periodic processes of the oxydation of malonic acid in solution. Biofizika, vol. 9, pp. 306-311.
37. Zhu, T.; Zhang, J.-D.; Atluri,S. N. (1998): A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach. Computational Mechanics, vol. 21, pp. 223-235. (Pubitemid 128566412)