Using 1D-IRBFN method for solving high-order nonlinear differential equations arising in models of active-dissipative systems

Strunin, Dmitry V. and Ngo-Cong, Duc and Bhanot, Rajeev (2015) Using 1D-IRBFN method for solving high-order nonlinear differential equations arising in models of active-dissipative systems. In: 1st Pan-American Congress on Computational Mechanics, (PANACM 2015) in conjunction with the XI Argentine Congress on Computational Mechanics (MECOM 2015), 27-29 April 2015, Buenos Aires, Argentina.

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We analyse a nonlinear partial differential equation modelling reaction-diffusion systems with nonlocal coupling and reaction fronts of gasless combustion. The equation is of active-dissipative type, nonlinear, with 6th-order spatial derivative. To numerically solve the equations we use the one-dimensional integrated radial basis function network (1D-IRBFN)method. The method has been previously developed and successfully applied to several problems
such as structural analysis, viscous and viscoelastic flows and fluid-structure interaction. A commonly used approach is to differentiate a function of interest to obtain approximate derivatives. However, this leads to a reduction in convergence rate for derivatives and this reduction is an increasing function of derivative order. Accordingly, differentiation magnifies errors. To avoid this problem and recognising that integration is a smoothing process, the proposed 1D-IRBFN method uses the integral formulation, where spectral approximants are utilised to represent highest-order derivatives under consideration and then integrated analytically to yield approximate expressions for lower-order derivatives and the function itself. Our preliminary results demonstrate good performance of the 1D-IRBFN algorithm for the equation under consideration.
Numerical solutions representing travelling waves are obtained, in agreement with the earlier studies.

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Item Type: Conference or Workshop Item (Commonwealth Reporting Category E) (Paper)
Refereed: Yes
Item Status: Live Archive
Additional Information: © The authors.
Faculty / Department / School: Current - Faculty of Health, Engineering and Sciences - School of Agricultural, Computational and Environmental Sciences
Date Deposited: 24 Jun 2016 01:40
Last Modified: 06 Sep 2016 04:06
Uncontrolled Keywords: 1D-IRBFN numerical method; nonlinear active-dissipative PDE
Fields of Research : 01 Mathematical Sciences > 0103 Numerical and Computational Mathematics > 010302 Numerical Solution of Differential and Integral Equations
01 Mathematical Sciences > 0102 Applied Mathematics > 010204 Dynamical Systems in Applications
Socio-Economic Objective: E Expanding Knowledge > 97 Expanding Knowledge > 970101 Expanding Knowledge in the Mathematical Sciences

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