A meshfree radial basis function method for nonlinear phi-four equation

Document Type : Research Paper

Authors

Department of Mathematics, Lovely Professional University, Phagwara, Punjab, India

Abstract

Radial basis function pseudospectral method is applied to obtain the solution for nonlinear Phi-four time dependant equation with nonhomogeneous initial and boundary conditions.  In this method, the efficient pseudospectral technique is combined with radial basis function to get the best of it. In the proposed method, the radial basis kernels are used to discretize the space derivatives in the Phi-four equation where as a time stepping technique is used to accord with the temporal part of the solution. The given Phi-four equation is transformed into a set of ordinary equations. An ode solver is used to solve the ordinary equations. An effective approach is used to choose the value of the shape parameter for radial basis function. Numerical results are presented to check the validity and accuracy of the method to solve the Phi-four equation.

Keywords

[1] E. J. Kansa, Multiquadrics—A scattered data approximation scheme with applications to computational fluiddynamic —II solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. Math. with Appl. 19(8) (1990) 147–161.
[2] J. Li and C. S. Chen, Some observations on unsymmetric radial basis function collocation methods for convectiondiffusion problems, Int. J. Numer. Meth. Engng. 57 (2003) 1085–1094.
[3] S. Chantasiriwan, Multiquadric collocation method for time-dependent heat conduction problems with temperaturedependent thermal properties, J Heat Transf Trans ASME 129(2) (2007) 109–113.
[4] Y. Duan, P.F. Tang, T.Z. Huang and S.J. Lai, Coupling projection domain decomposition method and Kansa’s method in electrostatic problems, Comput. Phys. Commun. 180(2) (2009) 209–214.
[5] W. Chen, L. Ye and H. Sun, Fractional diffusion equations by the Kansa method, Comput. Math. Appl. 59(5) (2010) 1614–1620.
[6] G. E. Fasshauer, RBF collocation methods as pseudospectral methods, WIT transactions on modelling and simulation. 2005.
[7] M. Uddin, S. Haq, and M. Ishaq, RBF-Pseudospectral Method for the Numerical Solution of Good Boussinesq Equation, Appl. Math. Sci. 6(49) (2012) 2403-2410.
[8] M. Uddin, RBF-PS scheme for solving the equal width equation, Appl. Math. Comput. 222 (2013) 619-631.
[9] A. Krowiak, Radial basis function-based pseudospectral method for static analysis of thin plates, Eng. Anal. Bound Elem. 71 (2016) 50-58 .
[10] D. Rostamy, M. Emamjome and S. Abbasbandy, A meshless technique based on pseudospectral radial basis functions method for solving the two- dimensional hyperbolic telegraph equation, Eur. Phys. J. Plus 132 (2017) 263.
[11] A. Chowdhury and A. Biswas, Singular solitons and numerical analysis of φ–four equation, Math. Sci. 6 (2012) 42.
[12] A.H. Bhrawy, L.M. Assas, and M.A. Alghamdi, An efficient spectral collocation algorithm for nonlinear Phi-four equations, Bound. Value Probl. 2013 (2013) 87.
[13] W. K. Zahra, W. A. Ouf, and M. S. El-Azab, A robust uniform B-spline collocation method for solving the generalized PHI-four equation, Appl. Appl. Math. 11(1) (2016) 384-396.
[14] H. Triki and AM. Wazwaz, Envelope solitons for generalized forms of the phi-four equation, J. King Saud Univ. Sci. 25 (2013) 129–133.
[15] M. Najafi, Using He’s Variational Method to Seek the Traveling Wave Solution of PHI-Four Equation, Int. J. Appl. Math. Res. 1(4) (2012) 659.
[16] S.T. Demiray and H. Bulut, Analytical solutions of Phi-four equation, Int. J. Optim. Control, Theor. Appl. 7(3) (2017) 275-280.
[17] G. E. Fasshauer and J. G. Zhang, On choosing “optimal” shape parameters for RBF approximation, Numer. Algorithms 45(1) (2007) 345-368.
[18] S. Rippa, An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Adv. Comput. Math. 11(2) (1999) 193-210.
Volume 13, Issue 1
March 2022
Pages 2043-2052
  • Receive Date: 14 February 2021
  • Accept Date: 18 October 2021