A new numerical treatment based on Lucas polynomials for 1D and 2D sinh-Gordon equation

Abstract In this paper, a new mixed method based on Lucas and Fibonacci polynomials is developed for numerical solutions of 1D and 2D sinh-Gordon equations. Firstly time variable discretized by central finite difference and then unknown function and its derivatives are expanded to Lucas series. With the help of these series expansion and Fibonacci polynomials, matrices for differentiation are derived. With this approach, finding the solution of sinh-Gordon equation transformed to finding the solution of an algebraic system of equations. Lucas series coefficients are acquired by solving this system of algebraic equations. Then by plugginging these coefficients into Lucas series expansion numerical solutions can be obtained consecutively. The main objective of this paper is to demonstrate that Lucas polynomial based method is convenient for 1D and 2D nonlinear problems. By calculating L2 and L∞ error norms of some 1D and 2D test problems efficiency and performance of the proposed method is monitored. Acquired accurate results confirm the applicability of the method.

[1]  Abdul-Majid Wazwaz,et al.  One and two soliton solutions for the sinh-Gordon equation in (1+1), (2+1) and (3+1) dimensions , 2012, Appl. Math. Lett..

[2]  I. Cabrera-Carnero,et al.  Noncommutative integrable field theories in 2d , 2003 .

[3]  Mehdi Dehghan,et al.  A Not-a-Knot meshless method using radial basis functions and predictor-corrector scheme to the numerical solution of improved Boussinesq equation , 2010, Comput. Phys. Commun..

[4]  Jibin Li,et al.  Bounded travelling wave solutions for the (n + 1)-dimensional sine- and sinh-Gordon equations , 2005 .

[5]  John D. Hunter,et al.  Matplotlib: A 2D Graphics Environment , 2007, Computing in Science & Engineering.

[6]  Liang Gao,et al.  Bifurcations of traveling wave solutions for a generalized Sinh-Gordon equation , 2008 .

[7]  Abdul-Majid Wazwaz,et al.  Exact solutions for the generalized sine-Gordon and the generalized sinh-Gordon equations , 2006 .

[8]  Abdul-Majid Wazwaz,et al.  Exact solutions to the double sinh-gordon equation by the tanh method and a variable separated ODE method , 2005 .

[9]  Mehdi Dehghan,et al.  A new operational matrix for solving fractional-order differential equations , 2010, Comput. Math. Appl..

[10]  Piero Filipponi,et al.  Derivative Sequences of Fibonacci and Lucas Polynomials , 1991 .

[11]  H. Kheiri,et al.  Exact solutions for the double sinh-Gordon and generalized form of the double sinh-Gordon equations by using (G'/G)-expansion method , 2010, Turkish Journal of Physics.

[12]  Mehdi Dehghan,et al.  A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions , 2008, Math. Comput. Simul..

[13]  K. Chow.  A class of doubly periodic waves for nonlinear evolution equations , 2002 .

[14]  Mehdi Dehghan,et al.  The numerical solution of the two–dimensional sinh-Gordon equation via three meshless methods , 2015 .

[15]  A. Kurnaz,et al.  A new Fibonacci type collocation procedure for boundary value problems , 2013 .

[16]  Boundary quantum field theories with infinite resonance states , 2001, hep-th/0107082.

[17]  A. A. Soliman,et al.  Numerical solutions of nonlinear evolution equations using variational iteration method , 2007 .

[18]  Xiaolin Li,et al.  Analysis and application of the element-free Galerkin method for nonlinear sine-Gordon and generalized sinh-Gordon equations , 2016, Comput. Math. Appl..