Analytical solutions to the pulsed Klein-Gordon equation using Modified Variational Iteration Method (MVIM) and Boubaker Polynomials Expansion Scheme (BPES)

In this study, we propose an analytical solution to the Klein-Gordon equation in a pulsed stationary regime. The performed protocols are based on the modified variational iteration method MVIM and Boubaker polynomials expansion scheme BPES. The results are presented, and compared with some solutions proposed later in order to confirm the good accuracy of the protocols used.

[1]  Mehdi Dehghan,et al.  The radial basis functions method for identifying an unknown parameter in a parabolic equation with overspecified data , 2007 .

[2]  P. Clarkson,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering: References , 1991 .

[3]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[4]  G. Rowlands,et al.  Nonlinear Waves, Solitons and Chaos , 1990 .

[5]  A. Polyanin,et al.  Handbook of Nonlinear Partial Differential Equations , 2003 .

[6]  Juan I. Ramos,et al.  On the variational iteration method and other iterative techniques for nonlinear differential equations , 2008, Appl. Math. Comput..

[7]  Yuepeng Wang,et al.  Generalized solitary wave solutions for the Klein-Gordon-Schrödinger equations , 2009, Comput. Math. Appl..

[8]  Dogan Kaya,et al.  An implementation of the ADM for generalized one-dimensional Klein-Gordon equation , 2005, Appl. Math. Comput..

[9]  M. Dehghan.  A computational study of the one‐dimensional parabolic equation subject to nonclassical boundary specifications , 2006 .

[10]  Abdul-Majid Wazwaz,et al.  The modified decomposition method for analytic treatment of differential equations , 2006, Appl. Math. Comput..

[11]  Muhammad Aslam Noor,et al.  An iterative method with cubic convergence for nonlinear equations , 2006, Appl. Math. Comput..

[12]  Modified variational iteration method , 2007 .

[13]  Benny Y. C. Hon,et al.  An efficient numerical scheme for Burgers' equation , 1998, Appl. Math. Comput..

[14]  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..

[15]  Sirendaoreji.  A new auxiliary equation and exact travelling wave solutions of nonlinear equations , 2006 .

[16]  Abdul-Majid Wazwaz,et al.  Compactons, solitons and periodic solutions for some forms of nonlinear Klein–Gordon equations , 2006 .

[17]  Muhammad Aslam Noor,et al.  Variational iteration technique for solving higher order boundary value problems , 2007, Appl. Math. Comput..

[18]  Hamid Reza Mohammadi Daniali,et al.  Application of the variational iteration method to the Whitham-Broer-Kaup equations , 2007, Comput. Math. Appl..

[19]  John D. Gibbon,et al.  The sine-Gordon equation as a model classical field theory , 1975 .

[20]  Abdul-Majid Wazwaz,et al.  New travelling wave solutions to the Boussinesq and the Klein–Gordon equations , 2008 .

[21]  M. Noor,et al.  Traveling Wave Solutions of Seventh-order Generalized KdV Equations Using He's Polynomials , 2009 .

[22]  Karem Boubaker,et al.  On Modified Boubaker Polynomials: Some Differential and Analytical Properties of the New Polynomials Issued from an Attempt for Solving Bi-varied Heat Equation , 2007 .

[23]  Holger Wendland,et al.  Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree , 1995, Adv. Comput. Math..

[24]  A. Belhadj,et al.  Boubaker Polynomials Expansion Scheme-Related Heat Transfer Investigation Inside Keyhole Model , 2009 .

[25]  Mehdi Dehghan,et al.  On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation , 2005 .

[26]  K. B. Ben Mahmoud.  Solution to Heat Equation Inside Cryogenic Vessels Using Boubaker Polynomials , 2009 .

[27]  Zuntao Fu,et al.  New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations , 2001 .

[28]  Karem Boubaker,et al.  A solution to Bloch NMR flow equations for the analysis of hemodynamic functions of blood flow system using m-Boubaker polynomials , 2009 .

[29]  C. S. Chen,et al.  On the use of boundary conditions for variational formulations arising in financial mathematics , 2001, Appl. Math. Comput..

[30]  Lin Jin,et al.  Application of variational iteration method and homotopy perturbation method to the modified Kawahara equation , 2009, Math. Comput. Model..

[31]  O. Awojoyogbe,et al.  Cut-off cooling velocity profiling inside a keyhole model using the Boubaker polynomials expansion scheme , 2009 .

[32]  J. Bessrour,et al.  Numerical Distribution of Temperature as a Guide to Investigation of Melting Point Maximal Front Spatial Evolution During Resistance Spot Welding Using Boubaker Polynomials , 2009 .

[33]  Mehdi Dehghan,et al.  On the solution of the non-local parabolic partial differential equations via radial basis functions , 2009 .

[34]  Mehdi Dehghan,et al.  Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices , 2006, Math. Comput. Simul..

[35]  Mehdi Dehghan,et al.  Numerical solution of the Klein–Gordon equation via He’s variational iteration method , 2007 .

[36]  Salah M. El-Sayed,et al.  The decomposition method for studying the Klein–Gordon equation , 2003 .

[37]  Tamer A. Abassy,et al.  Modified variational iteration method for Boussinesq equation , 2007, Comput. Math. Appl..

[38]  M. Amlouk,et al.  Enhancement of pyrolysis spray disposal performance using thermal time-response to precursor uniform deposition , 2007 .

[39]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[40]  Paul Abbott,et al.  The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations , 2002 .

[41]  Sirendaoreji.  Auxiliary equation method and new solutions of Klein-Gordon equations , 2007 .

[42]  S. Fridjine,et al.  A NEW PARAMETER: AN ABACUS FOR OPTIMIZING PV–T HYBRID SOLAR DEVICE FUNCTIONAL MATERIALS USING THE BOUBAKER POLYNOMIALS EXPANSION SCHEME , 2009 .

[43]  S. Mohyud-Din,et al.  Homotopy Perturbation Method for Solving Nonlinear Higher-order Boundary Value Problems , 2008 .

[44]  E. Kansa.  MULTIQUADRICS--A SCATTERED DATA APPROXIMATION SCHEME WITH APPLICATIONS TO COMPUTATIONAL FLUID-DYNAMICS-- II SOLUTIONS TO PARABOLIC, HYPERBOLIC AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS , 1990 .

[45]  B. Hasslacher,et al.  Nonperturbative methods and extended-hadron models in field theory. II. Two-dimensional models and extended hadrons , 1974 .

[46]  Carsten Franke,et al.  Convergence order estimates of meshless collocation methods using radial basis functions , 1998, Adv. Comput. Math..