A numerical algorithm based on modified cubic trigonometric B-spline functions for computational modelling of hyperbolic-type wave equations

Purpose The main purpose of this work is the development of a numerical algorithm based on modified cubic trigonometric B-spline functions for computational modelling of hyperbolic-type wave equations. These types of equations describe a variety of physical models in the vibrations of structures, nonlinear optics, quantum field theory and solid-state physics, etc. Design/methodology/approach Dirichlet boundary conditions cannot be handled easily by cubic trigonometric B-spline functions. Then, a modification is made in cubic trigonometric B-spline functions to handle the Dirichlet boundary conditions and a numerical algorithm is developed. The proposed algorithm reduced the hyperbolic-type wave equations into a system of first-order ordinary differential equations (ODEs) in time variable. Then, stability-preserving SSP-RK54 scheme and the Thomas algorithm are used to solve the obtained system. The stability of the algorithm is also discussed. Findings A different technique based on modified cubic trigonometric B-spline functions is proposed which is quite different from the schemes developed (Abbas et al., 2014; Nazir et al., 2016) and depicts the computational modelling of hyperbolic-type wave equations. Originality/value To the best of the authors’ knowledge, this technique is novel for solving hyperbolic-type wave equations and the developed algorithm is free from quasi-linearization process and finite difference operators for time derivatives. This algorithm gives better results than the results discussed in literature (Dehghan and Shokri, 2008; Batiha et al., 2007; Mittal and Bhatia, 2013; Jiwari, 2015).

[1]  Sailing He,et al.  Wave splitting of the telegraph equation in R3 and its application to inverse scattering , 1993 .

[2]  R. K. Mohanty.  An unconditionally stable finite difference formula for a linear second order one space dimensional hyperbolic equation with variable coefficients , 2005, Appl. Math. Comput..

[3]  Ram Jiwari,et al.  Numerical simulation of two dimensional quasilinear hyperbolic equations by polynomial differential quadrature method , 2013 .

[4]  Mehdi Dehghan,et al.  Fourth-order compact solution of the nonlinear Klein-Gordon equation , 2009, Numerical Algorithms.

[5]  X. Ai,et al.  A discontinuous finite element method for hyperbolic thermal wave problems , 2004 .

[6]  Jacek Banasiak,et al.  Singularly perturbed telegraph equations with applications in the random walk theory , 1998 .

[7]  Mehdi Dehghan,et al.  High order compact solution of the one‐space‐dimensional linear hyperbolic equation , 2008 .

[8]  Athanassios G. Bratsos,et al.  The solution of the sine-gordon equation using the method of lines , 1996, Int. J. Comput. Math..

[9]  Mehdi Dehghan,et al.  The boundary integral equation approach for numerical solution of the one‐dimensional Sine‐Gordon equation , 2008 .

[10]  R. K. Mohanty.  An unconditionally stable difference scheme for the one-space-dimensional linear hyperbolic equation , 2004, Appl. Math. Lett..

[11]  Mehdi Dehghan,et al.  A numerical method for solving the hyperbolic telegraph equation , 2008 .

[12]  Ram Jiwari,et al.  A Differential Quadrature Algorithm for the Numerical Solution of the Second-Order One Dimensional Hyperbolic Telegraph Equation , 2012 .

[13]  Muhammad Abbas,et al.  The numerical solution of advection–diffusion problems using new cubic trigonometric B-splines approach , 2016 .

[14]  Ashok Puri,et al.  Digital signal propagation in dispersive media. , 1999 .

[15]  L. Vázquez,et al.  A LEGENDRE SPECTRAL METHOD FOR SOLVING THE NONLINEAR KLEIN GORDON EQUATION , 1996 .

[16]  Stefania Tomasiello,et al.  Numerical stability of DQ solutions of wave problems , 2011, Numerical Algorithms.

[17]  Mehdi Dehghan,et al.  Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions , 2009 .

[18]  Z. Fei,et al.  Two energy conserving numerical schemes for the Sine-Gordon equation , 1991 .

[19]  Mehdi Dehghan,et al.  Application of the dual reciprocity boundary integral equation technique to solve the nonlinear Klein-Gordon equation , 2010, Comput. Phys. Commun..

[20]  Qin Sheng,et al.  A predictor‐‐corrector scheme for the sine‐Gordon equation , 2000 .

[21]  Kamy Sepehrnoori,et al.  Numerical integration of semidiscrete evolution systems , 1981 .

[22]  Ram Jiwari,et al.  A numerical scheme based on differential quadrature method for numerical simulation of nonlinear Klein-Gordon equation , 2014 .

[23]  G. Walz.  Identities for trigonometric B-splines with an application to curve design , 1997 .

[24]  Mehdi Dehghan,et al.  Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method , 2010 .

[25]  M. Dehghan,et al.  A meshless method for numerical solution of a linear hyperbolic equation with variable coefficients in two space dimensions , 2009 .

[26]  Mehdi Dehghan,et al.  Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method , 2010 .

[27]  J. Perring,et al.  A Model unified field equation , 1962 .

[28]  The cubic spline solution of practical problems modelled by hyperbolic partial differential equations , 1976 .

[29]  Mehrdad Lakestani,et al.  Numerical solution of telegraph equation using interpolating scaling functions , 2010, Comput. Math. Appl..

[30]  Stefania Tomasiello,et al.  DQ Based Methods: Theory and Application to Engineering and Physical Sciences , 2012 .

[31]  Bengisen Pekmen,et al.  Differential quadrature solution of nonlinear Klein-Gordon and sine-Gordon equations , 2012, Comput. Phys. Commun..

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

[33]  R. C. Mittal,et al.  Numerical solution of second order one dimensional hyperbolic telegraph equation by cubic B-spline collocation method , 2013, Appl. Math. Comput..

[34]  Mohamed El-Gamel,et al.  A numerical algorithm for the solution of telegraph equations , 2007, Appl. Math. Comput..

[35]  R. C. Mittal,et al.  Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method , 2012, Comput. Phys. Commun..

[36]  Y. Tourigny,et al.  Product approximation for nonlinear Klein-Gordon equations , 1990 .

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

[38]  Athanassios Nikolis.  Numerical Solutions Of Ordinary Differential Equations With Quadratic Trigonometric Splines , 2004 .

[39]  Ram Jiwari,et al.  Cosine expansion based differential quadrature algorithm for numerical simulation of two dimensional hyperbolic equations with variable coefficients , 2015 .

[40]  Mohd. Salmi Md. Noorani,et al.  Numerical solution of sine-Gordon equation by variational iteration method , 2007 .

[41]  Mehdi Dehghan,et al.  Combination of meshless local weak and strong (MLWS) forms to solve the two dimensional hyperbolic telegraph equation , 2010 .

[42]  Mehdi Dehghan,et al.  A method based on meshless approach for the numerical solution of the two‐space dimensional hyperbolic telegraph equation , 2012 .

[43]  R. K. Mohanty,et al.  On the use of high order difference methods for the system of one space second order nonlinear hyperbolic equations with variable coefficients , 1996 .

[44]  G. B. Whitham,et al.  Linear and Nonlinear Waves: Whitham/Linear , 1999 .

[45]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[46]  Mehdi Dehghan,et al.  The use of Chebyshev cardinal functions for solution of the second‐order one‐dimensional telegraph equation , 2009 .

[47]  J. Gibbon,et al.  Solitons and Nonlinear Wave Equations , 1982 .

[48]  Muhammad Abbas,et al.  The application of cubic trigonometric B-spline to the numerical solution of the hyperbolic problems , 2014, Appl. Math. Comput..

[49]  Tom Lyche,et al.  Control curves and knot insertion for trigonometric splines , 1995, Adv. Comput. Math..

[50]  Mehdi Dehghan,et al.  Collocation and finite difference-collocation methods for the solution of nonlinear Klein-Gordon equation , 2010, Comput. Phys. Commun..

[51]  Stefania Tomasiello,et al.  Solving 2D-wave problems by the iterative differential quadrature method , 2011, Int. J. Comput. Math..

[52]  Ram Jiwari,et al.  Lagrange interpolation and modified cubic B-spline differential quadrature methods for solving hyperbolic partial differential equations with Dirichlet and Neumann boundary conditions , 2015, Comput. Phys. Commun..