Numerical simulation of second-order hyperbolic telegraph type equations with variable coefficients

Abstract In this article, the authors proposed a numerical scheme based on Crank–Nicolson finite difference scheme and Haar wavelets to find numerical solutions of different types of second order hyperbolic telegraph equations (i.e. telegraph equation with constant coefficients, with variable coefficients, and singular telegraph equation). This work is an extension of the scheme by Jiwari (2012) for hyperbolic equations. The use of Haar basis function is made with multiresolution analysis to get the fast and accurate results on collocation points. The convergence of the proposed scheme is proved by doing its error analysis. Four test examples are considered to demonstrate the accuracy and efficiency of the scheme. The scheme is easy and very suitable for computer implementation and provides numerical solutions close to the exact solutions and available in the literature.

[1]  S. H. Behiry,et al.  Numerical integration using wavelets , 2009, Appl. Math. Comput..

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

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

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

[5]  Ram Jiwari,et al.  A differential quadrature method for numerical solutions of Burgers'‐type equations , 2012 .

[6]  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..

[7]  Sapna Pandit,et al.  Wavelet Transform and Wavelet Based Numerical Methods: an Introduction , 2012 .

[8]  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 .

[9]  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 .

[10]  R. C. Mittal,et al.  A differential quadrature algorithm to solve the two dimensional linear hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions , 2012, Appl. Math. Comput..

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

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

[13]  M. M. Hosseini,et al.  New Rothe-wavelet method for solving telegraph equations , 2012, Int. J. Syst. Sci..

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

[15]  Andrew K. Chan,et al.  Fundamentals of Wavelets: Theory, Algorithms, and Applications , 2011 .

[16]  Ram Jiwari,et al.  A Haar wavelet quasilinearization approach for numerical simulation of Burgers' equation , 2012, Comput. Phys. Commun..

[17]  Jalil Rashidinia,et al.  Spline methods for the solution of hyperbolic equation with variable coefficients , 2007 .

[18]  Siraj-ul-Islam,et al.  Haar wavelet collocation method for the numerical solution of boundary layer fluid flow problems , 2011 .

[19]  Sapna Pandit,et al.  Haar Wavelet Approach for Numerical Solution of Two Parameters Singularly Perturbed Boundary Value Problems , 2014 .

[20]  Su-Cheol Yi,et al.  Fourth-order compact difference and alternating direction implicit schemes for telegraph equations , 2012, Comput. Phys. Commun..

[21]  Ülo Lepik,et al.  Numerical solution of differential equations using Haar wavelets , 2005, Math. Comput. Simul..

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

[23]  David J. Evans,et al.  The numerical solution of the telegraph equation by the alternating group explicit (AGE) method , 2003, Int. J. Comput. Math..

[24]  Manoj Kumar,et al.  A composite numerical scheme for the numerical simulation of coupled Burgers' equation , 2014, Comput. Phys. Commun..

[25]  Yiming Chen,et al.  Error analysis for numerical solution of fractional differential equation by Haar wavelets method , 2012, J. Comput. Sci..

[26]  Mehmet Sezer,et al.  Taylor polynomial solution of hyperbolic type partial differential equations with constant coefficients , 2011, Int. J. Comput. Math..