Taylor polynomial solution of hyperbolic type partial differential equations with constant coefficients

The purpose of this study is to give a Taylor polynomial approximation for the solution of hyperbolic type partial differential equations with constant coefficients. The technique used is an improved Taylor matrix method, which has been given for solving ordinary differential, integral and integro-differential equations [M. Gülsu and M. Sezer, A method for the approximate solution of the high-order linear difference equations in terms of Taylor polynomials, Int. J. Comput. Math. 82(5) (2005), pp. 629–642; M. Gülsu and M. Sezer, On the solution of the Riccati equation by the Taylor matrix method, Appl. Math. Comput. 188 (2007), pp. 446–449; A. Karamete and M. Sezer, A Taylor collocation method for the solution of linear integro-differential equations, Int. J. Comput. Math. 79(9) (2002), pp. 987–1000; N. Kurt and M. Çevik, Polynomial solution of the single degree of freedom system by Taylor matrix method, Mech. Res. Commun. 35 (2008), pp. 530–536; N. Kurt and M. Sezer, Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients, J. Franklin Inst. 345 (2008), pp. 839–850; Ş. Nas, S. Yalçinbaş, and M. Sezer, A method for approximate solution of the high-order linear Fredholm integro-differential equations, Int. J. Math. Edu. Sci. Technol. 27(6) (1996), pp. 821–834; M. Sezer, Taylor polynomial solution of Volterra integral equations, Int. J. Math. Edu. Sci. Technol. 25(5) (1994), pp. 625–633; M. Sezer, A method for approximate solution of the second order linear differential equations in terms of Taylor polynomials, Int. J. Math. Edu. Sci. Technol. 27(6) (1996), pp. 821–834; M. Sezer, M. Gülsu, and B. Tanay, A matrix method for solving high-order linear difference equations with mixed argument using hybrid Legendre and Taylor polynomials, J. Franklin Inst. 343 (2006), pp. 647–659; S. Yalçinbaş, Taylor polynomial solutions of nonlinear Volterra–Fredholm integral equation, Appl. Math. Comput. 127 (2002), pp. 196–206; S. Yalçinbaş and M. Sezer, The approximate solution of high-order linear Volterra–Fredholm integro-differential equations in terms of Taylor polynomials, Appl. Math. Comput. 112 (2000), pp. 291–308]. Some numerical examples, which consist of initial and boundary conditions, are given to illustrate the reliability and efficiency of the method. Also, the results obtained are compared by the known results; the error analysis is performed and the accuracy of the solution is shown.

[1]  Mehmet Sezer,et al.  Taylor polynomial solutions of Volterra integral equations , 1994 .

[2]  Mehmet Sezer,et al.  On the solution of the Riccati equation by the Taylor matrix method , 2006, Appl. Math. Comput..

[3]  Mehmet Sezer,et al.  A method for the approximate solution of the second‐order linear differential equations in terms of Taylor polynomials , 1996 .

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

[5]  Jafar Biazar,et al.  An approximation to the solution of hyperbolic equations by Adomian decomposition method and comparison with characteristics method , 2005, Appl. Math. Comput..

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

[7]  Allaberen Ashyralyev,et al.  Two new approaches for construction of the high order of accuracy difference schemes for hyperbolic differential equations , 2005 .

[8]  R. K. Mohanty,et al.  An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation , 2001 .

[9]  Mehmet Sezer,et al.  A method for the approximate solution of the high-order linear difference equations in terms of Taylor polynomials , 2005, Int. J. Comput. Math..

[10]  Mehmet Sezer,et al.  A matrix method for solving high-order linear difference equations with mixed argument using hybrid legendre and taylor polynomials , 2006, J. Frankl. Inst..

[11]  Mehmet Sezer,et al.  Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients , 2008, J. Frankl. Inst..

[12]  Mehmet Emir Koksal,et al.  On the Second Order of Accuracy Difference Scheme for Hyperbolic Equations in a Hilbert Space , 2005 .

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

[14]  Mehmet Sezer,et al.  The approximate solution of high-order linear Volterra-Fredholm integro-differential equations in terms of Taylor polynomials , 2000, Appl. Math. Comput..

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

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

[17]  Mehmet Emir Koksal,et al.  On the numerical solution of hyperbolic PDEs with variable space operator , 2009 .

[18]  Mehmet Sezer,et al.  A Taylor Collocation Method for the Solution of Linear Integro-Differential Equations , 2002, Int. J. Comput. Math..

[19]  S Yalcinbas.  TAYLOR POLYNOMIAL SOLUTION OF NON-LINEAR VOLTERRA–FREDHOLM INTEGRAL EQUATION , 2002 .

[20]  Feng Gao,et al.  Unconditionally stable difference schemes for a one-space-dimensional linear hyperbolic equation , 2007, Appl. Math. Comput..

[21]  Mehmet Çevik,et al.  Polynomial solution of the single degree of freedom system by Taylor matrix method , 2008 .

[22]  Esmail Babolian,et al.  Numerical solutions of the nonlinear Volterra-Fredholm integral equations by using homotopy perturbation method , 2007, Appl. Math. Comput..

[23]  Mehdi Dehghan,et al.  The use of cubic B‐spline scaling functions for solving the one‐dimensional hyperbolic equation with a nonlocal conservation condition , 2007 .

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

[25]  Salih Yalçinbas.  Taylor polynomial solutions of nonlinear Volterra-Fredholm integral equations , 2002, Appl. Math. Comput..

[26]  Sohrab Ali Yousefi.  Legendre multiwavelet Galerkin method for solving the hyperbolic telegraph equation , 2009 .

[27]  R. K. Mohanty,et al.  An Unconditionally Stable ADI Method for the Linear Hyperbolic Equation in Three Space Dimensions , 2002, Int. J. Comput. Math..