Block Unification Scheme for Elliptic, Telegraph, and Sine-Gordon Partial Differential Equations


In this paper, we use the method of lines to convert elliptic and hyperbolic partial differential equations (PDEs) into systems of boundary value problems and initial value problems in ordinary differential equations (ODEs) by replacing the appropriate derivatives with central difference methods. The resulting system of ODEs is then solved using an extended block Numerov-type method (EBNUM) via a block unification technique. The accuracy and speed advantages of the EBNUM over the finite difference method (FDM) are established numerically.

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Jator, S. (2015) Block Unification Scheme for Elliptic, Telegraph, and Sine-Gordon Partial Differential Equations. American Journal of Computational Mathematics, 5, 175-185. doi: 10.4236/ajcm.2015.52014.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Lambert, J.D. (1973) Computational Methods in Ordinary Differential Equations. John Wiley, New York.
[2] Vigo-Aguiar, J. and Ramos, H. (2007) A Family of A-Stable Collocation Methods of Higher Order for Initial-Value Problems. IMA Journal of Numerical Analysis, 27, 798-817.
[3] Brugnano, L. and Trigiante, D. (1998) Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon and Breach Science Publishers, Amsterdam.
[4] D’Ambrosio, R. and Paternoster, B. (2014) Numerical Solution of a Diffusion Problem by Exponentially Fitted Finite Difference Methods. Springer Plus, 3, 425-431.
[5] Sun, H. and Zhang, J. (2004) A High Order Finite Difference Discretization Strategy Based on Extrapolation for Convection Diffusion Equations. Numerical Methods for Partial Differential Equations, 20, 18-32.
[6] Jator, S.N. (2008) A Class of Initial Value Methods for the Direct Solution of Second Order Initial Value Problems. International Journal of Pure and Applied Mathematics, 46, 225-230.
[7] Adee, S.O., Onumanyi, P., Sirisena, U.W. and Yahaya, Y.A. (2005) Note on Starting the Numerov Method More Accurately by a Hybrid Formula of Order Four for an Initial Value Problem. Journal of Computational and Applied Mathematics, 175, 369-373.
[8] Jator, S.N. and Li, J. (2012) An Algorithm for Second Order Initial and Boundary Value Problems with an Automatic Error Estimate Based on a Third Derivative Method. Numerical Algorithms, 59, 333-346.
[9] Coleman, J.P. and Ixaru, L.G.R. (1996) P-Stability and Exponential-Fitting Methods for . IMA Journal of Numerical Analysis, 16, 179-199.
[10] Coleman, J.P. and Duxbury, S.C. (2000) Mixed Collocation Methods for . Journal of Computational and Applied Mathematics, 126, 47-75.]
[11] Burden, R.L. and Faires, J.D. (1985) Numerical Analysis. 3rd Edition, Prindle, Weber and Schmidt, Boston.
[12] Zill, D.G. and Cullen, M.R. (2001) Differential Equations with Boundary-Value Problems. 5th Edition, Brooks/Cole, California.
[13] Cheney, W. and Kincaid, D. (1985) Numerical Mathematics and Computing. Brooks/Cole, California.
[14] Dehghan, M. and Shokri, A. (2008) A Numerical Method for One-Dimensional Nonlinear Sine-Gordon Equation Using Collocation and Radial Basis Functions. Numerical Methods for Partial Differential Equations, 24, 687-698.
[15] Ding, H., Zhang, Y., Cao, J. and Tian, J. (2012) A Class of Difference Scheme for Solving Telegraph Equation by New Non-Polynomial Spline Methods. Applied Mathematics and Computation, 218, 4671-4683.

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