Septic B-spline Collocation Method for Solving One-dimensional Hyperbolic Telegraph Equation
Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion for such branches of sciences. In this paper, a numerical solution for the one-dimensional hyperbolic telegraph equation by using the collocation method using the septic splines is proposed. The scheme works in a similar fashion as finite difference methods. Test problems are used to validate our scheme by calculate L2-norm and L∞-norm. The accuracy of the presented method is demonstrated by two test problems. The numerical results are found to be in good agreement with the exact solutions.
[1] D. M. Pozar, Microwave Engineering. NewYork: Addison-Wesley, 1990.
[2] A. Mohebbi, M. Dehghan, High order compact solution of the one-spacedimensional
linear hyperbolic equation, Numerical Methods for Partial
Differential Equations, 24 (2008) 1222-1235.
[3] A. Jeffrey, Advanced Engineering Mathematics. Harcourt Academic
Press, 2002.
[4] A. Jeffrey, Applied Partial Differential Equations. NewYork: Academic
Press, 2002.
[5] R. K. Mohanty, New unconditionally stable difference schemes for the
solution of multi-dimensional telegraphic equations, Computer Mathematics,
86 (2008) 2061 - 2071.
[6] H. Pascal, Pressure wave propagation in a fluid flowing through a
porous medium and problems related to interpretation of Stoneley-s wave
attenuation in acoustical well logging, Engineering Science, 24 (1986)
1553-1570.
[7] G. Bohme, Non-Newtonian Fluid Mechanics, NewYork: North-Holland,
1987.
[8] D. J. Evans, H. Bulut, Thenumerical solution of thetelegraph equation by
the alternating group explicit method, Computer Mathematics, 80 (2003)
1289- 1297.
[9] P. M. Jordan, M. R. Meyer, A. Puri, Causal implications of viscous
damping in compressible fluid flows. Phys Rev, 62 (2000) 7918-7926.
[10] L. L. Schumaker, Spline Functions: Basic Theory, Krieger Publishing
Company, Florida, 1981.
[11] J. M. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and
Their Applications, Academic Press, NewYork, 1967.
[12] P. M. Prenter, Splines and Variational Methods, New York: John Wiley,
1975.
[13] C. De Boor, A Practical Guide to Splines, Springer-Verlag, New York,
1978.
[14] G. Micula, Handbook of Splines, Kluwer Academic Publishers, Dordrecht,
The Netherlands, 1999.
[15] M. A. Ramadan, T. S. El-Danaf, F. E.I. Abd Alaal, A numerical solution
of the Burgers- equation using septic B-splines, Chaos, Solitons and
Fractals, 26 (2005) 795-804.
[16] T. S. A. El-Danaf, Septic B-spline method of the Korteweg-de Vries-
Burger-s equation, Communications in Nonlinear Science and Numerical
Simulation, 13 (2008) 554-566.
[17] A. A. Soliman, M. H. Hussien, Collocation solution for RLW equation
with septic spline, Applied Mathematics and Computation, 161 (2005)
623-636.
[18] A.Quarteroni, R. Sacco, F. Saleri, Numerical mathematics, Springer-
Verlag, Berlin, 2007.
[1] D. M. Pozar, Microwave Engineering. NewYork: Addison-Wesley, 1990.
[2] A. Mohebbi, M. Dehghan, High order compact solution of the one-spacedimensional
linear hyperbolic equation, Numerical Methods for Partial
Differential Equations, 24 (2008) 1222-1235.
[3] A. Jeffrey, Advanced Engineering Mathematics. Harcourt Academic
Press, 2002.
[4] A. Jeffrey, Applied Partial Differential Equations. NewYork: Academic
Press, 2002.
[5] R. K. Mohanty, New unconditionally stable difference schemes for the
solution of multi-dimensional telegraphic equations, Computer Mathematics,
86 (2008) 2061 - 2071.
[6] H. Pascal, Pressure wave propagation in a fluid flowing through a
porous medium and problems related to interpretation of Stoneley-s wave
attenuation in acoustical well logging, Engineering Science, 24 (1986)
1553-1570.
[7] G. Bohme, Non-Newtonian Fluid Mechanics, NewYork: North-Holland,
1987.
[8] D. J. Evans, H. Bulut, Thenumerical solution of thetelegraph equation by
the alternating group explicit method, Computer Mathematics, 80 (2003)
1289- 1297.
[9] P. M. Jordan, M. R. Meyer, A. Puri, Causal implications of viscous
damping in compressible fluid flows. Phys Rev, 62 (2000) 7918-7926.
[10] L. L. Schumaker, Spline Functions: Basic Theory, Krieger Publishing
Company, Florida, 1981.
[11] J. M. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and
Their Applications, Academic Press, NewYork, 1967.
[12] P. M. Prenter, Splines and Variational Methods, New York: John Wiley,
1975.
[13] C. De Boor, A Practical Guide to Splines, Springer-Verlag, New York,
1978.
[14] G. Micula, Handbook of Splines, Kluwer Academic Publishers, Dordrecht,
The Netherlands, 1999.
[15] M. A. Ramadan, T. S. El-Danaf, F. E.I. Abd Alaal, A numerical solution
of the Burgers- equation using septic B-splines, Chaos, Solitons and
Fractals, 26 (2005) 795-804.
[16] T. S. A. El-Danaf, Septic B-spline method of the Korteweg-de Vries-
Burger-s equation, Communications in Nonlinear Science and Numerical
Simulation, 13 (2008) 554-566.
[17] A. A. Soliman, M. H. Hussien, Collocation solution for RLW equation
with septic spline, Applied Mathematics and Computation, 161 (2005)
623-636.
[18] A.Quarteroni, R. Sacco, F. Saleri, Numerical mathematics, Springer-
Verlag, Berlin, 2007.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:53411", author = "Marzieh Dosti and Alireza Nazemi", title = "Septic B-spline Collocation Method for Solving One-dimensional Hyperbolic Telegraph Equation", abstract = "Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion for such branches of sciences. In this paper, a numerical solution for the one-dimensional hyperbolic telegraph equation by using the collocation method using the septic splines is proposed. The scheme works in a similar fashion as finite difference methods. Test problems are used to validate our scheme by calculate L2-norm and L∞-norm. The accuracy of the presented method is demonstrated by two test problems. The numerical results are found to be in good agreement with the exact solutions.
", keywords = "B-spline, collocation method, second-order hyperbolic telegraph equation, difference schemes.", volume = "5", number = "8", pages = "1214-5", }
