Hermite Matrix Polynomial Collocation Method for Linear Complex Differential Equations and Some Comparisons

Abstract

In this paper, we introduce a Hermite operational matrix collocation method for solving higher-order linear complex differential equations in rectangular or elliptic domains. We show that based on a linear algebra theorem, the use of different polynomials such as Hermite, Bessel and Taylor in polynomial collocation methods for solving differential equations leads to an equal solution, and the difference in the numerical results arises from the difference in the coefficient matrix of final linear systems of equations. Some numerical examples will also be given.

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Bagherpoorfard, M. and Ghassabzade, F. (2013) Hermite Matrix Polynomial Collocation Method for Linear Complex Differential Equations and Some Comparisons. Journal of Applied Mathematics and Physics, 1, 58-64. doi: 10.4236/jamp.2013.15009.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] G. Barsegian, “Gamma-Lines: On the Geometry of Real and Complex Functions,” Taylor and Frencis, London, 2002.
[2] G. Barsegian and D. T. Le, “On a Topological Description of Solutions of Complex Differential Equations,” Complex Variables, Vol. 50, No. 7-11, 2005, pp. 307-318.
http://dx.doi.org/10.1080/02781070500086743
[3] K. Ishisaki and K. Tohge, “On the Complex Oscillation of Some Linear Differential Equations,” Journal of Mathematical Analysis and Applications, Vol. 206, No. 2, 1997, pp. 503-517.
http://dx.doi.org/10.1006/jmaa.1997.5247
[4] J. Heittokangas, R. Korhonen and J. Rattya, “Growth Estimates for Solutions of Linear Complex Differential Equations,” Annales Academiae Scientiarum Fennicae Mathematica, Vol. 29, No. 1, 2004, pp. 233-246.
[5] V. A. Prokhorov, “On Best Rational Approximation of Analytic Functions,” Journal of Approximation Theory, Vol. 133, No. 2, 2005, pp. 284-296.
http://dx.doi.org/10.1016/j.jat.2004.12.007
[6] V. Andrievskii, “Polynomial Approximation of Analytic Functions on Finite Number of Continua in the Complex Plane,” Journal of Approximation Theory, Vol. 133, No. 2, 2005, pp. 238-244.
http://dx.doi.org/10.1016/j.jat.2004.12.016
[7] M. Sezer and M. Güsu, “Approximate Solution of Complex Differential Equations for a Rectangular Domain with Taylor Collocation Method,” Applied Mathematics and Computation, Vol. 177, No. 2, 2006, pp. 844-851.
http://dx.doi.org/10.1016/j.amc.2005.11.035
[8] M. Güsu and M. Sezer, “Approximate Solution to Linear Complex Differential Equations by a New Approximate Approach,” Applied Mathematics and Computation, Vol. 185, No. 1, 2007, pp. 636-645.
http://dx.doi.org/10.1016/j.amc.2006.07.050
[9] M. Sezer and S. Yalcinbas, “A Collocation Method to Solve Higher Order Linear Complex Differential Equations in Rectangular Domains,” Numerical Methods for Partial Differential Equations, Vol. 26, No. 3, 2010, pp. 596-611.
[10] M. Sezer, B. Tanay and M. Güsu, “Numerical Solution of a Class of Complex Differential Equations by the Taylor Collocation Method in Elliptic Domains,” Numerical Methods for Partial Differential Equations, Vol. 26, No. 5, 2010, pp. 1191-1205.
[11] S. Yüzbasi, N. Sahin and M. Sezer, “A Collocation Approach for Solving Linear Complex Differential Equations in Rectangular Domain,” Mathematical Methods in the Applied Sciences, Vol. 35, No. 10, 2012, pp. 1126-1139. http://dx.doi.org/10.1002/mma.1590
[12] M. Güsu, H. Yalman, Y. Ozturk and M. Sezer, “A New Hermite Collocation Method for Solving Differential Difference Equations,” Applications and Applied Mathematics, Vol. 6, No. 1, 2011, pp. 116-129.
[13] K. Haffman and R. Kunze, “Linear Algebra,” 2nd Edition, Prentice-Hall, Upper Saddle River, 1971.

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