SeMA Journal

, Volume 76, Issue 1, pp 1–13

# Numerical solution of high order linear complex differential equations via complex operational matrix method

• Farshid Mirzaee
• Sahar Alipour
Article

## Abstract

In this work, a numerical method based on a complex operational matrix is utilized to solve high order linear complex differential equations under mixed initial conditions. For this aim, we introduce orthonormal Bernstein polynomials (OBPs), and we obtain their complex operational matrix of differentiation. The main advantage of the proposed method is that by using this method complex differential equations reduce to a linear system of algebraic equations which can be solved by using an appropriate iterative method. To, some results concerning the error analysis associated with the present method are discussed. Finally, we give some numerical examples to reveal accuracy and efficiency of the proposed method. Also, the numerical results obtained by this method are compared with numerical results achieved from other existing methods.

## Keywords

Complex differential equations Orthonormal Bernstein polynomials Operational matrix method Error analysis

## Mathematics Subject Classification

34Mxx 14F10 65R99 65Gxx

## Notes

### Acknowledgements

The authors would like to express our very great appreciation to editor and anonymous reviewers for their valuable comments and constructive suggestions which have helped to improve the quality and presentation of this paper.

## References

1. 1.
Ince, E.L.: Ordinary Differential Equations. Dover Publications, Mineola, New York (1958)Google Scholar
2. 2.
Ishisaki, K., Tohge, K.: On the complex oscillation of some linear differential equations. J. Math. Anal. Appl. 206, 503–517 (1997)
3. 3.
Barsegian, G.A.: Gamma-Lines: On the Geometry of Real and Complex Functions. Taylor and Francis, London (2002)
4. 4.
Lingyun, G.: The growth of solutions of systems of complex nonlinear algebraic differential equations. Acta Math. Sci. 30B, 932–938 (2010)
5. 5.
Prokhorov, V.A.: On best rational approximation of analytic functions. J. Approx. Theory. 133, 284–296 (2005)
6. 6.
Cveticanin, L.: Analytic approach for the solution of the complex-valued strong non-linear differential equation of Duffing type. Phys. A 297, 348–360 (2001)
7. 7.
Sezer, M., Yalcinas, S.: A collocation method to solve higher order linear complex differential equations in rectangular domains. Numer. Methods Partial Differ. Equ. 26, 596–611 (2010)
8. 8.
Sezer, M., Gulsu, M.: Approximate solution of complex differential equations for a rectangular domain with Taylor collocation method. Appl. Math. Comput. 177, 844–851 (2006)
9. 9.
Yüzbaşı, Ş., Sezer, M.: A collocation method to find solutions of linear complex diferential equations in circular domains. Appl. Math. Comput. 219, 10610–10626 (2013)
10. 10.
Mirzaee, F., Bimesl, S., Tohidi, E.: A new complex-valued method and its applications in solving differential equations. Sci. Iran. 22, 2424–2431 (2015)Google Scholar
11. 11.
Rostamy, D., Jafari, H., Alipour, M., Khalique, C.M.: Computational method based on Bernstein operational matrices for multi-order fractional differential equations. Filomat 28(3), 591–601 (2014)
12. 12.
Mirzaee, F., Samadyar, N.: Application of orthonormal Bernstein polynomials to construct a efficient scheme for solving fractional stochastic integro-differential equation. Optik Int. J. Light Electron Opt. 132, 262–273 (2017)
13. 13.
Javadi, Sh, Babolian, E., Taheri, Z.: Solving generalized pantograph equations by shifted orthonormal Bernstein polynomials. J. Appl. Comput. Math. 303, 1–14 (2016)
14. 14.
Maleknejad, K., Hashemizadeh, E., Basirat, B.: Computational method based on Bernstein operational matrices for nonlinear Volterra-Fredholm-Hammerstein integral equations. Commun. Nonlinear Sci. Numer. Simul. 17(1), 52–61 (2012)
15. 15.
Pandey, R.K., Kumar, N.: Solution of Lane–Emden type equations using Bernstein operational matrix of differentiation. New Astron. 17(3), 303–308 (2012)
16. 16.
Yousefi, S.A., Behroozifar, M.: Operational matrices of Bernstein polynomials and their applications. Int. J. Syst. Sci. 41, 709–716 (2010)
17. 17.
Saadatmandi, A.: Bernstein operational matrix of fractional derivatives and its applications. Appl. Math. Model. 38(4), 1365–1372 (2014)
18. 18.
Spiegel, M.R.: Theory and Problems of Complex Variables. McGraw-Hill Inc., New York (1974)Google Scholar
19. 19.
Ahlfors, L.V.: Complex Analysis. McGraw-Hill Inc., Tokyo (1966)
20. 20.
Sezer, M., Akyuz Dascoglu, A.: Taylor polynomial solutions of general linear differential-difference equations with variable coefficients. Appl. Math. Comput. 174, 1526–1538 (2006)
21. 21.
Chiang, Y.M., Wang, S.: Oscillation results of certain higher-order linear differential equations with periodic coefficients in the complex plane. J. Math. Anal. Appl. 215, 560–576 (1997)
22. 22.
Gulsu, M., Sezer, M.: Approximate solution to linear complex differential equation by a new approximate approach. Appl. Math. Comput. 185, 636–645 (2007)
23. 23.
Sezer, M., Gulsu, M., Tanay, B.: A Taylor collocation method for the numerical solution of complex differential equations with mixed conditions in elliptic domains. Appl. Math. Comput. 182, 498–508 (2006)
24. 24.
Mirzaee, F., Samadyar, N.: Parameters estimation of HIV infection model of CD4$$^+$$ T-cells by applying orthonormal Bernstein collocation method. Int. J. Biomath. 11(2), 1850020 (2018)

## Authors and Affiliations

• Farshid Mirzaee
• 1