Solving the First-Order Linear Matrix Differential Equations Using Bernstein Matrix Approach

Subject Areas : International Journal of Industrial Mathematics

زهرا لرکوجوری ¹ ( Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran )
ناصر میکائیل وند ² ( Department of Mathematics, Ardebil Branch, Islamic Azad Unilversity, Ardebil, Iran. )
اسماعیل بابلیان ³ ( Department of Computer Science, Kharazmi University, Tehran, Iran. )

Received: 2020-03-23 Accepted : 2020-05-17 Published : 2021-04-01

Keywords: Operational matrix of derivative, Error ‎estimation.‎, M‎atrix differential equation, Matrix differential equation, Bernstein polynomials,

Abstract :

This paper uses a new framework for solving a class of linear matrix differential equations. For doing so, the operational matrix of the derivative based on the shifted Bernstein polynomials together with the collocation method are exploited to decrease the principal problem to system of linear matrix equations. An error estimation of this method is provided. Numerical experiments are reported to show the applicably and efficiency of the propounded method.

References:

[1] Z. A. A. Al-Zhour, New techniques for solving some matrix and matrix differential equations,Ain Shams Eng. J. 6 (2015) 347-354.
[2] S. U. Altinbasak, M. Demiralp, Solutions to linear matrix ordinary differential equations via minimal, regular, and excessive space extension based universalization, J. Math. Chem. 48 (2010) 2266-86.
[3] F. P. A. Beik, D. K. Salkuyeh, On the global Krylov subspace methods for solving general coupled matrix equation, Comput. Math. Appl. 62 (2011) 4605-4613.
[4] F. P. A. Beik, D. K. Salkuyeh, The coupled Sylvester-transpose matrix equations over
generalized centrosymmetric matrices,Int. J. Comput. Math. 90 (2013) 1546-1566.
[5] F. P. A. Beik, D. K. Salkuyeh, M. M. Moghadam, Gradient-based iterative algorithm for solving the generalized coupled Sylvester-transpose and conjugate matrix equations over reflexive (anti-reflexive) matrices,Trans. Inst. Measurement. Control. 36 (2014) 99-110.
[6] D. S. Bernstein, Matrix Mathematics theory, facts, and formulas, Second edition, Princeton University Press, New Jersey, (2009).
[7] M. Bhatti, P. Bracken, Solutions of differential equations in a Bernstein Polynomial basis, Journal of Computational and Applied Mathematics 205 (2007) 272-280.
[8] J. P. Boyd, Chebyshev and Fourier spectral methods, Second edition, Dover, New York, (2000).
[9] C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral methods in single domisns, Springer-Verlag, (2006).
[10] J. P. Chehab, Matrix differential equations and inverse preconditioners, Comp. Appl. Math. 26 (2007) 95-128.
[11] E. Defez, A. Hervs, J. Ibez, M. M. Tung, Numerical solutions of matrix differential models using higher-order matrix splines,Mediterr J. Math. 9 (2012) 865-882.
[12] E. Defez, L. Solera, A. Hervs, C. Santamaria, Numerical solution of matrix differential  models using cubic matrix splines, Computers and Mathematics with Applications 50 (2005)  693-699.
[13] F. Ding, T. Chen, Iterative least-squares solutions of coupled Sylvester matrix equations,Systems Control Lett. 54 (2005) 95-107.
[14] F. Ding, T. Chen, On iterative solutions of general coupled matrix equations, SIAM J. Control Optim. 44 (2006) 2269-2284.
[15] T. M. Flett, Differential Analysis, Cambridge University Press, (1980).
[16] A. Golbabai, S. Panjeh Ali Beik, An efficient method based on operational matrices of Bernoulli polynomials for solving matrix differential equations, Comp. Appl. Math. 34 (2015) 159-175.
[17] G. H. Golub, C. F. V. Loan, Matrix computations, 2nd edn. The Johns Hopkins University Press, Baltimore, (1989).
[18] E. Kreyszig, Introductory functional analysis with applications, John Wiley and Sons, Inc., (1978).
[19] S. Mashayekhi, Y. Ordokhani, M. Razzaghi, Hybrid functions approach for optimal control of systems described by integrodifferential equations, Appl. Math. Model. 37 (2013)  3355-3368.
[20] K. Nouri, S. Panjeh Ali Beik, L. Torkzadeh, Operational Matrix Approach for SecondOrder Matrix Differential Models, Iranian Journal of Science and Technology Transactions A: Science (2019).
[21] M. Paripour, M. Kamyar, Numerical solution of nonlinear Volterra-Fredholm integral  equations by using new basis functions, Communications in Numerical Analysis 17 (2013) 1-12.
[22] Y. Saad, Iterative Methods for Sparse linear Systems, PWS press, New York, (1995).
[23] D. K. Salkuyeh, F. P. A. Beik, On the gradient based algorithm for solving the general coupled matrix equations, Trans. Inst. Measurement. Control. 36 (2014) 375-381.
[24] E. Tohidi, A. H. Bhrawy, K. Erfani, A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation, Appl. Math. Model. 37  (2013) 4283-4294.
[25] S. Yousefi, M. Behroozifar, Operational matrices of Bernstein polynomials and their applications, Int. J. Syst. Sci. 41 (2010) 709-716.
[26] H. Zheng, W. Han, On some discretization methods for solving a linear matrix ordinary differential equation, J. Math. Chem. 49 (2011) 1026-1041.
[27] S. Abbasbandy, T. Allahviranloo, Numerical solutions of fuzzy differential equations by taylor method, Computational Methods in Applied Mathematics 2 (2002) 113-124.
[28] D. Dubois, H. Prade, Towards fuzzy differential calculus, Fuzzy Sets and Systems 8 (1982) 1-7.
[29] Y. Lai, C. L. Hwang, Fuzzy Mathematical programming theory and applications, Springer, Belin, (1992).