# An approach based on matrix polynomials for linear systems of partial differential equations

Special Matrices (2013)

• Volume: 1, page 42-48
• ISSN: 2300-7451

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## Abstract

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In this paper, an approach based on matrix polynomials is introduced for solving linear systems of partial differential equations. The main feature of the proposed method is the computation of the Smith canonical form of the assigned matrix polynomial to the linear system of PDEs, which leads to a reduced system. It will be shown that the reduced one is an independent system of PDEs having only one unknown in each equation. A comparison of the results for several test problems reveals that the method is very effective and convenient. The basic idea described in this paper can be employed to solve other linear functional systems.

## How to cite

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N. Shayanfar, and M. Hadizadeh. "An approach based on matrix polynomials for linear systems of partial differential equations." Special Matrices 1 (2013): 42-48. <http://eudml.org/doc/266867>.

@article{N2013,
abstract = {In this paper, an approach based on matrix polynomials is introduced for solving linear systems of partial differential equations. The main feature of the proposed method is the computation of the Smith canonical form of the assigned matrix polynomial to the linear system of PDEs, which leads to a reduced system. It will be shown that the reduced one is an independent system of PDEs having only one unknown in each equation. A comparison of the results for several test problems reveals that the method is very effective and convenient. The basic idea described in this paper can be employed to solve other linear functional systems.},
journal = {Special Matrices},
keywords = {linear system of partial differential equations; linear operator; matrix polynomial; Smith canonical form; numerical treatment; reduced system},
language = {eng},
pages = {42-48},
title = {An approach based on matrix polynomials for linear systems of partial differential equations},
url = {http://eudml.org/doc/266867},
volume = {1},
year = {2013},
}

TY - JOUR
AU - N. Shayanfar
TI - An approach based on matrix polynomials for linear systems of partial differential equations
JO - Special Matrices
PY - 2013
VL - 1
SP - 42
EP - 48
AB - In this paper, an approach based on matrix polynomials is introduced for solving linear systems of partial differential equations. The main feature of the proposed method is the computation of the Smith canonical form of the assigned matrix polynomial to the linear system of PDEs, which leads to a reduced system. It will be shown that the reduced one is an independent system of PDEs having only one unknown in each equation. A comparison of the results for several test problems reveals that the method is very effective and convenient. The basic idea described in this paper can be employed to solve other linear functional systems.
LA - eng
KW - linear system of partial differential equations; linear operator; matrix polynomial; Smith canonical form; numerical treatment; reduced system
UR - http://eudml.org/doc/266867
ER -

## References

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1. [1] M.S. Boudellioua and A. Quadrat, Serre 0s reduction of linear functional systems, Math. Comput. Sci. 4 (2010), 289–312. Zbl1275.16003
2. [2] M.V. Bulatov and M.G. Lee, Application of matrix polynomials to the analysis of linear differential-algebraic equations of higher order, Differ. Equ. 44 (2008), 1353–1360. [WoS] Zbl1194.34008
3. [3] T. M. Elzaki and S.M. Elzaki, On the Elzaki transform and system of partial differential equations, Adv. Theor. Appl. Math. 6 (2011), 115–123. Zbl1251.34004
4. [4] M.G. Frost and C. Storey, Equivalence of a matrix over R[s; z] with its Smith form, Internat. J. Control 28 (1978), 665–671. Zbl0394.93013
5. [5] M.G. Frost and M.S. Boudellioua, Some further results concerning matrices with elements in a polynomial ring, Int. J. Control 43 (1986), 1543–1555. Zbl0587.15015
6. [6] F.R. Gantmacher, The Theory of Matrices, Chelsea Publishing Company, New York, 1960. Zbl0088.25103
7. [7] I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982. Zbl0482.15001
8. [8] J.H. He and X.H. Wu, Exp-function method for nonlinear wave equations, Chaos Solitons Fractals 30 (2006), 700– 708. Zbl1141.35448
9. [9] T. Kailath, Linear Systems, Englewood Cliffs, Prentice Hall, 1980. Zbl0454.93001
10. [10] V. N. Kublanovskaya, On some factorizations of two parameter polynomail matrices, J. Math. Sci. 86 (1997), 2866– 2879.
11. [11] R.C. McOwen, Partial Differential Equations: Methods and Applications, Prentice Hall, Upper Saddle River, 2002. Zbl0849.35001
12. [12] W.H. Neven and C. Praagman, Column reduction of polynomial matrices, Linear Algebra Appl. 188 (1993), 569–589.
13. [13] H.H. Rosenbrock, State Space and Multivariable Theory, Wiley-Interscience, New York, 1970.
14. [14] A. Sami Bataineh, M.S.M. Noorani and I. Hashim, Approximate analytical solutions of systems of PDEs by homotopy analysis method, Comput. Math. Appl. 55 (2008), 2913–2923. [WoS] Zbl1142.65423
15. [15] N. Shayanfar and M. Hadizadeh, Splitting a linear system of operator equations with constant coefficients: A matrix polynomial approach, Filomat 27(8) (2013), 1447–1454. Zbl06451488
16. [16] N. Shayanfar, M. Hadizadeh and A. Amiraslani, Integral operators acting as variables of the matrix polynomial: application to system of integral equations, Ann. Funct. Anal. 3(2) (2012), 170–182. Zbl1268.47059
17. [17] A.M. Wazwaz, The decomposition method applied to systems of partial diferential equations and to the reaction diffusion Brusselator model, Appl. Math. Comput. 110 (2000), 251–264. Zbl1023.65109
18. [18] J. Wilkening and J. Yu, A local construction of the Smith normal form of a matrix polynomial, J. Symbolic Comput. 46 (2011), 1–22. [WoS] Zbl1206.65146

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