 |
 |
 |
 |
 |
 |
Sains Malaysiana 46(2)(2017): 335347 http://dx.doi.org/10.17576/jsm-2017-4602-19 A Novel Collocation Method Based
on Residual Error Analysis for Solving Integro-Differential Equations
Using Hybrid Dickson and Taylor Polynomials (Kaedah Novel Kolokasi Berdasarkan Analisis
Sisa Ralat untuk Menyelesaikan Persamaan Integro-Pembezaan yang Menggunakan Hibrid
Dickson dan Polinomial Taylor) ÖMÜR KIVANÇ
KÜRKÇÜ1*,
ERSIN
ASLAN2
& MEHMET SEZER3 1Department of Mathematics,
Faculty of Science, Celal Bayar University, Manisa 45140 Turkey 2Turgutlu Vocational
Training School, Celal Bayar University, Manisa, Turkey 3Department of Mathematics,
Faculty of Science, Celal Bayar University, Manisa 45140 Turkey Diserahkan: 1 Mei
2015/Diterima: 18 Jun 2016 ABSTRACT In this study, a novel matrix
method based on collocation points is proposed to solve some linear
and nonlinear integro-differential equations with variable coefficients
under the mixed conditions. The solutions are obtained by means
of Dickson and Taylor polynomials. The presented method transforms
the equation and its conditions into matrix equations which comply
with a system of linear algebraic equations with unknown Dickson
coefficients, via collocation points in a finite interval. While
solving the matrix equation, the Dickson coefficients and the
polynomial approximation are obtained. Besides, the residual error
analysis for our method is presented and illustrative examples
are given to demonstrate the validity and applicability of the
method. Keywords: Collocation and matrix
methods; Dickson and Taylor polynomials; integro-differential
equations; nonlinear equations; pseudocode ABSTRAK Dalam kajian ini, kaedah matriks
novel berdasarkan titik kolokasi adalah dicadangkan untuk menyelesaikan
persamaan integro-pembezaan bagi sesetengah linear dan tak linear
dengan pekali pemboleh ubah dalam keadaan bercampur-campur. Penyelesaian
yang diperoleh dengan cara polinomial Dickson dan Taylor. Kaedah
yang dibentangkan mengubah persamaan serta keadaannya ke dalam
persamaan matriks yang bertepatan dengan sistem persamaan algebra
linear dengan pekali Dickson tidak diketahui, melalui titik kolokasi
dalam selang terhingga. Semasa menyelesaikan persamaan matriks
ini, pekali Dickson dan penganggaran polinomial diperoleh. Selain
itu, analisis sisa ralat bagi kaedah kami ini telah dikemukakan
dan contoh ilustrasi diberi untuk menunjukkan kesahihan dan penerapan
kaedah. Kata kunci: Kolokasi dan kaedah matriks; polinomial Dickson dan Taylor;
persamaan integro-pembezaan; persamaan tak linear; tatasusunan RUJUKAN Akyüz-Daşcıoğlu,
A. & Sezer, M. 2007. A Taylor polynomial approach for solving
high-order linear Fredholm integro-differential equations in the
most general form. Int. J. Comput. Math. 84(4): 527-539.
Akyüz-Daşcıoğlu,
A. 2006. A Chebyshev polynomial approach for linear Fredholm-Volterra
integro-differential equations in the most general form. Appl.
Math. Comput. 181: 103-112. Babolian, E., Lotfi,
T. & Paripour, M. 2007. Wavelet moment method for solving
Fredholm integral equations of the first kind. Appl. Math.
Comput. 186: 1467-1471. Bhargava, M. &
Zieve, M.E. 1999. Factorizing Dickson polynomials over finite
fields. Finite Fields Appl. 5: 103-111. Brewer, B.W. 1961.
On certain character sums. Transactions of the American Mathematical
Society 99(2): 241-245. Dickson, L.E. 1896.
The analytic representation of substitutions on a power of a prime
number of letters with a discussion of the linear group I-II.
Annals of Mathematics 11(1/6): 65-120. Evans, D.J. &
Raslan, K.R. 2005. The Adomian decomposition method for solving
delay differential equation. Int. J. Comput. Math.
82: 49-54. Fernando, N. 2013.
A study of permutation polynomials over finite fields. University
of South Florida. Kürkçü, Ö.K., Ersin,
A. & Sezer, M. 2016. A numerical approach with error estimation
to solve general integro-differential-difference equations using
Dickson polynomials. Appl. Math. Comput. 276: 324-339.
Karamete, A. &
Sezer, M. 2002. A Taylor collocation method for the solution of
linear integro-differential equations. Int. J. Comput. Math.
79(9): 987-1000. Kurt, N. &
Sezer, M. 2008. Polynomial solution of high-order linear Fredholm
integro-differential equations with constant coefficients. J.
Franklin Inst. 345: 839-850. Lidl, R., Mullen,
G.L. & Turnwald, G. 1993. Dickson polynomials, Pitman monographs
and surveys in pure and applied mathematics. Longman Scientific
and Technical. Harlow, Essex. p. 65. Sezer, M. 1996.
A method for approximate solution of the second order linear differential
equations in terms of Taylor polynomials. Int. J. Math. Educ.
Sci. Technol. 27(6): 821-834. Sezer, M. 1994.
Taylor polynomial solutions of Volterra integral equations. Int.
J. Math. Educ. Sci. Technol. 25(5): 625-633. Stoll, T. 2007.
Complete decomposition of Dickson-type recursive polynomials and
a related Diophantine equation. Formal Power Series and Algebraic
Combinatorics, Tianjin, China. Wei, P., Liao,
X. & Wong, K-W. 2011. Key exchange based on Dickson polynomials
over finite field with 2^m. Journal of Computers 6(12):
2546-2551. Yalçınbaş,
S., Sezer, M. & Sorkun, H. 2009. Legendre polynomial solutions
of high-order linear Fredholm integro-differential equations.
Appl. Math. Comput. 210: 334-349. Yalçınbaş,
S. & Sezer, M. 2000. The approximate solution of high-order
linear Volterra-Fredholm integro-differential equations in terms
of Taylor polynomials. Appl. Math. Comput. 112: 291-308.
Yüzbaşı,
Ş., Şahin, N. & Sezer, M. 2011. Bessel polynomial
solutions of high-order linear Volterra integro-differential equations.
Computers and Mathematics with Applications 62: 1940-1956.
*Pengarang untuk surat-menyurat; email: omurkivanc@outlook.com
|
|||
|
