Sains Malaysiana 52(8)(2023): 2337-2351
http://doi.org/10.17576/jsm-2023-5208-13
Hybrid Multistep Block Method for Solving Neutral Volterra
Integro-Differential Equation with Proportional and Mixed Delays
(Kaedah Berbilang Langkah Blok Hibrid untuk Menyelesaikan Persamaan
Kamiran-Pembezaan Neutral Volterra dengan Kelengahan Berkadar dan Bercampur)
NUR INSHIRAH NAQIAH ISMAIL1 &
ZANARIAH ABDUL MAJID1,2,*
1Institute for Mathematical Research, Universiti Putra Malaysia,
43400 UPM Serdang, Selangor, Malaysia
2Department of Mathematics and
Statistics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang,
Selangor, Malaysia
Diserahkan: 20 April 2023/Diterima: 11 Julai 2023
Abstract
The neutral Volterra integro-differential
equation with proportional and mixed delays (NDVIDE) is being solved by a newly
proposed technique in numerical method, namely, the two-point one off-point
block multistep method (1OBM3). The method is also known as a hybrid multistep
block method. Subsequently, Lagrange interpolating polynomial is utilized in
order to develop the hybrid block method. The foundation of the technique is
taken from predictor and corrector formulae. The proposed method will solve
NDVIDE in two steps simultaneously, with three predictor formulae including one
off-point. The NDVIDE problems are solved via the constant step size technique.
In order to solve the integral and differential parts of the problems, two
alternative numerical approaches are applied. The differentiation part is
approximated by deriving the divided difference formula, while the integration
part is interpolated using composite Simpson’s rule. Note that the proposed
method has been analysed thoroughly regarding its order, consistency, zero stability
and convergence of the method. The stability region for 1OBM3 has been
constructed based on the stability polynomial obtained. Consequently, numerical
results are presented to demonstrate the effectiveness of the proposed method,
1OBM3.
Keywords: Hybrid multistep block method; mixed delay; neutral delay
Volterra integro-differential equations; proportional delay
Abstrak
Persamaan kamiran-pembezaan neutral
Volterra dengan kelengahan berkadar dan bercampur (NDVIDE) diselesaikan dengan
teknik baharu yang dicadangkan dalam kaedah berangka iaitu, kaedah blok
berbilang langkah dua titik dan satu luar-titik (1OBM3). Kaedah ini juga dikenali
sebagai kaedah blok berbilang langkah hibrid. Interpolasi polinomial Lagrange
dimanfaatkan bagi membangunkan kaedah blok hibrid. Asas kepada kaedah ini
diambil daripada formula peramal-pembetul. Kaedah yang dicadangkan akan
menyelesaikan NDVIDE dalam dua langkah serentak dengan tiga formula peramal
termasuk satu luar-titik. Masalah NDVIDE diselesaikan melalui teknik saiz
langkah malar. Untuk menyelesaikan masalah di bahagian kamiran dan pembezaan,
dua pendekatan berangka alternatif digunakan. Bahagian pembezaan dianggarkan
dengan memperoleh formula perbezaan terbahagi manakala bahagian kamiran di
interpolasi dengan menggunakan peraturan Simpson komposit. Kaedah yang
dicadangkan telah dianalisis dengan teliti dari segi peringkat, ketekalan,
kestabilan sifar dan penumpuan. Kawasan kestabilan untuk 1OBM3 telah dibina
berdasarkan polinomial kestabilan yang diperoleh. Keputusan berangka
dibentangkan untuk menunjukkan keberkesanan kaedah 1OBM3 yang dicadangkan.
Kata
kunci: Kaedah blok berbilang langkah hybrid; kelengahan bercampur; kelengahan
berkadar; kelengahan neutral persamaan kamiran-pembezaan Volterra
RUJUKAN
Altun, Y.
2021a. Asymptotic behaviours of the solutions of neutral type Volterra
integro-differential equations and some numerical solutions via differential
transform method. MANAS Journal of
Engineering 9(Special 1): 49-57.
Altun, Y.
2021b. A new result on the global exponential stability of nonlinear neutral
Volterra integro-differential equation with variable lags. Mathematics in Natural Science 5(2019): 29-43.
Baharum,
N.A., Majid, Z.A. & Senu, N. 2018. Solving Volterra integrodifferential
equations via diagonally implicit multistep block method. International Journal of Mathematics and Mathematical Sciences 2018:
7392452.
Gurbuz,
B. 2021. A numerical scheme for the solution of neutral integro-differential
equations including variable delay. Mathematical
Sciences 16(1-9): 13-21.
Ismail,
N.I.N., Majid, Z.A. & Senu, N. 2022. Numerical solution for neutral delay
differential equation of constant or proportional type using hybrid block
method. Advanced in Applied Mathematics
and Mechanics 14(5): 1138-1160.
Ismail,
N.I.N., Majid, Z.A. & Senu, N. 2020. Solving neutral delay differential
equation of pantograph type. Malaysian
Journal of Mathematical Sciences (ICoAIMS2019) 14(S): 107-121.
Jator,
S.N. 2010. On the hybrid method with three off-step points for initial value
problems. International Journal of
Mathematical Education in Science and Technology 41(1): 110-118.
Laib, H.,
Bellour, A. & Boulmerka, A. 2022. Taylor collocation method for high-order
neutral delay Volterra integro-differential equations. Journal of Innovative Applied Mathematics and Computational Sciences 2(1): 43-77.
Lambert,
J.D. 1991. Numerical Methods for Ordinary
Differential Systems. New York: John Wiley & Sons.
Lambert,
J.D. 1973. Computational Methods in
Ordinary Differential Equations. New York: John Wiley & Sons.
Li, Y.
& Li, S. 2021. Classical theory of linear multistep methods for Volterra
functional differential equations. Discrete
Dynamics in Nature and Society 2021:
1-15.
Mirzaee,
F., Bimesl, S. & Tohidi, E. 2016. A numerical framework for solving
high-order pantograph-delay Volterra integro-differential equations. Kuwait Journal of Science 43(1): 69-83.
Mohamed,
N.A. & Majid, Z.A. 2016. Multistep block method for solving Volterra
integro-differential equations. Malaysian
Journal of Mathematical Sciences 10(2016):
33-48.
Reutskiy,
S.Y. 2016. The backward substitution method for multipoint problems with linear
volterra–fredholm integro-differential equations of the neutral type. Journal of Computational and Applied
Mathematics 296: 724-738.
Rihan,
F.A., Doha, E.H., Hassan, M.I. & Kamel, N. 2009. Numerical treatments for
Volterra delay integro-differential equations. Computational Methods in Applied Mathematics 9(3): 292-318.
Sedaghat,
S., Ordokhani, Y. & Dehghan, M. 2014. On spectral method for Volterra
functional integro-differential equations of neutral type. Numerical Functional Analysis and Optimization 35(2): 223-239.
Vijayakumar,
V. 2018, Approximate controllability results for abstract neutral integrodifferential
inclusions with infinite delay in Hilbert spaces. IMA Journal of Mathematical Control and Information 5(1): 297-314.
Wen, L.
& Zhou, Y. 2017. Convergence of one-leg methods for neutral delay
integro-differential equations. Journal
of Computational and Applied Mathematics 317: 432-446.
Wen, L.
& Yu, Y. 2016. Convergence of Runge–Kutta methods for neutral delay
integro-differential equations. Applied
Mathematics and Computation 282(2016):
84-96.
Wu, S. & Gan, S. 2008. Analytical and numerical
stability of neutral delay integro-differential equations and neutral delay
partial differential equations. Computers & Mathematics with
Applications 55(11): 2426-2443.
Yuzbasi,
S. 2014. Laguerre approach for solving pantograph-type Volterra
integro-differential equations. Applied
Mathematics and Computation 232: 1183-1199.
Yuzbasi,
S. & Karacayir, M. 2017. A numerical approach for solving high-order linear
delay Volterra integro-differential equations. International Journal of Computational Methods 15(5): 1850042.
*Pengarang untuk surat-menyurat; email:
am_zana@upm.edu.my
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