Sains Malaysiana 48(3)(2019): 677–684

http://dx.doi.org/10.17576/jsm-2019-4803-22

 

Fifth Order Multistep Block Method for Solving Volterra Integro-Differential Equations of Second Kind

(Kaedah Blok Berbilanglangkah Peringkat Lima bagi Penyelesaian Persamaan Pembezaan - Kamiran Volterra Jenis Kedua)

 

ZANARIAH ABDUL MAJID1,2* & NURUL ATIKAH MOHAMED1

 

1Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia

 

2Mathematics Department, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia

 

Received: 3 July 2018/Accepted: 21 November 2018

 

ABSTRACT

In the present paper, the multistep block method is proposed to solve the linear and non-linear Volterra integro-differential equations (VIDEs) of the second kind using constant step size. The proposed block method of order five consists of two point block method presented as in the simple form of Adams Moulton type. The numerical solutions are obtained at two new values simultaneously at each of the integration step. In VIDEs, the unknown function appears in the form of derivative and under the integral sign. The approximation of the integral part is estimated using the Boole’s quadrature rule. The stability region is shown, and the numerical results are presented to illustrate the performance of the proposed method in terms of accuracy, total function calls and execution times compared to the existing method.

 

Keywords: Block method; quadrature rule; Volterra integro-differential equation

 

ABSTRAK

Dalam makalah ini, kaedah blok berbilanglangkah dicadangkan bagi menyelesaikan persamaan pembezaan-kamiran Volterra (PPKV) linear dan tak linear daripada jenis kedua menggunakan saiz langkah yang malar. Kaedah blok peringkat lima yang dicadangkan terdiri daripada dua titik blok yang dibentangkan dalam bentuk yang mudah daripada jenis Adams Moulton. Penyelesaian berangka diperoleh dalam dua nilai baru pada masa yang sama di setiap langkah kamiran. Dalam PPKV, fungsi yang tidak diketahui muncul dalam bentuk terbitan dan tanda kamiran. Penghampiran bahagian kamiran dianggarkan dengan menggunakan peraturan kuadratur Boole. Rantau kestabilan ditunjukkan dan keputusan berangka dibentangkan untuk menggambarkan prestasi kaedah yang dicadangkan daripada segi kejituan, jumlah panggilan fungsi dan masa pelaksanaan berbanding kaedah sedia ada.

 

Kata kunci: Aturan kuadratur; kaedah blok; persamaan pembezaan-kamiran Volterra

REFERENCES

Chang, S.H. 1982. On certain extrapolation methods for the numerical solution of integro-differential equations. Mathematics of Computation 39(159): 165-171.

Chen, H. & Zhang, C. 2011. Boundary value methods for Volterra integral and integro-differential equations. Applied Mathematics and Computation 218: 2619-2630.

Day, J.T. 1967. Note on the numerical solution of integro-differential equations. The Computer Journal 9(4): 394-395.

Dehghan, M. & Salehi, R. 2012. The numerical solution of the non-linear integro-differential equations based on the meshless method. Journal of Computational and Applied Mathematics 236: 2367- 2377.

Faires, D. & Burden, R.L. 2005. Numerical Analysis. Belmont, CA: Thomson Brooks/Cole.

Feldstein, A. & Sopka, J.R. 1974. Numerical methods for nonlinear Volterra integro-differential equations. Siam J. Numer. Anal. 11: 826-846.

Filiz, A. 2014. Numerical solution of linear Volterra integro-differential equation using Runge-Kutta-Fehlberg method. Applied and Computational Mathematics 3(1): 9-14.

Filiz, A. 2013. A fourth-order robust numerical method for integro-differential equations. Asian Journal of Fuzzy and Applied Mathematics 1(1): 28-33.

Ishak, F. & Ahmad, S.N. 2016. Development of extended trapezoidal method for numerical solution of Volterra integro-differential equations. International Journal of Mathematics, Computational, Physical, Electrical and Computer Engineering 10(11): 52856.

Kürkçü, Ö.K., Aslan, E. & Sezer, M. 2017. A novel collocation method based on residual error analysis for solving integro-differential equations using hybrid Dickson and Taylor polynomials. Sains Malaysiana 46(2): 335-347.

Lambert, L.D. 1973. Computational Methods in Ordinary Differential Equations. New York: John Wiley & Sons, Inc.

Linz, P. 1969. Linear multistep methods for Volterra integro-differential equations. Journal of the Association for Computing Machinery 16(2): 295-301.

Lubich, C. 1982. Runge-Kutta theory for Volterra integro-differential equations. Numer. Math. 40: 119-135.

Majid, Z.A. & Suleiman, M. 2011. Predictor-corrector block iteration method for solving ordinary differential equations. Sains Malaysiana 40(6): 659-664.

Makroglou, A. 1982. Hybrid methods in the numerical solution of Volterra integro-differential equations. IMA Jounal of Numerical Analysis 2: 21-35.

Mocarsky, W.L. 1971. Convergence of step-by-step methods for non-linear integro-differential equations. IMA Journal of Applied Mathematics 8(2): 235-239.

Mohamed, N.A. & Majid, Z.A. 2016. Multistep block method for solving Volterra integro-differential equations. Malaysian Journal of Mathematical Sciences 10: 33-48.

Yuan, W. & Tang, T. 1990. The numerical analysis of implicit Runge-Kutta methods for a certain nonlinear integro-differential equation. Mathematics of Computation 54(189): 155-168.

 

*Corresponding author; email: zana_majid99@yahoo.com

 

 

 

 

 

previous