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 MAJID^1,2* & 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

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*Corresponding author; email: zana_majid99@yahoo.com