Sains Malaysiana 38(5)(2009): 723–728

Solving Directly Two Point Boundary Value Problems Using

Direct Multistep Method

(Penyelesaian Langsung Masalah Nilai Sempadan Dua Titik

Menggunakan Kaedah Multilangkah Secara Langsung)

Mahanum Diana Jafri¹, Mohamed Suleiman², Zanariah Abdul Majid¹* & Zarina Bibi Ibrahim²

¹Institute for Mathematical Research,

Universiti Putra Malaysia, 43400 Serdang, Selangor D.E. Malaysia

Mathematics Department, Faculty Science, Universiti Putra Malaysia

43400 Serdang, Selangor D.E. Malaysia

Received: 11 August 2008 / Accepted: 18 November 2008

ABSTRACT

In this paper we consider solving directly two point boundary value problems (BVPs) for second-order ordinary differential equations (ODEs). We are concerned with solving this problem using multistep method in term of backward difference formula and approximating the solutions with the shooting method. Most of the existence researches involved BVPs will reduce the problem to a system of first order ODEs. This approach is very well established but it obviously will enlarge the system of first order equations. However, the direct multistep method in this paper will be utilised to obtain a series solution of the initial value problems directly without reducing to first order equations. The numerical results show that the proposed method with shooting method can produce good results.

Keywords: Backward difference formula; boundary value problem; shooting method

ABSTRAK

Dalam makalah ini, penyelesaian masalah nilai sempadan dua titik untuk sistem persamaan peringkat kedua telah diambilkira. Masalah ini diselesaikan menggunakan kaedah multilangkah secara langsung dalam sebutan rumus beza ke belakang dan penghampiran kepada penyelesaian menggunakan kaedah tembakan. Penyelidikan yang sedia ada yang melibatkan masalah nilai sempadan akan diturunkan ke sistem persamaan peringkat pertama. Pendekatan ini sangat dikenali tetapi ia akan meningkatkan saiz sistem persamaan peringkat pertama. Manakala, kaedah multilangkah secara langsung di dalam makalah ini akan menghasilkan siri penyelesaian untuk masalah nilai awal tanpa diturunkan ke sistem persamaan peringkat pertama. Hasil berangka menunjukkan kaedah yang dicadangkan bersama kaedah tembakan dapat menghasilkan keputusan yang baik.

Kata kunci: Kaedah tembakan; masalah nilai sempadan; rumus beza ke belakang

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*Corresponding author;  email: zanariah@science.upm.edu.my