Sains Malaysiana 47(9)(2018): 2179–2186

http://dx.doi.org/10.17576/jsm-2018-4709-28

Penyelesaian Terus Persamaan Pembezaan Biasa Peringkat Tiga menggunakan Kaedah Blok Hibrid Kolokasi Diperbaiki

(Solution of Third Order Ordinary Differential Equation using Improved Block Hybrid Collocation Method)

LEE KHAI CHIEN, UMMUL KHAIR SALMA DIN* & ROKIAH ROZITA AHMAD

Pusat Pengajian Sains Matematik, Fakulti Sains dan Teknologi, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor Darul Ehsan, Malaysia

Diserahkan: 8 Jun 2017/Diterima: 5 Mei 2018 

ABSTRACT

The improved block hybrid collocation method (KBHK(B)) with two off-step points and four collocation points is proposed for the direct solution of general third order ordinary differential equations. Modification is done by adding first derivative of third order function into the general form of block hybrid collocation method to yield KBHK(B). Both main and additional methods are derived via interpolation (power series function) and collocation of the basic polynomial. An improved block method is derived to provide the approximation at five points concurrently. Zero stability, consistency, convergence and absolute stability region of KBHK(B) are investigated. Some numerical examples with exact solution are tested to illustrate the efficiency of the method. KBHK(B) is compared by other block hibrid collocation method in term of global error and step number. It is shown that KBHK(B) generates minimum global error with minimum step number.

Keywords: Block hibrid collocation method; consistency; convergence; third order ordinary differential equations; zero stability

ABSTRAK

Kaedah blok hibrid kolokasi diperbaiki, KBHK(B) dengan dua langkah dan empat titik kolokasi dalam satu blok dibangunkan untuk menyelesaikan masalah persamaan pembezaan biasa peringkat tiga (PPBPT) secara langsung. Penambahan fungsi terbitan pertama bagi PPBPT ke dalam rumus kaedah blok hibrid kolokasi sedia ada dilakukan untuk menghasilkan rumus KBHK(B). Kaedah utama dan kaedah sampingan dihasilkan melalui kaedah interpolasi, iaitu fungsi siri kuasa dan kolokasi terhadap rumus polinomial asas. Kaedah tersebut dibentuk dalam blok untuk menghasilkan anggaran penyelesaian bagi empat titik penyelesaian dengan serentak. Sifat kaedah tersebut diuji daripada segi kestabilan sifar, ketekalan, penumpuan dan rantau kestabilan mutlak. Perbandingan KBHK(B) dengan kaedah blok hibrid kolokasi sedia ada dengan menyelesaikan beberapa masalah berangka peringkat tiga dijalankan untuk menguji keberkesanan kaedah tersebut. Didapati KBHK(B) menghasilkan ralat sejagat yang minimum dengan jumlah bilangan langkah yang paling minimum.

Kata kunci: Kaedah blok hibrid kolokasi; kestabilan sifar; ketekalan; penumpuan; persamaan pembezaan biasa peringkat tiga

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*Pengarang untuk surat-menyurat; email: ummul@ukm.edu.my