Sains Malaysiana 43(12)(2014): 1951–1959

Modified Exponential-rational Methods for the Numerical Solution of First Order

Initial Value Problems

(Kaedah Eksponen-Nisbah Terubah Suai bagi Penyelesaian Masalah Nilai Awal Peringkat

Pertama Secara Berangka)

TEH YUAN YING*, ZURNI OMAR & KAMARUN HIZAM MANSOR

School of Quantitative Sciences, College of Arts and Sciences, Universiti Utara Malaysia

06010 UUM Sintok, Kedah Darul Aman, Malaysia

Diserahkan: 1 Januari 2014/Diterima: 17 April 2014

ABSTRACT

Exponentially-fitted numerical methods are appealing because L-stability is guaranteed when solving initial value problems of the form y' = λy, y(a) = η, λ ∈ , Re(λ) < 0. Such numerical methods also yield the exact solution when solving the above-mentioned problem. Whilst rational methods have been well established in the past decades, most of them are not ‘completely’ exponentially-fitted. Recently, a class of one-step exponential-rational methods (ERMs) was discovered. Analyses showed that all ERMs are exponentially-fitted, hence implying L-stability. Several numerical experiments showed that ERMs are more accurate than existing rational methods in solving general initial value problem. However, ERMs have two weaknesses: every ERM is non-uniquely defined and may return complex values. Therefore, the purpose of this study was to modify the original ERMs so that these weaknesses will be overcome. This study discusses the generalizations of the modified ERMs and the theoretical analyses involved such as consistency, stability and convergence. Numerical experiments showed that the modified ERMs and the original ERMs are found to have comparable accuracy; hence modified ERMs are preferable to original ERMs.

Keywords: Exponential function; initial value problem; modified exponential-rational method; problem whose solution possesses singularity; rational function

ABSTRAK

Kaedah berangka yang bersesuaian secara eksponen adalah menarik kerana kestabilan L adalah terjamin apabila menyelesaikan masalah nilai awal yang berbentuk y’ = λy, y(a) = η, λ ∈ , Re(λ) < 0. Kaedah berangka yang sedemikian juga menghasilkan penyelesaian tepat apabila menyelesaikan masalah yang dinyatakan sebelum ini. Walaupun kaedah nisbah telah menjadi mantap dalam beberapa dekad yang lalu, sebahagian besar daripada kaedah ini tidak bersesuaian secara eksponen sepenuhnya. Baru-baru ini, suatu kelas kaedah eksponen-nisbah satu-langkah (ERM) telah ditemui. Beberapa analisis menunjukkan bahawa semua ERM adalah bersesuaian secara eksponen, maka mengimplikasikan kestabilan L. Beberapa pengujian berangka menunjukkan bahawa ERM adalah lebih tepat berbanding dengan kaedah nisbah yang sedia ada dalam menyelesaikan masalah nilai awal umum. Walau bagaimanapun, ERM mempunyai dua kelemahan: setiap ERM tidak ditakrifkan secara unik dan boleh mengembalikan nilai-nilai yang kompleks. Oleh itu, tujuan kajian ini adalah untuk mengubah suai ERM yang asal supaya kelemahan tersebut dapat diatasi. Kajian ini membincangkan pengitlakan bagi ERM yang diubah suai dan analisis teori yang terlibat seperti kekonsistenan, kestabilan dan penumpuan. Pengujian secara berangka menunjukkan bahawa ERM yang telah diubah suai dan ERM yang asal didapati mempunyai ketepatan yang setara; maka ERM yang diubah suai lebih sesuai berbanding dengan ERM yang asal.

Kata kunci: Fungsi eksponen; fungsi nisbah; kaedah eksponen-nisbah diubah suai; masalah dengan penyelasaian yang mempunyai ketunggalan; masalah nilai awal

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*Pengarang untuk surat-menyurat; email: yuanying@uum.edu.my