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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
Received:
1 January 2014/Accepted: 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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*Corresponding author; email: yuanying@uum.edu.my
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