Sains Malaysiana 40(6)(2011): 651–657

A New Fuzzy Version of Euler’s Method for Solving Differential Equations with Fuzzy Initial Values

(Versi Baru Kaedah Euler Kabur untuk Menyelesaikan Persamaan Pembezaan dengan Nilai-Nilai Awal Kabur)

M. Z. Ahmad*

Institute for Engineering Mathematics, Universiti Malaysia Perlis, 02000 Kuala Perlis, Perlis, Malaysia

M. K. Hasan

School of Information Technology, Faculty of Technology and Information Science

Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia

Diserahkan: 9 March 2010 / Diterima: 11 Oktober 2010

ABSTRACT

This paper proposes a new fuzzy version of Euler’s method for solving differential equations with fuzzy initial values. Our proposed method is based on Zadeh’s extension principle for the reformulation of the classical Euler’s method, which takes into account the dependency problem that arises in fuzzy setting. This problem is often neglected in numerical methods found in the literature for solving differential equations with fuzzy initial values. Several examples are provided to show the advantage of our proposed method compared to the conventional fuzzy version of Euler’s method proposed in the literature.

Keywords: Euler’s method; fuzzy initial value; fuzzy set; optimization

ABSTRAK

Kertas ini mencadangkan satu versi baru kaedah Euler kabur untuk menyelesaikan persamaan pembezaan dengan nilai awal kabur. Pendekatan yang digunakan adalah berasaskan kepada prinsip perluasan Zadeh dengan mengambil kira masalah kebergantungan yang wujud dalam kaedah Euler klasik. Masalah ini sentiasa diabaikan oleh penyelidik-penyelidik dalam menyelesaikan persamaan pembezaan dengan nilai awal kabur. Beberapa contoh diberikan untuk menunjukkan kelebihan kaedah yang dicadangkan dan perbandingan juga dilakukan dengan versi kabur konvensional.

Kata kunci: Kaedah Euler; nilai awal kabur; pengoptimuman; set kabur

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*Pengarang untuk surat-menyurat, email: mzaini@unimap.edu.my