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

Received: 9 March 2010 / Accepted: 11 October 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

REFERENCES

Ahmad, M.Z. & De Baets, B. 2009. A predator-prey model with fuzzy initial populations. Proceedings of the Joint 13th IFSA World Congress and 6th EUSFLAT Conference pp. 1311-1314.

Ahmad, M.Z. & Hasan, M.K. 2010. Incorporating optimisation technique into Euler method for solving differential equations with fuzzy initial values. Proceeding of the 1st Regional Conference on Applied and Engineering Mathematics: 2-3 June, Penang, Malaysia.

Ahmad, M.Z., Hasan, M.K. & De Baets, B. 2010. A new method for computing continuous function with fuzzy variable.Proceeding of International Conference on Fundamental and Appplied Sciences: 15-17 June, Kuala Lumpur, Malaysia

Bonarini, A. & Bontempi, G. 1994. A qualitative simulation approach for fuzzy dynamical models. ACM Transaction Modelling and Computer Simulation 4:285-313.

Buckley, J.J. & Feuring, T. 2000. Fuzzy differential equations. Fuzzy Sets and Systems 110: 43-54.

Diniz, G. L., Fernandes, J.F.R., Meyer, J.F.C.A. & Barros, L.C. 2001. A fuzzy Cauchy problem modelling the decay of the biochemical oxygen demand in water. Proceding of the Joint 9th IFSA World Congress and 20th NAFIPS International Conferences pp. 512–516.

Hüllermeier, E. 1997. An approach to modelling and simulation of uncertain dynamical systems. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 5: 117-137.

Kaleva, O. 1987. Fuzzy differential equations. Fuzzy Sets and Systems 24: 301-317.

Laksmikantham, V. 2004. Set differential equations versus fuzzy differential equations. Applied Mathematics and Computation 164: 277-294.

Ma, M., Friedman, M. & Kandel, A. 1999. Numerical solution of fuzzy differential equations. Fuzzy Sets and Systems 105:133-138.

Mizukoshi, M.T., Barros, L.C., Chalco-Cano, Y. Román-Flores, H. & Bassanezi, R.C. 2007. Fuzzy differential equations and the extension principle. Information Sciences 177: 3627-3635.

Román-Flores, H., Barros, L. & Bassanezi, R. 2001. A note on Zadeh’s extension principle. Fuzzy Sets and Systems 117: 327-331.

Seikkala, S. 1987. On the fuzzy initial value problem. Fuzzy Sets and Systems 24: 319-330.

Zadeh, L.A. 1965. Fuzzy sets. Information and Control 8: 338-353.

Zadeh, L.A. 1975a. The concept of linguistic variables and its application to approximate reasoning I, Information Sciences 8: 199-249.

Zadeh, L.A. 1975b. The concept of linguistic variables and its application to approximate reasoning II, Information Sciences 8: 301-357.

Zadeh, L.A. 1975c. The concept of linguistic variables and its application to approximate reasoning III, Information Sciences 9: 43-80.

*Corresponding author, email: mzaini@unimap.edu.my