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Incorporating Fuzziness in the Traditional Runge–Kutta Cash–Karp Method and Its Applications to Solve Autonomous and Non-Autonomous Fuzzy Differential Equations

Author

Listed:
  • Nurain Zulaikha Husin

    (Institute of Engineering Mathematics, Universiti Malaysia Perlis, Arau 02600, Perlis, Malaysia)

  • Muhammad Zaini Ahmad

    (Institute of Engineering Mathematics, Universiti Malaysia Perlis, Arau 02600, Perlis, Malaysia)

  • Mohd Kamalrulzaman Md Akhir

    (Institute of Engineering Mathematics, Universiti Malaysia Perlis, Arau 02600, Perlis, Malaysia
    Oasis Integrated Group, Institute for Integrated Engineering, Universiti Tun Hussein Onn Malaysia, Parit Raja, Batu Pahat 86400, Johor, Malaysia
    ANNA Systems LLC, Moscow Region, Dubna, 9 Maya Street, Building 7B, Building 2 Office 10.141707, Moscow, Dolgoprudnenskoe Highway, 3, Fiztekhpark, Moscow 141980, Russia)

Abstract

The study of the fuzzy differential equation is a topic that researchers are interested in these days. By modelling, this fuzzy differential equation can be used to resolve issues in the real world. However, finding an analytical solution to this fuzzy differential equation is challenging. Thus, this study aims to present the fuzziness in the traditional Runge–Kutta Cash–Karp of the fourth-order method to solve the first-order fuzzy differential equation. Later, this method is referred to as the fuzzy Runge–Kutta Cash–Karp of the fourth-order method. There are two types of fuzzy differential equations to be solved: autonomous and non-autonomous fuzzy differential equations. This fuzzy differential equation is divided into the (i) and (ii)–differentiability on the basis of the characterization theorem. The convergence analysis of the fuzzy Runge–Kutta Cash–Karp of the fourth-order method is also presented. By implementing the fuzzy Runge–Kutta Cash–Karp of the fourth-order method, the approximate solution is compared with the analytical and numerical solutions obtained from the fuzzy Runge–Kutta of the fourth-order method. The results demonstrated that the approximate solutions of the proposed method are accurate with an analytical solution, when compared with the solutions of the fuzzy Runge–Kutta of the fourth-order method.

Suggested Citation

  • Nurain Zulaikha Husin & Muhammad Zaini Ahmad & Mohd Kamalrulzaman Md Akhir, 2022. "Incorporating Fuzziness in the Traditional Runge–Kutta Cash–Karp Method and Its Applications to Solve Autonomous and Non-Autonomous Fuzzy Differential Equations," Mathematics, MDPI, vol. 10(24), pages 1-25, December.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:24:p:4659-:d:997891
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    References listed on IDEAS

    as
    1. Smita Tapaswini & S. Chakraverty, 2012. "A new approach to fuzzy initial value problem by improved Euler method," Fuzzy Information and Engineering, Springer, vol. 4(3), pages 293-312, September.
    2. Atanaska Georgieva, 2020. "Double Fuzzy Sumudu Transform to Solve Partial Volterra Fuzzy Integro-Differential Equations," Mathematics, MDPI, vol. 8(5), pages 1-13, May.
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