Hybridization of the shooting and Runge-Kutta Cash-Karp methods for solving Fuzzy Boundary Value Problems

Nurain Zulaikha Husin; Muhammad Zaini Ahmad; Nurain Zulaikha Husin; Muhammad Zaini Ahmad

doi:10.3934/math.20241529

AIMS Mathematics

2024, Volume 9, Issue 11: 31806-31847. doi: 10.3934/math.20241529

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Research article

Hybridization of the shooting and Runge-Kutta Cash-Karp methods for solving Fuzzy Boundary Value Problems

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

Received: 15 July 2024 Revised: 21 October 2024 Accepted: 23 October 2024 Published: 08 November 2024
MSC : 34K28, 03E72, 34L99

Fuzzy Differential Equations (FDEs) have attracted great interest among researchers. These FDEs have been used to develop a mathematical model for everyday life problems. In this study, we propose a solution method for a second-order Fuzzy Boundary Value Problem (FBVP). Four systems of FBVPs were developed based on the generalized fuzzy derivative. The second-order FBVP for each system was divided into two parts: Fuzzy non-homogeneous and fuzzy homogeneous equations. Using the shooting method, these two equations were then reduced to first-order FDEs. By implementing the Fuzzy Runge-Kutta Cash-Karp of the fourth-order method (FRKCK4), the approximate solution was compared with the analytical solution and the solution from the Fuzzy Runge-Kutta of the fourth-order method (FRK4).
- fuzzy boundary value problem,
- Runge-Kutta Cash-Karp,
- fuzzy differential equations
Citation: Nurain Zulaikha Husin, Muhammad Zaini Ahmad. Hybridization of the shooting and Runge-Kutta Cash-Karp methods for solving Fuzzy Boundary Value Problems[J]. AIMS Mathematics, 2024, 9(11): 31806-31847. doi: 10.3934/math.20241529

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Abstract

Fuzzy Differential Equations (FDEs) have attracted great interest among researchers. These FDEs have been used to develop a mathematical model for everyday life problems. In this study, we propose a solution method for a second-order Fuzzy Boundary Value Problem (FBVP). Four systems of FBVPs were developed based on the generalized fuzzy derivative. The second-order FBVP for each system was divided into two parts: Fuzzy non-homogeneous and fuzzy homogeneous equations. Using the shooting method, these two equations were then reduced to first-order FDEs. By implementing the Fuzzy Runge-Kutta Cash-Karp of the fourth-order method (FRKCK4), the approximate solution was compared with the analytical solution and the solution from the Fuzzy Runge-Kutta of the fourth-order method (FRK4).

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