Numerical solution of system of fuzzy fractional order Volterra integro-differential equation using optimal homotopy asymptotic method

Sumbal Ahsan; Rashid Nawaz; Muhammad Akbar; Saleem Abdullah; Kottakkaran Sooppy Nisar; Velusamy Vijayakumar; Sumbal Ahsan; Rashid Nawaz; Muhammad Akbar; Saleem Abdullah; Kottakkaran Sooppy Nisar; Velusamy Vijayakumar

doi:10.3934/math.2022726

AIMS Mathematics

2022, Volume 7, Issue 7: 13169-13191. doi: 10.3934/math.2022726

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Numerical solution of system of fuzzy fractional order Volterra integro-differential equation using optimal homotopy asymptotic method

1.
Department of Mathematics, Abdul Wali Khan University Mardan, Khyber Pakhtunkhwa, Pakistan
2.
Department of Mathematics, College of Arts and Sciences, Wadi Aldawaser, 11991, Prince Sattam bin Abdulaziz University, Saudi Arabia
3.
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore -632 014, Vellore, Tamilnadu, India

Received: 23 December 2021 Revised: 26 March 2022 Accepted: 05 April 2022 Published: 10 May 2022
MSC : 34K28, 47Gxx, 45Dxx

In this paper, an efficient technique called Optimal Homotopy Asymptotic Method has been extended for the first time to the solution of the system of fuzzy integro-differential equations of fractional order. This approach however, does not depend upon any small/large parameters in comparison to other perturbation method. This method provides a convenient way to control the convergence of approximation series and allows adjustment of convergence regions where necessary. The series solution has been developed and the recurrence relations are given explicitly. The fuzzy fractional derivatives are defined in Caputo sense. It is followed by suggesting a new result from Optimal Homotopy Asymptotic Method for Caputo fuzzy fractional derivative. We then construct a detailed procedure on finding the solutions of system of fuzzy integro-differential equations of fractional order and finally, we demonstrate a numerical example. The validity and efficiency of the proposed technique are demonstrated via these numerical examples which depend upon the parametric form of the fuzzy number. The optimum values of convergence control parameters are calculated using the well-known method of least squares, obtained results are compared with fractional residual power series method. It is observed from the results that the suggested method is accurate, straightforward and convenient for solving system of fuzzy Volterra integrodifferential equations of fractional order.
- OHAM,
- fuzzy Caputo operator,
- Riemann-Liouville fractional integral operator,
- approximate solution,
- system of fuzzy fractional order,
- Volterra integro-differential equations
Citation: Sumbal Ahsan, Rashid Nawaz, Muhammad Akbar, Saleem Abdullah, Kottakkaran Sooppy Nisar, Velusamy Vijayakumar. Numerical solution of system of fuzzy fractional order Volterra integro-differential equation using optimal homotopy asymptotic method[J]. AIMS Mathematics, 2022, 7(7): 13169-13191. doi: 10.3934/math.2022726

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Abstract

In this paper, an efficient technique called Optimal Homotopy Asymptotic Method has been extended for the first time to the solution of the system of fuzzy integro-differential equations of fractional order. This approach however, does not depend upon any small/large parameters in comparison to other perturbation method. This method provides a convenient way to control the convergence of approximation series and allows adjustment of convergence regions where necessary. The series solution has been developed and the recurrence relations are given explicitly. The fuzzy fractional derivatives are defined in Caputo sense. It is followed by suggesting a new result from Optimal Homotopy Asymptotic Method for Caputo fuzzy fractional derivative. We then construct a detailed procedure on finding the solutions of system of fuzzy integro-differential equations of fractional order and finally, we demonstrate a numerical example. The validity and efficiency of the proposed technique are demonstrated via these numerical examples which depend upon the parametric form of the fuzzy number. The optimum values of convergence control parameters are calculated using the well-known method of least squares, obtained results are compared with fractional residual power series method. It is observed from the results that the suggested method is accurate, straightforward and convenient for solving system of fuzzy Volterra integrodifferential equations of fractional order.

References

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