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International Journal of Pure & Applied Mathematical Research

ON NUMERICALLY SOLVING FIRST-ORDER FUZZY DIFFERENTIAL EQUATIONS BY THE THIRD ORDER RUNGE-KUTTA METHOD BASED ON LINEAR COMBINATION OF VARIOUS KINDS OF MEANS

R. Diwahar, A. Hari Ganesh , V. Parimala

R. Diwahar*, A. Hari Ganesh** and V. Parimala**

Research Scholar* (Full Time), Assistant Professor**, PG & Research Department of Mathematics, Poompuhar College (Autonomous), (Affiliated to Bharathidasan University), Melaiyur – 609 107, Mayiladuthurai (Dt.), Tamil Nadu, India.

Assistant Professor***, Department of Mathematics, AJK College of Arts and Science (Autonomous,), Navakkari, Coimbatore –641 105

DOI : https://doi.org/10.5281/zenodo.15469375 Page No : 01-19

Published Online : 2025-05-20

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ABSTRACT

Using the RK Method of Order Three, which is based on the combination of various types of means in pair such as Geometric Mean, Harmonic Mean, and Heronian Mean in a linear form, an attempt has been made in this work to discover a numerical solution for fuzzy differential equations of the first order. This methodology was used in order to accomplish this goal. Through the use of the γ-cut of triangular fuzzy numbers, the objective of this study is to find a solution to the first-order fuzzy initial value problem. Doing this test proves that the proposed method is both accurate and efficient.

Keywords: Fuzzy Initial Value Problem, Runge – Kutta Method, Triangular Fuzzy Number, Geometric Mean, Heronian Mean, Harmonic Mean

REFERENCES

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Abbasbandy. S, Allahviranloo. T, Numerical solution of fuzzy differential equations by Taylor method, Journal of Computational Methods and Applied Mathematics, 2(2002)113-124.
Abdul-MajidWazwaz, A comparision of modified Runge-Kutta formulas based on a variety of means, International Journal of Computer Mathematics, vol.50, pp-105-112.
Balachandran. K, Prakash. P, Existence of solutions of fuzzy delay differential equations with nonlocal condition, Journal of the Korea Society for Industrial and Applied Mathematics, 6(2002)81-89.
Balachandran. K, Kanagarajan. K, Existence of solutions of fuzzy delay integro-differential equations with nonlocal condition, Journal of the Korea Society for Industrial and Applied Mathematics, 9(2005)65-74.
Chang. S.L, Zadeh. L.A, On fuzzy mapping and control, IEEE Trans, Systems Man Cybernet. 2(1972)30-34.
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GethsiSharmila . R and E.C. Henry Amirtharaj, Implementation of a new third order weighted Runge-Kutta formula based on Centroidal Mean for solving stiff initial value problems, Recent Research in Science and Technology 3(10) (2011) 91-97.
Gethsi Sharmila. R and E.C. Henry Amirtharaj, Numerical Solution of Fuzzy Initial Value Problems by Fourth Order Runge-Kutta Method Based on Contraharmonic Mean, Indian journal of applied research 3(4) (2013) 339-343.
Gethsi Sharmila. R and E.C. Henry Amirtharaj, Numerical Solution of Nth - Order Fuzzy Initial Value Problems by Fourth Order Runge-Kutta Method Basesd On Centroidal Mean IOSR Journal of Mathematics (IOSR-JM) Volume 6, Issue 3 (May. - Jun. 2013), PP 47-63
Kaleva. O, Fuzzy differential equations, Fuzzy Sets and Systems, 24(1987)301-317.
Kaleva. O, The Cauchy problem for fuzzy differential equations, Fuzzy Sets and Systems, 35(1990)389 -396.
Kanagarajan. K, Sambath. M , Runge- Kutta Nystrom method of order three for solving fuzzy differential equations, Computational methods in Applied Mathematics, Vol.10(2010) , No.2, pp.195-203.
Kanagarajan. K, Sambath. M , Numerical solution of fuzzy differential equations by third order Runge-Kutta method, International journal of Applied Mathematics and Computaion, 2(4) (2010) pp.1-8.
Khattri S. K. (2012), Euler’s Number and Some Means, Tamsui Oxford Journalof Information and Mathematical Sciences, pp369-377.
Ma. M, Friedman. M, Kandel. A, Numerical solutions of fuzzy differential equations, Fuzzy Sets and Systems, 105(1999)133-138.
Nirmala. V, ChenthurPandian. S, “Numerical Solution of Fuzzy Differential Equation by Fourth Order Runge-Kutta Method with Higher Order Derivative Approximations”, European Journal of Scientific Research, Vol.62 N0.2 (2011),pp.198-206.
Palligkinis. S.ch., Papageorgiou. G, Famelis, I. TH., “Runge-Kutta methods for fuzzy differential equations”, App. Math.Comp 209(2009),97-105.
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Cite this Article as : R. Diwahar, A. Hari Ganesh, & V. Parimala (2025), “On Numerically Solving First-Order Fuzzy Differential Equations By The Third Order Runge-Kutta Method Based On Linear Combination Of Various Kinds Of Means”, International Journal of Pure & Applied Mathematical Research, Vol. 9, Issue 1, 2025, pp 1-19, DOI: https://doi.org/10.5281/zenodo.15469375

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REFERENCES

Abbasbandy. S, Allah Viranloo. T, Numerical solution of fuzzy differential equations by Runge-Kutta method, Nonlinear studies.11(2004) N0.1,117-129.
Abbasbandy. S, Allahviranloo. T, Numerical solution of fuzzy differential equations by Taylor method, Journal of Computational Methods and Applied Mathematics, 2(2002)113-124.
Abdul-MajidWazwaz, A comparision of modified Runge-Kutta formulas based on a variety of means, International Journal of Computer Mathematics, vol.50, pp-105-112.
Balachandran. K, Prakash. P, Existence of solutions of fuzzy delay differential equations with nonlocal condition, Journal of the Korea Society for Industrial and Applied Mathematics, 6(2002)81-89.
Balachandran. K, Kanagarajan. K, Existence of solutions of fuzzy delay integro-differential equations with nonlocal condition, Journal of the Korea Society for Industrial and Applied Mathematics, 9(2005)65-74.
Chang. S.L, Zadeh. L.A, On fuzzy mapping and control, IEEE Trans, Systems Man Cybernet. 2(1972)30-34.
Dubois. D, Prade. H, Towards fuzzy differential calculus part 3: Differentiation, Fuzzy Sets and Systems, 8(1982)225-233.
Evans, D.J. and Yaacub, A.R. (1993). A new fourth order Runge – Kutta Method based on the Centroidal Mean (CeM) formula. Computer Studies 851. Department of Computer Studies, University of Technology, Loughborough, U.K
GethsiSharmila . R and E.C. Henry Amirtharaj, Implementation of a new third order weighted Runge-Kutta formula based on Centroidal Mean for solving stiff initial value problems, Recent Research in Science and Technology 3(10) (2011) 91-97.
Gethsi Sharmila. R and E.C. Henry Amirtharaj, Numerical Solution of Fuzzy Initial Value Problems by Fourth Order Runge-Kutta Method Based on Contraharmonic Mean, Indian journal of applied research 3(4) (2013) 339-343.
Gethsi Sharmila. R and E.C. Henry Amirtharaj, Numerical Solution of Nth - Order Fuzzy Initial Value Problems by Fourth Order Runge-Kutta Method Basesd On Centroidal Mean IOSR Journal of Mathematics (IOSR-JM) Volume 6, Issue 3 (May. - Jun. 2013), PP 47-63
Kaleva. O, Fuzzy differential equations, Fuzzy Sets and Systems, 24(1987)301-317.
Kaleva. O, The Cauchy problem for fuzzy differential equations, Fuzzy Sets and Systems, 35(1990)389 -396.
Kanagarajan. K, Sambath. M , Runge- Kutta Nystrom method of order three for solving fuzzy differential equations, Computational methods in Applied Mathematics, Vol.10(2010) , No.2, pp.195-203.
Kanagarajan. K, Sambath. M , Numerical solution of fuzzy differential equations by third order Runge-Kutta method, International journal of Applied Mathematics and Computaion, 2(4) (2010) pp.1-8.
Khattri S. K. (2012), Euler’s Number and Some Means, Tamsui Oxford Journalof Information and Mathematical Sciences, pp369-377.
Ma. M, Friedman. M, Kandel. A, Numerical solutions of fuzzy differential equations, Fuzzy Sets and Systems, 105(1999)133-138.
Nirmala. V, ChenthurPandian. S, “Numerical Solution of Fuzzy Differential Equation by Fourth Order Runge-Kutta Method with Higher Order Derivative Approximations”, European Journal of Scientific Research, Vol.62 N0.2 (2011),pp.198-206.
Palligkinis. S.ch., Papageorgiou. G, Famelis, I. TH., “Runge-Kutta methods for fuzzy differential equations”, App. Math.Comp 209(2009),97-105.
Puri. M.L, Ralescu. D.A, Differentials of fuzzy functions, Journal of Mathematical Analysis and Applications, 91(1983)552-558.

Int. J.of Pure & App Math. Res.

ON NUMERICALLY SOLVING FIRST-ORDER FUZZY DIFFERENTIAL EQUATIONS BY THE THIRD ORDER RUNGE-KUTTA METHOD BASED ON LINEAR COMBINATION OF VARIOUS KINDS OF MEANS

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