The Solutions of Caputo-Fabrizio Random Fractional Ordinary Differential Equations by Aboodh Transform Method

Year 2025, Volume: 18 Issue: 1, 149 - 170, 28.03.2025

Yasin Şahin , Mehmet Merdan

https://doi.org/10.18185/erzifbed.1543499

Cited By: 1

Abstract

The primary objective of the paper is to employ the Aboodh transformation to solve Caputo-Fabrizio ordinary fractional differential equations. Furthermore, the methodology described is used to solve some common fractional differential equations. Analyzed using the Aboodh Transformation Method are random ordinary differential equations derived by randomly picking the coefficients or initial conditions of Caputo-Fabrizio fractional ordinary differential equations. The initial conditions or coefficients of the equations are converted into random variables distributed according to Normal, Uniform, and Exponential distributions. The probability characteristics, including expected value, variance, coefficient of variation, and confidence interval, of the random ordinary differential equations solved are computed using the MATLAB (2013a) package software. The acquired results are then shown and discussed.

Keywords

Caputo-Fabrizio fractional derivative , Aboodh Transform Method , Continuous probability distributions , variance

Project Number

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The Solutions of Caputo-Fabrizio Random Fractional Ordinary Differential Equations by Aboodh Transform Method

Abstract

Project Number

no

References

• [1] Aboodh, K.S. (2013). The New Integral Transform ''Aboodh Transform''. Global Journal of Pure and Applied Mathematics, 9(1), 35-43
• [2] Aboodh, K.S. (2014). Application of New Transform "Aboodh Transform" to Partial Differential Equations. Global Journal of Pure and Applied Mathematics, 10(2), 249-254
• [3] Benattıa, M.E., Kacem Belghaba, K. (2019). A Varıational Iteration Method for Solving Ordinary Differential Equations Using the Aboodh Transform, Global Journal of Pure and Applied Scıences, 25, 55-61
• [4] Şahin Y., (2024). Solutions of Random Fractional Differential Equations Using Aboodh and Aboodh-Adomian Decomposition Method. Master's Thesis, Gümüşhane University, Postgraduate Education Institute
• [5] Abdelilah K., Sedeeg, H., and Abdelrahim Mahgoub, Mohand M., (2016). Aboodh Transform Homotopy Perturbation Method For Solving System Of Nonlinear Partial Differential Equations, Mathematical Theory and Modeling 6,8
• [6] Abdelbagy, A., Alshikhand Mohand M., Abdelrahim Mahgoub, (2016). A Comparative Study Between Laplace Transform and Two New Integrals “ELzaki” Transform and “Aboodh” Transform, Pure and Applied Mathematics Journal, 5(5), 145-150
• [7] Verma, D., Alam, A., Singh,A. P. (2020). Applications of Aboodh Transform, International Research Journal of Innovations in Engineering and Technology, 4(6), 47-51
• [8] Almardy, I. A., Farah, R. A., Alkeer, M. A., Aboodh, K. S., Osman, A. K., Mohammed, M. A. (2023). Aboodh Adomian Decomposition Method Applied To Logistic Differential Model, Journal of Positive School Psychology, 7(2), 482-488
• [9] Asai, Y., Kloeden, P.E. (2013). Numerical schemes for random ODEs via stochastic differential equations. Commun. Appl. Anal, 17(3 & 4), 511–528
• [10] Bekiryazıcı, Z., Merdan, M., Kesemen, T., Najmuldeen, M. (2016). Mathematical Modeling of Dengue Disease under Random Effects Mathematıcal Scıences and Applıcatıons E-Notes, 4 (2), 58-70.
• [11] Merdan, M., Bekiryazıcı, Z., Kesemen, T., Khaniyev, T. (2017a). Deterministic stability and random behavior of a Hepatitis C model, PLoS ONE 12(7): e0181571
• [12] Merdan, M., Bekiryazıcı, Z., Kesemen, T., Khaniyev, T. (2017b). Comparison of stochastic and random models for bacterial resistance Advances in Difference Equations, 133.
• [13] Anaç, H., Merdan, M., Kesemen, T. (2020). Solving for the random component time-fractional partial differential equations with the new Sumudu transform iterative method SN Applied Sciences, , 2,1112
• [14] Bekiryazıcı, Z., Merdan, M., Kesemen, T. (2021). Modification of the random differential transformation method and its applications to compartmental models, Communications in Statistics - Theory and Methods, 50:18, 4271-4292
• [15] Merdan, M., Anaç, H., Kesemen, T. (2019). The new Sumudu transform iterative method for studying the random component time-fractional Klein-Gordon equation. Sigma J Eng Nat Sci 10(3), 343–354
• [16] Bekiryazici Z., Kesemen T., Merdan M., Khaniyev T. (2022). A modification of approximate random characteristics for a model of Zika virus transmission, Thermal Science , 26(4A),3067-3077
• [17] Anaç, H., Merdan, M., Bekiryazıcı, Z., Kesemen, T.(2019). Solution of Some Random Partial Differential Equations Using Differential Transform Method and Laplace-Padé Method. Gümüşhane University Science Institute Journal. 9(1), 108–118
• [18] Alkan, A.,(2024). Analysis of Fractional Advection Equation with Improved Homotopy Analysis Method. Osmaniye Korkut Ata University Journal of Science Institute, 7(3), 1215-1229
• [19] Alkan, A., Anaç, H.(2024). A new study on the Newell-Whitehead-Segel equation with Caputo-Fabrizio fractional derivative. AIMS Mathematics, 9(10), 27979-27997
• [20] Alkan, A., Anaç, H.(2024). The novel numerical solutions for time-fractional Fornberg-Whitham equation by using fractional natural transform decomposition method. AIMS Mathematics, 9(9), 25333-25359
• [21] Alkan, A., (2022). Improving homotopy analysis method with an optimal parameter for time-fractional Burgers equation. Karamanoğlu Mehmetbey University Journal of Engineering and Natural Sciences, 4(2), 117-134
• [22] Al-Refai, M., Pal, K. (2019). New Aspects of Caputo-Fabrizio Fractional Derivative, Progress in Fractional Differentiation and Applications an International Journal, Progr. Fract. Differ. Appl. 5(2), 157-166
• [23] Sania Qureshi, S., A. Rangaig, N., Baleanu, D. (2019). New Numerical Aspects of Caputo-Fabrizio Fractional Derivative Operator, Mathematics, 7(4), 374
• [24] Owolabi, M., Atangana, K. A.(2017). Analysis and Application of New Fractional Adams–Bashforth Scheme With Caputo–Fabrizio Derivative, Chaos, Solitons & Fractals, 111-119
• [25] Bhangale, N., B.Kachhia, K., Gómez-Aguilar, J. F.(2022). A New Iterative Method with ρ Laplace Transform for Solving Fractional Differential Equations with Caputo Generalized Fractional Derivative. Engineering with Computers, 38(3) ,2125–2138
• [26] Caputo, M., Fabrizio, M. (2016). Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels. Progr. Fract. Differ. Appl. 2, 1-11
• [27] Mboro Nchama, G. A. (2020). Properties of Caputo-Fabrizio Fractional Operators. New Trends in Mathematical Sciences. 1(8), 001-025
• [28] Benattia, M.E., Belghaba, K. (2020). Application of The Aboodh Transform for Solving Fractional Delay Differential Equations. Universal Journal of Mathematics and Applications. 3(3), 93-101
• [29] Ziane, D., Belgacem, R., Bokhari, A. (2019). A New Modified Adomian Decomposition Method for Nonlinear Partial Differential Equations. Open Journal of Mathematical Analysis.3(2),81-90
• [30] Khaniyev, T., Ünver, İ., Küçük, Z., Kesemen T. (2017). Olasılık Kuramında Çözümlü Problemler, Nobel Akademik Yayıncılık.
• [31] Akdeniz, F. (2014). Olasılık ve İstatistik, Akedemisyen Kitabevi, Ankara, 602s.