SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS WITH RANDOM VARIABLE COEFFICIENTS BY FOURIER TRANSFORM METHOD

Year 2026, Volume: 27 Issue: 1 , 149 - 165 , 27.03.2026

Hatice Ezber , Mehmet Merdan

https://doi.org/10.18038/estubtda.1785373

https://izlik.org/JA36UA93WW

Abstract

The primary aim of this work is to utilize the Fourier Transform in addressing random ordinary differential equations characterized by stochastic coefficients. The Fourier Transform Method was employed to analyze random ordinary differential equations arising from the stochastic selection of coefficients or initial conditions. The coefficients or beginning conditions were treated as random variables by applying uniform, exponential, and beta probability distributions to them. MATLAB (2013a) was employed to compute the statistical characteristics of the derived random solutions, encompassing expected value, variance, and confidence intervals. The results acquired are illustrated graphically and examined comprehensively.

Keywords

Fourier Transform Method , Continuous probability distribution , Expected value , Variance

References

• [1] Ceylan B. Boundary Solutions of Time-Delayed Nonlinear Parabolic Equations. Trakya Üniversitesi, Fen Bilimleri Enstitüsü,Yüksek Lisans Tezi, 79s, Edirne.2016.
• [2] Hanbay K, Talu M, Özgüven Ö. Real-time fabric defect detection using Fourier transform. Journal of the Faculty of Engineering and Architecture of Gazi University, 2017,32(1), 151-158.
• [3] Düz M, Avezov S, Issa A. Solutions to Differential-Differential Difference Equations with Variable. Süleyman Demirel University Faculty of Arts and Sciences Journal of Science, 2023, 18(3), 259-267.
• [4] Avezov S, Issa A, Düz M. Solving difference equations using fourier transform method. Sigma Journal of Engineering and Natural Sciences, 2024, 42(4),1239−1244.
• [5] Yalaz S. Distribution Function and Fourier Transform in Statistics. Osmaniye Korkut Ata Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 2024,7(2), 581-591.
• [6] Merdan M, Anaç H, Bekiryazıcı Z, Kesemen T. Solving of Some Random Partial Differential Equations by Using Differential Transformation Method and Laplace- Padé Method, Gümüşhane Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 2019, 9(1), 108-118.
• [7] Merdan M, Atasoy N. On the solutions of fractional random ordinary differential equations with the Residual power series method. Alexandria Engineering Journal, 2023,70, 169-177.
• [8] Şahin Y. Solutions of arbitrary fractional-order differential equations using the aboodh and aboodh-Adomian decomposition methods. Gümüşhane Üniversitesi, Lisansüstü Eğitim Enstitüsü, Yüksek Lisans Tezi, 110s, Gümüşhane.2024.
• [9] Merdan M, Öktem, S. On solutions of time fractional order random HIV/AIDS modelling. Sigma Journal of Engineering and Natural Sciences, 2024, 42(6), 1899-1906.
• [10] Cort´es JC, J´odar L. Villafuerte, Numerical solution of random differential equations: A mean square approach, Mathematical and Computer Modelling, 2007, 45, 757–765.
• [11] Bekiryazıcı Z, Merdan M, Kesemen T. Modification of the random differential transformation method and its applications to compartmental models, Communications in Statistics - Theory and Methods, 2021, 50:18, 4271-4292.
• [12] Bekiryazıcı Z, Merdan M, Kesemen, T, Najmulden, M. Mathematical Modeling of Dengue Disease under Random Effects, Mathematical Sciences and Applications E-Notes, 2016, 4(2), 58-70.
• [13] Samadyar N, Mirzaee F. Innovative techniques for the stochastic time-fractional telegraph equation: a meshless method for complex geometries. Iranian Journal of Science, 2025,50 (4).
• [14] Mirzaee F, Rezaei S, Samadyar N. Solution of time‐fractional stochastic nonlinear sine‐Gordon equation via finite difference and meshfree techniques. Mathematical Methods in the Applied Sciences, 2022, 45 (7), 3426-3438.
• [15] Mirzaee F, Rezaei S, Samadyar N. Application of combination schemes based on radial basis functions and finite difference to solve stochastic coupled nonlinear time fractional sine-Gordon equations. Computational and Applied Mathematics, 2022,41 (1), 10.
• [16] Mirzaee, F., Rezaei, S., Samadyar, N. Numerical solution of two-dimensional stochastic time-fractional Sine–Gordon equation on non-rectangular domains using finite difference and meshfree methods. Engineering Analysis with Boundary Elements, 2021,127, 53-63.
• [17] Altın A. Fourier Analysis. Gazi Kitapevi, Ankara, 126s, 2011.
• [18] Feller W. An Introduction to Probability Theory and Its Applications. John Wiley, Hoboken, 683s.1968.
• [19] Khaniyev T, Ünver İ, Küçük Z, Kesemen T. Solved Problems in Probability Theory Nobel Akademik, Ankara, 378s.2017.