Kesir mertebeden rastgele adi diferansiyel denklemlerin Adomian Ayrıştırma Yöntemi ile analizi

Öz

Bu çalışmada, adi diferansiyel denklemlerin katsayılarının veya başlangıç koşullarının rasgele seçilmesiyle elde edilen rasgele adi diferansiyel denklemler, Adomian Ayrıştırma Yöntemi ile analiz edilecektir. Denklemlerin başlangıç koşulları veya katsayıları, normal ve üstel dağılıma sahip rasgele değişkenlere dönüştürülecektir. Elde edilen rastgele adi diferansiyel denklemlerin beklenen değeri, varyansı ve güven aralığı gibi olasılık özellikleri hesaplanacaktır. Elde edilen sonuçlar MATLAB (2013a) paket programı yardımıyla çizilecek ve rastgele sonuçlar yorumlanacaktır.

Anahtar Kelimeler

• Adomian ayrıştırma yöntemi
• kesirli türev
• lineer olmayan adi diferansiyel denklem

Destekleyen Kurum

yok

Analysis of fractional random ordinary differential equations by Adomian Decomposition Method

Öz

In this study, random ordinary differential equations obtained by randomly choosing the coefficients or initial conditions of the ordinary differential equations will be analyzed by the Adomian Decomposition Method. The initial conditions or coefficients of the equations will be converted to random variables with normal and exponential distribution. Probability characteristics such as expected value, variance and confidence interval of the obtained random ordinary differential equations will be calculated. Obtained results will be drawn with the help of MATLAB (2013a) package program and random results will be interpreted.

Anahtar Kelimeler

• Adomian decomposition method
• fractional derivative
• nonlinear ordinary differantial equation

Kaynakça

1. Adomian, G., Nonlinear Stochastic Operator Equation, USA: Academic Press., (1986).
2. Adomian, G., A review of decomposition method and some recent results for nonlinear equation, Math. Comput. Model., 5, 101-127, (1991).
3. Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method, Dordrecht: Kluwer Academic, (1993).
4. Babolian, E., Biazar, J. and Vahidi, A. R., Solution of a system of nonlinear equations by Adomian decomposition method. Applied Mathematics and Computation, 150(3), 847-854, (2004).
5. Hamoud, A.A., Ghadle, K. and Atshan, S., The approximate solutions of fractional integro-differential equations by using modified Adomian decomposition method. Khayyam J. Math. 5(1), 21–39, (2019).
6. Odibat, Z., An optimized decomposition method for nonlinear ordinary and partial differential equations, Physica A 541, Article ID 123323 (2019).
7. Turkyilmazoglu, M., Accelerating the convergence of Adomian decomposition method (ADM), J. Comput. Sci., 31, 54–59 (2019).
8. Li, W. and Pang, Y., Application of Adomian decomposition method to nonlinear systems. Advances in Difference Equations, 2020(1), 1-17, (2020).

9. Zeidan, D., Chau, C. K., Lu, T. T. and Zheng, W. Q., Mathematical studies of the solution of Burgers' equations by Adomian decomposition method. Mathematical Methods in the Applied Sciences, 43(5), 2171-2188, (2020).
10. Ali, A., Gul, Z., Khan, W. A., Ahmad, S. and Zeb, S., Investigation of fractional order sine-Gordon equation using Laplace Adomian decomposition method, Fractals, 29(05), 2150121, (2021).
11. Lu, T. T. and Zheng, W. Q., Adomian decomposition method for first order PDEs with unprescribed data. Alexandria Engineering Journal, 60(2), 2563-2572, (2021).
12. Sayed, Y., M Abdelgaber, K., R Elmahdy, A. and L El-Kalla, I., Solution of the telegraph equation using adomian decomposition method with accelerated formula of adomian polynomials. Information Sciences Letters, 10(1), 6, (2021).
13. Kumar, M., Numerical solution of singular boundary value problems using advanced Adomian decomposition method. Engineering with Computers, 37(4), 2853-2863, (2021).
14. Lin, M. X., Tseng, C. H. and Chen, C. K., Numerical solution of large deflection beams by using the Laplace Adomian decomposition method, Engineering Computations, (2021).
15. Maturi, D. A. and Malaikah, H. M., The adomian decomposition method for solving nonlinear partial differential equation using maple. Advances in Pure Mathematics, 11(6), 595-603, (2021).
16. Hussein, M. A., A Review on Algorithms of Sumudu Adomian Decomposition Method for FPDEs. Journal of Research in Applied Mathematics, 8(8), (2022).
17. Kumar, M., Recent development of Adomian decomposition method for ordinary and partial differential equations, International Journal of Applied and Computational Mathematics, 8(2), 1-25, (2022).
18. Hussein, M. A., A Review on Algorithms of Laplace Adomian Decomposition Method for FPDEs, Scientific Research Journal of Multidisciplinary, 2, 1-10, (2022).
19. Habib, U., Zeb, S., Shah, K. and Hussain, S. M., KdV Equation Solution by Double Laplace Adomian Decomposition Method and Its Convergence Analysis, Bioinorganic Chemistry & Applications, (2022).
20. Bairwa, R. K., Priyanka, S. B. and Tyagi, S., Analytical Approach to Fractional Fisher Equations by Laplace-Adomian Decomposition Method, Annals of Pure and Applied Mathematics, 26(2), 55-66, (2022).
21. Mulenga, J. and Phiri, P. A., Adomian Decomposition Method Applied to Covid-19 Model. Applied Mathematical Sciences, 16(2), 59-70, (2022).
22. Gaxiola, O. G., Solution of nonlinear partial differential Equations by adomian decomposition method: Solução de diferencial parcial não-linear Equações pelo método de decomposição adomiana. Studıes In Engıneerıng And Exact Scıences, 3(1), 61-78, (2022).
23. Bekiryazici, Z., Merdan, M. and Kesemen, T., Modification of the random differential transformation method and its applications to compartmental models. Communications in Statistics-Theory and Methods, 50(18), 4271-4292, (2021).
24. Keskin, A. Ü., Boundary Value Problems for Engineers With MATLAB Solutions, Switzerland: Springer, (2019).
25. Kiymaz, O., An algorithm for solving initial value problems using Laplace Adomian decomposition method. Applied Mathematical Sciences, 3(30), 1453-1459, (2009).
26. Wazwaz, A. M., A new algorithm for calculating Adomian polynomials for nonlinear operators, Applied Mathematics and Computation, 111(1), 53–69, (2000).
27. Merdan M., Altay Ö. and Bekiryazıcı Z., Behaivours of Random Effected Volterra and Fredholm Integral Equation, International Conference on Mathematics and Mathematics Education, Ordu, Türkiye, 27 - 29 Haziran 2018, ss.259-260, (2018).
28. Merdan, M. , Altay, Ö. and Bekiryazıcı, Z., Investigation of the Behaviour of Volterra Integral Equations with Random Effects, Gümüşhane Üniversitesi Fen Bilimleri Dergisi, 10 (2020 ): 205-216, (2020).
29. Anaç, H., Merdan, M. and Kesemen, T., Rastgele bileşenli zaman-fraksiyonel kısmi diferansiyel denklemlerin yeni Sumudu dönüşümü yinelemeli yöntemiyle çözülmesi, SN Applied Sciences, 2: 1-11, (2020).
30. Anaç, H., Merdan, M., Bekiryazıcı, Z. and Kesemen, T., Bazı Rastgele Kısmi Diferansiyel Denklemlerin Diferansiyel Dönüşüm Metodu ve Laplace-Padé Metodu Kullanarak Çözümü, Gümüşhane Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 9 (1): 108-118, (2019).
31. Sengul, S., Bekiryazıcı, Z. and Merdan, M., The performance of Wong-Zakai approximations for the investigation of stochastic differential equation models with nonlinear multiplicative noise, Acta Mathematica Universitatis Comenianae 90 (2), 231-243, (2021).
32. Şişman, Ş. and Merdan, M., Global stability of Susceptible Diabetes Complication (SDC) model in discrete time, Sigma Journal of Engineering and Natural Sciences 39 (3), 290-312, (2021).
33. Merdan, M. and Şişman, Ş., Investigation of linear difference equations with random effects, Advances in Difference Equations (1), 1-19, (2020).
34. Yavuz M., Novel solution methods for initial boundary value problems of fractional order with conformable differentiation. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 8(1), 1-7, (2018)
35. Duran, S., Durur, H., Yavuz, M. and Yokus, A., Discussion of numerical and analytical techniques for the emerging fractional order Murnaghan model in materials science, Optical and Quantum Electronics, 55(6), 571, (2023)
36. Evirgen, F. and Özdemir, N., Multistage Adomian Decomposition Method for Solving NLP Problems Over a Nonlinear Fractional Dynamical System, Journal of computational and nonlinear dynamics, 6(2).021003, (2011)
37. Yel, G., Kayhan, M. and Ciancio, A., A new analytical approach to the (1+1)-dimensional conformable Fisher equation, Mathematical Modelling and Numerical Simulation with Applications, 2(4), 211-220, (2022)
38. Isah, M.A. and Yokuş, A., The investigation of several soliton solutions to the complex Ginzburg-Landau model with Kerrlaw nonlinearity, Mathematical Modelling and Numerical Simulation with Applications, 2(3),147-163, (2022)
39. Yavuz, M. and Ozdemir, N., A quantitative approach to fractional option pricing problems with decomposition series, Konuralp Journal of Mathematics, 6(1), 102-109, (2018)
40. Yavuz, M., European option pricing models described by fractional operators with classical and generalized Mittag‐Leffler kernels, Numerical Methods for Partial Differential Equations, 38(3), 434-456, (2020)
41. Chen Y.,Liu F., Yu Q., Li T.,Review of fractional epidemic models, Applied Mathematical Modelling, 97 (2021) 281–307
42. Hanert E.,Schumacher E., Front dynamics in fractional-order epidemic modes, J. Theor. Biol., 279 (2011) 9–16 .
43. Podlubny, I., Fractional differential equations, Academic Press (1998)
44. Abdelrazec, A. and Pelinovsky, D., Convergence of ADM for initial value problems, Wiley Periodicals; 2009. DOI: 10.1002/num.20549