Research Article
BibTex RIS Cite

KESİRLİ MERTEBELİ DİFERANSİYEL DENKLEMLERİN YENİ SAYISAL ÇÖZÜMLERİ

Year 2024, , 43 - 52, 29.06.2024
https://doi.org/10.46460/ijiea.1459659

Abstract

Kesirli stokastik diferansiyel denklemler, çok çeşitli mühendislik ve bilimsel olguları simüle etmek için yaygın olarak kullanılan araçlardır. Bu makalede, belirsiz katsayılar yaklaşımının çeşitli kesirli stokastik modellere uygulanabilirliği incelenmiştir. Bu modeller kesirli beyaz gürültü terimine sahiptir ve çoğunlukla kesirli dereceli türev operatörleri tarafından üretilir. Ayrıca polinom kaos algoritmasının stokastik Lotka-Volterra ve Benney sistemlerine uygulamalarını da araştırıyoruz. Kesirli stokastik denklemler, çok çeşitli bilimsel ve mühendislik problemleri için model olarak işlev görme potansiyeline sahip tamamen yeni sistemlerdir. Bu makalede Galerkin tipi yaklaşımların etkin kullanımından ve belirsizlik veya gürültü terimi içeren kesirli dereceli sistemlere uygulanabilirliği araştırılmıştır.

References

  • Baleanu, D., Diethelm, K., Scalas, E., & Trujillo, J.J. (2012). Fractional Calculus: Models and Numerical Methods, World Scientific Publishing.
  • Özüpak, Y. (2023). Design and Analysis of Permanent Magnet DC Machines with FEM Based ANSYS-Maxwell, International Journal of Innovative Engineering Applications, 7(1), 7-12.
  • Cavlak Aslan, E., & Gürgöze, L. (2022). Soliton and Other Function Solutions of The Potential KdV Equation with Jacobi Elliptic Function Method, International Journal of Innovative Engineering Applications, 6(2), 183-188.
  • Dung, N.T. (2013). Fractional Stochastic Differential Equations with Applications to Finance, Journal of Mathematical Analysis and Applications, 397(1), 334-348.
  • Rihan, A.F., Rajivganthi, C., & Muthukumar, P. (2017). Fractional Stochastic Differential Equations with Hilfer Fractional Derivative: Poisson Jumps and Optimal Control, Discrete Dynamics in Nature and Society, Article ID 5394528.
  • Klebaner, F. C., & Liptser, R. (2001). Asymptotic analysis and extinction in a stochastic Lotka-Volterra model, The Annals of Applied Probability, 11(4), 1263-1291.
  • Kinoshita, S. (2013). 1-Introduction to Nonequilibrium Phenomena Pattern Formations and Oscillatory Phenomena, 1-59.
  • Zakharov, V. E. (1981). On the Benney equations, Physica D: Nonlinear Phenomena, 3(1-2), 193-202.
  • Pellegrini, G. (2014). Polynomial Chaos Expansion with applications to PDEs (Master dissertation, University of Verona).
  • Nualart, D. (2006). Fractional Brownian motion. The Malliavin Calculus and Related Topics. Probability, its applications. Springer, Berlin, Heidelberg.
  • Mahmud, A.A., Muhamad, K.A., Tanriverdi, T., & Baskonus, H.M. (2024). An investigation of Fokas system using two new modifications for the trigonometric and hyperbolic trigonometric function methods, Optical and Quantum Electronics, 56:717.
  • Mahmud, A. A., Tanriverdi, T., Muhamad, K. A., & Baskonus, H. M. (2023). Characteristic of ion-acoustic waves described in the solutions of the (3+1)-dimensional generalized Korteweg-de Vries-Zakharov-Kuznetsov equation, Journal of Applied Mathematics and Computational Mechanics, 22, 2, 36-48.
  • Mahmud, A. A., Baskonus, H. M., Tanriverdi, T., & Muhamad, K. A. (2023). Optical Solitary Waves and Soliton Solutions of the (3+1)-Dimensional Generalized Kadomtsev–Petviashvili–Benjamin–Bona–Mahony Equation, Partial Differential Equations, 63, 1085-1102.
  • Baskonus, H. M., Mahmud, A. A., Muhamad, K. A., & Tanriverdi, T. (2022). A study on Caudrey–Dodd–Gibbon–Sawada–Kotera partial differential equation, Mathematical Methods in the Applied Sciences.
  • Tanriverdi, T., Baskonus, H. M., Mahmud, A. A., & Muhamad, K. A. (2021). Explicit solution of fractional order atmosphere-soil-land plant carbon cycle system, Ecological Complexity, 48, 100966.

NUMERICAL SOLUTIONS TO THE STOCHASTIC SYSTEMS WITH FRACTIONAL OPERATORS

Year 2024, , 43 - 52, 29.06.2024
https://doi.org/10.46460/ijiea.1459659

Abstract

Fractional-stochastic differential equations are widely used tools to simulate a wide - range of engineering and scientific phenomena. In this paper, the applicability of the approach of indeterminate coefficients to various fractional-stochastic models is examined. These models have a fractional white noise term and are mostly produced by fractional-order derivative operators. We also investigate applications of a polynomial chaos algorithm to stochastic Lotka-Volterra and Benney systems. Fractional-stochastic equations are entirely novel systems that have the potential to function as models for a wide range of scientific and engineering phenomena. It is noted that fractional-order systems with uncertainty or a noise term can benefit from the effective use of Galerkin-type approaches in this article.

Ethical Statement

Ethics committee approval document is not required for this study.

References

  • Baleanu, D., Diethelm, K., Scalas, E., & Trujillo, J.J. (2012). Fractional Calculus: Models and Numerical Methods, World Scientific Publishing.
  • Özüpak, Y. (2023). Design and Analysis of Permanent Magnet DC Machines with FEM Based ANSYS-Maxwell, International Journal of Innovative Engineering Applications, 7(1), 7-12.
  • Cavlak Aslan, E., & Gürgöze, L. (2022). Soliton and Other Function Solutions of The Potential KdV Equation with Jacobi Elliptic Function Method, International Journal of Innovative Engineering Applications, 6(2), 183-188.
  • Dung, N.T. (2013). Fractional Stochastic Differential Equations with Applications to Finance, Journal of Mathematical Analysis and Applications, 397(1), 334-348.
  • Rihan, A.F., Rajivganthi, C., & Muthukumar, P. (2017). Fractional Stochastic Differential Equations with Hilfer Fractional Derivative: Poisson Jumps and Optimal Control, Discrete Dynamics in Nature and Society, Article ID 5394528.
  • Klebaner, F. C., & Liptser, R. (2001). Asymptotic analysis and extinction in a stochastic Lotka-Volterra model, The Annals of Applied Probability, 11(4), 1263-1291.
  • Kinoshita, S. (2013). 1-Introduction to Nonequilibrium Phenomena Pattern Formations and Oscillatory Phenomena, 1-59.
  • Zakharov, V. E. (1981). On the Benney equations, Physica D: Nonlinear Phenomena, 3(1-2), 193-202.
  • Pellegrini, G. (2014). Polynomial Chaos Expansion with applications to PDEs (Master dissertation, University of Verona).
  • Nualart, D. (2006). Fractional Brownian motion. The Malliavin Calculus and Related Topics. Probability, its applications. Springer, Berlin, Heidelberg.
  • Mahmud, A.A., Muhamad, K.A., Tanriverdi, T., & Baskonus, H.M. (2024). An investigation of Fokas system using two new modifications for the trigonometric and hyperbolic trigonometric function methods, Optical and Quantum Electronics, 56:717.
  • Mahmud, A. A., Tanriverdi, T., Muhamad, K. A., & Baskonus, H. M. (2023). Characteristic of ion-acoustic waves described in the solutions of the (3+1)-dimensional generalized Korteweg-de Vries-Zakharov-Kuznetsov equation, Journal of Applied Mathematics and Computational Mechanics, 22, 2, 36-48.
  • Mahmud, A. A., Baskonus, H. M., Tanriverdi, T., & Muhamad, K. A. (2023). Optical Solitary Waves and Soliton Solutions of the (3+1)-Dimensional Generalized Kadomtsev–Petviashvili–Benjamin–Bona–Mahony Equation, Partial Differential Equations, 63, 1085-1102.
  • Baskonus, H. M., Mahmud, A. A., Muhamad, K. A., & Tanriverdi, T. (2022). A study on Caudrey–Dodd–Gibbon–Sawada–Kotera partial differential equation, Mathematical Methods in the Applied Sciences.
  • Tanriverdi, T., Baskonus, H. M., Mahmud, A. A., & Muhamad, K. A. (2021). Explicit solution of fractional order atmosphere-soil-land plant carbon cycle system, Ecological Complexity, 48, 100966.
There are 15 citations in total.

Details

Primary Language English
Subjects Numerical Methods in Mechanical Engineering
Journal Section Articles
Authors

Mehmet Ali Akınlar 0000-0002-7005-8633

Early Pub Date June 29, 2024
Publication Date June 29, 2024
Submission Date March 27, 2024
Acceptance Date May 20, 2024
Published in Issue Year 2024

Cite

APA Akınlar, M. A. (2024). NUMERICAL SOLUTIONS TO THE STOCHASTIC SYSTEMS WITH FRACTIONAL OPERATORS. International Journal of Innovative Engineering Applications, 8(1), 43-52. https://doi.org/10.46460/ijiea.1459659
AMA Akınlar MA. NUMERICAL SOLUTIONS TO THE STOCHASTIC SYSTEMS WITH FRACTIONAL OPERATORS. ijiea, IJIEA. June 2024;8(1):43-52. doi:10.46460/ijiea.1459659
Chicago Akınlar, Mehmet Ali. “NUMERICAL SOLUTIONS TO THE STOCHASTIC SYSTEMS WITH FRACTIONAL OPERATORS”. International Journal of Innovative Engineering Applications 8, no. 1 (June 2024): 43-52. https://doi.org/10.46460/ijiea.1459659.
EndNote Akınlar MA (June 1, 2024) NUMERICAL SOLUTIONS TO THE STOCHASTIC SYSTEMS WITH FRACTIONAL OPERATORS. International Journal of Innovative Engineering Applications 8 1 43–52.
IEEE M. A. Akınlar, “NUMERICAL SOLUTIONS TO THE STOCHASTIC SYSTEMS WITH FRACTIONAL OPERATORS”, ijiea, IJIEA, vol. 8, no. 1, pp. 43–52, 2024, doi: 10.46460/ijiea.1459659.
ISNAD Akınlar, Mehmet Ali. “NUMERICAL SOLUTIONS TO THE STOCHASTIC SYSTEMS WITH FRACTIONAL OPERATORS”. International Journal of Innovative Engineering Applications 8/1 (June 2024), 43-52. https://doi.org/10.46460/ijiea.1459659.
JAMA Akınlar MA. NUMERICAL SOLUTIONS TO THE STOCHASTIC SYSTEMS WITH FRACTIONAL OPERATORS. ijiea, IJIEA. 2024;8:43–52.
MLA Akınlar, Mehmet Ali. “NUMERICAL SOLUTIONS TO THE STOCHASTIC SYSTEMS WITH FRACTIONAL OPERATORS”. International Journal of Innovative Engineering Applications, vol. 8, no. 1, 2024, pp. 43-52, doi:10.46460/ijiea.1459659.
Vancouver Akınlar MA. NUMERICAL SOLUTIONS TO THE STOCHASTIC SYSTEMS WITH FRACTIONAL OPERATORS. ijiea, IJIEA. 2024;8(1):43-52.