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Investigation of Solutions of 𝜷 −conformable Fractional Ordinary Differential Equation With Artificial Neural Network

Yıl 2023, Cilt: 13 Sayı: 2, 1266 - 1274, 01.06.2023
https://doi.org/10.21597/jist.1230287

Öz

İn this study, we present a method in order to get initial value fractional differential equations with artificial neural networks. On the basis of the function approach of feedforward neural networks, this method is a general method that is written in an implicit analytical form and results in the creation of a differentiable solution. The first part of the created trial solution which is stated as the sum of the two parts, with no controllable parameters, gives the initial conditions. The second part, unaffected by the initial conditions, consists of a feedforward neural network with controllable parameters (weights). The applicability of this approach is demonstrated in systems of both fractional single ODEs and fractional coupled ODEs.

Kaynakça

  • Aderyani, S. R., Saadati, R., Vahidi, J., & Allahviranloo, T. (2022). The exact solutions of the conformable time-fractional modified nonlinear Schrödinger equation by the Trial equation method and modified Trial equation method. Advances in Mathematical Physics, 2022.
  • Ain, Q. T., Nadeem, M., Karim, S., Akgül, A., & Jarad, F. (2022). Optimal variational iteration method for parametric boundary value problem. AIMS Mathematics, 7(9), 16649-16656.
  • Akinyemi, L., Mirzazadeh, M., Amin Badri, S., & Hosseini, K. (2022). Dynamical solitons for the perturbated Biswas–Milovic equation with Kudryashov's law of refractive index using the first integral method. Journal of Modern Optics, 69(3), 172-182.
  • Akram, G., Sadaf, M., & Zainab, I. (2022). The dynamical study of Biswas–Arshed equation via modified auxiliary equation method. Optik, 255, 168614.
  • Atangana, A. (2015). Derivative with a new parameter: Theory, methods and applications. Academic Press.
  • Chen, T. Q., Rubanova, Y., Bettencourt, J., & Duvenaud, D. (2018). Neural ordinary differential equations, in ‘Advances in neural information processing systems. La Jolla.
  • Dufera, T. T. (2021). Deep neural network for system of ordinary differential equations: Vectorized algorithm and simulation. Machine Learning with Applications, 5, 100058.
  • Esen, A., Ucar, Y., Yagmurlu, N., & Tasbozan, O. (2013). A Galerkin finite element method to solve fractional diffusion and fractional diffusion-wave equations. Mathematical Modelling and Analysis, 18(2), 260-273.
  • Gao, Y., Liu, H., Wang, X., & Zhang, K. (2022). On an artificial neural network for inverse scattering problems. Journal of Computational Physics, 448, 110771.
  • Jafarian, A., Mokhtarpour, M., & Baleanu, D. (2017). Artificial neural network approach for a class of fractional ordinary differential equation. Neural Computing and Applications, 28(4), 765-773.
  • Karatas Akgül, E., & Akgül, A. (2022). New applications of Sumudu transform method with different fractional derivatives. International Journal of Applied and Computational Mathematics, 8(5), 1-12.
  • Kocak, Z. F., Bulut, H., & Yel, G. (2014). The solution of fractional wave equation by using modified trial equation method and homotopy analysis method. In AIP Conference Proceedings, 1637(1), 504-512.
  • Kumar, M. (2022). Recent development of Adomian decomposition method for ordinary and partial differential equations. International Journal of Applied and Computational Mathematics, 8(2), 1-25.
  • Lee, H., & Kang, I. S. (1990). Neural algorithm for solving differential equations. Journal of Computational Physics, 91(1), 110-131.
  • Liu, Z., Yang, Y., & Cai, Q. (2019). Neural network as a function approximator and its application in solving differential equations. Applied Mathematics and Mechanics, 40(2), 237-248.
  • Omidi, M., Arab, B., Rasanan, A. H., Rad, J. A., & Parand, K. (2022). Learning nonlinear dynamics with behavior ordinary/partial/system of the differential equations: looking through the lens of orthogonal neural networks. Engineering with Computers, 38(2), 1635-1654.
  • Shakeel, M., El-Zahar, E. R., Shah, N. A., & Chung, J. D. (2022). Generalized Exp-Function Method to Find Closed Form Solutions of Nonlinear Dispersive Modified Benjamin–Bona–Mahony Equation Defined by Seismic Sea Waves. Mathematics, 10(7), 1026.
  • Tang, Y., Ma, J., Zhou, B., & Zhou, J. (2021). From 2Mth-order wronskian determinant solutions to Mth-order lump solutions for the (2+1)-Dimensional Kadomtsev–Petviashvili I equation. Wave Motion, 104, 102746.
  • Tian, Y., & Liu, J. (2021). A modified exp-function method for fractional partial differential equations. Thermal Science, 25(2 Part B), 1237-1241.
  • Yang, D. Y., Tian, B., Hu, C. C., & Zhou, T. Y. (2022). The generalized Darboux transformation and higher-order rogue waves for a coupled nonlinear Schrödinger system with the four-wave mixing terms in a birefringent fiber. The European Physical Journal Plus, 137(11), 1-11.
  • Yang, Y., Hou, M., & Luo, J. (2018). A novel improved extreme learning machine algorithm in solving ordinary differential equations by Legendre neural network methods. Advances in Difference Equations, 2018(1), 1-24.
  • Zhan, R., Chen, W., Chen, X., & Zhang, R. (2022). Exponential Multistep Methods for Stiff Delay Differential Equations. Axioms, 11(5), 185. .
Yıl 2023, Cilt: 13 Sayı: 2, 1266 - 1274, 01.06.2023
https://doi.org/10.21597/jist.1230287

Öz

Kaynakça

  • Aderyani, S. R., Saadati, R., Vahidi, J., & Allahviranloo, T. (2022). The exact solutions of the conformable time-fractional modified nonlinear Schrödinger equation by the Trial equation method and modified Trial equation method. Advances in Mathematical Physics, 2022.
  • Ain, Q. T., Nadeem, M., Karim, S., Akgül, A., & Jarad, F. (2022). Optimal variational iteration method for parametric boundary value problem. AIMS Mathematics, 7(9), 16649-16656.
  • Akinyemi, L., Mirzazadeh, M., Amin Badri, S., & Hosseini, K. (2022). Dynamical solitons for the perturbated Biswas–Milovic equation with Kudryashov's law of refractive index using the first integral method. Journal of Modern Optics, 69(3), 172-182.
  • Akram, G., Sadaf, M., & Zainab, I. (2022). The dynamical study of Biswas–Arshed equation via modified auxiliary equation method. Optik, 255, 168614.
  • Atangana, A. (2015). Derivative with a new parameter: Theory, methods and applications. Academic Press.
  • Chen, T. Q., Rubanova, Y., Bettencourt, J., & Duvenaud, D. (2018). Neural ordinary differential equations, in ‘Advances in neural information processing systems. La Jolla.
  • Dufera, T. T. (2021). Deep neural network for system of ordinary differential equations: Vectorized algorithm and simulation. Machine Learning with Applications, 5, 100058.
  • Esen, A., Ucar, Y., Yagmurlu, N., & Tasbozan, O. (2013). A Galerkin finite element method to solve fractional diffusion and fractional diffusion-wave equations. Mathematical Modelling and Analysis, 18(2), 260-273.
  • Gao, Y., Liu, H., Wang, X., & Zhang, K. (2022). On an artificial neural network for inverse scattering problems. Journal of Computational Physics, 448, 110771.
  • Jafarian, A., Mokhtarpour, M., & Baleanu, D. (2017). Artificial neural network approach for a class of fractional ordinary differential equation. Neural Computing and Applications, 28(4), 765-773.
  • Karatas Akgül, E., & Akgül, A. (2022). New applications of Sumudu transform method with different fractional derivatives. International Journal of Applied and Computational Mathematics, 8(5), 1-12.
  • Kocak, Z. F., Bulut, H., & Yel, G. (2014). The solution of fractional wave equation by using modified trial equation method and homotopy analysis method. In AIP Conference Proceedings, 1637(1), 504-512.
  • Kumar, M. (2022). Recent development of Adomian decomposition method for ordinary and partial differential equations. International Journal of Applied and Computational Mathematics, 8(2), 1-25.
  • Lee, H., & Kang, I. S. (1990). Neural algorithm for solving differential equations. Journal of Computational Physics, 91(1), 110-131.
  • Liu, Z., Yang, Y., & Cai, Q. (2019). Neural network as a function approximator and its application in solving differential equations. Applied Mathematics and Mechanics, 40(2), 237-248.
  • Omidi, M., Arab, B., Rasanan, A. H., Rad, J. A., & Parand, K. (2022). Learning nonlinear dynamics with behavior ordinary/partial/system of the differential equations: looking through the lens of orthogonal neural networks. Engineering with Computers, 38(2), 1635-1654.
  • Shakeel, M., El-Zahar, E. R., Shah, N. A., & Chung, J. D. (2022). Generalized Exp-Function Method to Find Closed Form Solutions of Nonlinear Dispersive Modified Benjamin–Bona–Mahony Equation Defined by Seismic Sea Waves. Mathematics, 10(7), 1026.
  • Tang, Y., Ma, J., Zhou, B., & Zhou, J. (2021). From 2Mth-order wronskian determinant solutions to Mth-order lump solutions for the (2+1)-Dimensional Kadomtsev–Petviashvili I equation. Wave Motion, 104, 102746.
  • Tian, Y., & Liu, J. (2021). A modified exp-function method for fractional partial differential equations. Thermal Science, 25(2 Part B), 1237-1241.
  • Yang, D. Y., Tian, B., Hu, C. C., & Zhou, T. Y. (2022). The generalized Darboux transformation and higher-order rogue waves for a coupled nonlinear Schrödinger system with the four-wave mixing terms in a birefringent fiber. The European Physical Journal Plus, 137(11), 1-11.
  • Yang, Y., Hou, M., & Luo, J. (2018). A novel improved extreme learning machine algorithm in solving ordinary differential equations by Legendre neural network methods. Advances in Difference Equations, 2018(1), 1-24.
  • Zhan, R., Chen, W., Chen, X., & Zhang, R. (2022). Exponential Multistep Methods for Stiff Delay Differential Equations. Axioms, 11(5), 185. .
Toplam 22 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Matematik / Mathematics
Yazarlar

Sadullah Bulut 0000-0001-5026-6534

Muhammed Yiğider 0000-0003-4255-5760

Erken Görünüm Tarihi 27 Mayıs 2023
Yayımlanma Tarihi 1 Haziran 2023
Gönderilme Tarihi 6 Ocak 2023
Kabul Tarihi 23 Şubat 2023
Yayımlandığı Sayı Yıl 2023 Cilt: 13 Sayı: 2

Kaynak Göster

APA Bulut, S., & Yiğider, M. (2023). Investigation of Solutions of 𝜷 −conformable Fractional Ordinary Differential Equation With Artificial Neural Network. Journal of the Institute of Science and Technology, 13(2), 1266-1274. https://doi.org/10.21597/jist.1230287
AMA Bulut S, Yiğider M. Investigation of Solutions of 𝜷 −conformable Fractional Ordinary Differential Equation With Artificial Neural Network. Iğdır Üniv. Fen Bil Enst. Der. Haziran 2023;13(2):1266-1274. doi:10.21597/jist.1230287
Chicago Bulut, Sadullah, ve Muhammed Yiğider. “Investigation of Solutions of 𝜷 −conformable Fractional Ordinary Differential Equation With Artificial Neural Network”. Journal of the Institute of Science and Technology 13, sy. 2 (Haziran 2023): 1266-74. https://doi.org/10.21597/jist.1230287.
EndNote Bulut S, Yiğider M (01 Haziran 2023) Investigation of Solutions of 𝜷 −conformable Fractional Ordinary Differential Equation With Artificial Neural Network. Journal of the Institute of Science and Technology 13 2 1266–1274.
IEEE S. Bulut ve M. Yiğider, “Investigation of Solutions of 𝜷 −conformable Fractional Ordinary Differential Equation With Artificial Neural Network”, Iğdır Üniv. Fen Bil Enst. Der., c. 13, sy. 2, ss. 1266–1274, 2023, doi: 10.21597/jist.1230287.
ISNAD Bulut, Sadullah - Yiğider, Muhammed. “Investigation of Solutions of 𝜷 −conformable Fractional Ordinary Differential Equation With Artificial Neural Network”. Journal of the Institute of Science and Technology 13/2 (Haziran 2023), 1266-1274. https://doi.org/10.21597/jist.1230287.
JAMA Bulut S, Yiğider M. Investigation of Solutions of 𝜷 −conformable Fractional Ordinary Differential Equation With Artificial Neural Network. Iğdır Üniv. Fen Bil Enst. Der. 2023;13:1266–1274.
MLA Bulut, Sadullah ve Muhammed Yiğider. “Investigation of Solutions of 𝜷 −conformable Fractional Ordinary Differential Equation With Artificial Neural Network”. Journal of the Institute of Science and Technology, c. 13, sy. 2, 2023, ss. 1266-74, doi:10.21597/jist.1230287.
Vancouver Bulut S, Yiğider M. Investigation of Solutions of 𝜷 −conformable Fractional Ordinary Differential Equation With Artificial Neural Network. Iğdır Üniv. Fen Bil Enst. Der. 2023;13(2):1266-74.