Dirichlet Sınır Koşullu Diferansiyel Denklemlerin Bir Sınıfının Geometrik Ortalamalar Optimizasyonu Çözümleri

Yıl 2024, Cilt: 1 Sayı: 1, 19 - 29, 06.07.2024

İclal Gör Türkan Özyörük

Öz

Bu çalışma, matematiksel modellemede yaygın olarak karşılaşılan Dirichlet sınır koşullarını sağlayan diferansiyel denklemlerin çözümüne odaklanmaktadır. Bu denklemleri çözmek için Geometrik Ortalama Optimizasyonu (GMO) algoritması kullanılmıştır. Bu amaçla öncelikle ileri beslemeli yapay sinir ağı ile sınır koşullarının sağlanması için ön eğitimden geçirilmiş ve maliyet fonksiyonu minimize edilerek yapay sinir ağına ait ağırlık güncelleme işlemi gerçekleştirilmiştir. GMO algoritmasının kullanımı, geleneksel çözüm yöntemlerine bir alternatif olarak incelenmiş ve deneysel çalışmalarda benzer performans sağlamıştır. Bu amaçla, Dirichlet koşullarına sahip diferansiyel denklemlerin çözümü araştırılmıştır. Elde edilen sonuçlar, GMO algoritmasının Dirichlet koşullu diferansiyel denklemlerin çözümü için bir alternatif sunduğunu önermektedir.

Anahtar Kelimeler

• Küresel optimizasyon, meta-sezgisel algoritma, geometrik ortalama optimizasyonu, dirichlet sınır koşulllu diferansiyel denklem

Teşekkür

Çalışmaya değerli katkılarından ötürü Doç. Dr. Korhan GÜNEL'e teşekkür ederiz.

Kaynakça

• Ahmad, J. ve Rauf, H. T. (2021). Comparison of Different Bat Initialization Techniques for Global Optimization Problems. International Journal of Applied Metaheuristic Computing, 12-1. doi: 10.4018/IJAMC.2021010109
• Alomoush A. A., Alsewari. A. A., Alamri, H.S., Zamli, K. Z., Alomoush, W. ve Younis. M. I. (2019). Modified Opposition Based Learning to Improve Harmony Search Variants Exploration. In: Saeed, F., Mohammed, F., Gazem, N. (eds) Emerging Trends in Intelligent Computing and Informatics. IRICT 2019. Advances in Intelligent Systems and Computing, 1073. Springer, Cham. doi: 10.1007/978-3-030-33582-3_27
• Chen, J., He, M. ve Huang, Y. (2020). A fast multiscale Galerkin method for solving second order linear Fredholm integro-differential equation with Dirichlet boundary conditions. Journal of Computational and Applied Mathematics, 364, 112352. doi: 10.1016/j.cam.2019.112352
• Chen, D., Liu, J., Yao, C., Zhang, Z. ve Du, X. (2022). Multi-strategy improved salp swarm algorithm and its application in reliability optimization, Mathematical Biosciences and Engineering. 19(5): 5269–5292. doi: 10.3934/mbe.2022247
• Cheng, C. ve Zhang, G.T. (2021). Deep Learning Method Based on Physics Informed Neural Network with Resnet Block for Solving Fluid Flow Problems. Water, 13, 423. doi: 10.3390/w13040423
• Deng, X., He, D. ve Qu, L.. (2024). A Multi-strategy Enhanced Arithmetic Optimization Algorithm and Its Application in Path Planning of Mobile Robots. Neural Processing Letters, 56:18. doi: 10.1007/s11063-024-11467-6
• Fang, J., Liu, C., Simos, T. E., Famelis, I. T. (2020). Neural Network Solution of Single-Delay Differential Equations. Mediterranean Journal of Mathematics, 5-15. doi: 10.1007/s00009-019-1452-5
• Gör, I. (2020). Diferansiyel Denklemlerin Yapay Sinir Ağları ile Nümerik Çözümleri. Aydın Adnan Menderes Üniversitesi, Aydın.
• Graef, J. R., Heidarkkhani, S., Kong, L. (2018), Existence of Solutions to An Impulsive Dirichlet Boundary Value Problem, Fixed Point Theory, 19, 1: 225-234. doi: 10.24193/fpt-ro.2018.1.18
• Günel, K., Gör, İ. ve Tekeli, K. (2020). ICA-RD: The Regional Domination Policy for Imperialist Competitive Algorithm from Imperialism to Internationalism. Arab Journal for Science and Engineering, 45, 10529–10589. doi: 10.1007/s13369-020-04787-x
• Günel, K., Isman, G. ve Kocakula, M. (2018). Simple recurrent neural networks for the numerical solutions of ODEs with Dirichlet boundary conditions, J. BAUN Inst. Sci. Technol., 20(3) Special Issue, 143-153. doi: 10.25092/baunfbed.483922
• Günel, K. ve Gör, I. (2022). Solving Dirichlet boundary problems for ODEs via swarm intelligence. Mathematical Sciences. 16:325–341. doi: 10.1007/s40096-021-00424-2
• Huang, Z. S. ve Li, W.-L. (2020). Novel Multi-Strategy Enhanced Whale Optimization Algorithm. 2nd IEEE Eurasia Conference on IOT, Communication and Engineering 2020. ISBN: 978-1-7281-8060-1. doi: 10.1109/ECICE50847.2020.9301990
• Khan, I., Raja, M. A. Z., Shoaib, M., Kumam, P., Alrabaiah, H., Shah, Z. ve Islam, S. (2020). Design of Neural Network with Levenberg-Marquardt and Bayesian Regularization Backpropagation for Solving Pantograph Delay Differential Equations. IEEE Access. 8: 137918-137933 doi: 10.1109/ACCESS.2020.3011820
• Kowalski, P. (2013). Dirichlet boundary value problem for Duffing’s equation. Electronic Journal of Qualitative Theory of Differential Equations. 37, 1-10. doi: 10.14232/ejqtde.2013.1.37
• Li, K., Fu, X., Wang, F. ve Jalil, H. (2021). Survey of Lévy Flight-Based Metaheuristics for Optimization. Mathematics. 10(15), 2785. doi: 10.3390/math10152785
• Li, K., An, Q., Deng, Q. ve Wang, G-G. (2022). A dynamic population reduction differential evolution algorithm combining linear and nonlinear strategy piecewise functions. Concurrency Computat Pract Exper. 34, 6773. doi: 10.1002/cpe.6773
• Liu, M. Peng, W., Hou, M. ve Tian, Z. (2023). Radial basis function neural network with extreme learning machine algorithm for solving ordinary differential equations. Soft Computing, 27:3955–3964. doi: 10.1007/s00500-022-07529-3
• Liu, X. ve Qin, X. (2020). A probability-based core dandelion guided dandelion algorithm and application to traffic flow prediction. Engineering Applications of Artificial Intelligence, 96: 103922. doi: 10.1016/j.engappai.2020.103922
• Malek, A. ve Beidokhti, R. S. (2006). Numerical solution for high order differential equations using a hybrid neural network Optimization method. Applied Mathematics and Computation. 183, 260-271. doi: 10.1016/j.amc.2006.05.068
• Navarro, M. A., Oliva, D. Ramos‐Michel, A., Morales‐Castañeda, B., Zaldívar, D. ve Luque−Chang, A. (2022). A Review of the Use of Quasi‐random Number Generators to Initialize the Population in Meta‐heuristic Algorithms. Archives of Computational Methods in Engineering. 29:5149–5184. doi: 10.1007/s11831-022-09759-y
• Panghal, S. ve Kumar, M. (2022). Neural network method: delay and system of delay differential equations. Engineering with Computers 38 (Suppl 3):S2423–S2432. doi: 10.1007/s00366-021-01373-z
• Rezaei, F., Safavi, H. R., Elaziz, M. A. ve Mirjalili, S. (2023). GMO: geometric mean optimizer for solving engineering problems. Soft Computing. 27(15):10571–10606. doi: 10.1007/s00500-023-08202-z
• Sabir, Z., Wahab, H. A., Umar, M., Sakar, M. G. ve Raja, M. A. Z. (2020). Novel design of Morlet wavelet neural network for solving second order Lane–Emden equation. Mathematics and Computers in Simulation, 172, 1–14. doi: 10.1016/j.matcom.2020.01.005
• Sabir, Z., Umar, M., Guirao, J. L. G., Shoaib, M. ve Raja, M. A. Z (2021). Integrated intelligent computing paradigm for nonlinear multi- singular third-order Emden–Fowler equation. Neural Computing and Applications, 33:3417–3436. doi: 10.1007/s00521-020-05187-w
• Sheng, Y. Zhang, H. ve Zeng, Z. (2018). Stability and Robust Stability of Stochastic Reaction–Diffusion Neural Networks With Infinite Discrete and Distributed Delays. IEEE Transactions On Systems, Man, and Cybernetics: Systems, 50(5): 1721-1732. doi: 10.1109/TSMC.2017.2783905
• Sirsant, S.ve Reddy, M. J. (2022). Improved MOSADE algorithm incorporating Sobol sequences for multi-objective design of Water Distribution Networks. Applied Soft Computing. 120. 108682. doi: 10.1016/j.asoc.2022.108682
• Sousa, J.V.d.C., Nyamoradi, N. ve Lamine, M. (2022). Nehari manifold and fractional Dirichlet boundary value problem. Anal.Math.Phys. 12, 143. doi: 10.1007/s13324-022-00754-x
• Wu, M., Zhang, J., Huang, Z., Li, X. ve Dong, Y. (2021). Numerical solutions of wavelet neural networks for fractional differential equations. Math Meth Appl Sci. 2023;46:3031–3044. doi: 10.1002/mma.7449
• Zhang, Z., Cai, Y. ve Zhang, D. (2020). Solving Ordinary Differential Equations with Adaptive Differential Evolution. IEEE Access, 8: 128908 - 128922. doi: 10.1109/ACCESS.2020.3008823.
• Zadeh, L. A. (1965). Fuzzy sets. Information and control. 8(3):338–353. doi: 10.1016/S0019-9958(65)90241-X