Approximate Solutions of Hyperbolic Telegraph Equations Arising in Circuit Theory

Abstract

In this study, the hyperbolic telegraph differential equation describing the behavior of certain wave-like phenomena, such as the transmission of signals along a telegraph line, is considered. Numerical solutions of a telegraph equation are computed by using various operator-difference schemes and physic-informed neural networks. The error analysis is performed, and the results are compared.

Keywords

• Hyperbolic Equations
• Numerical Solution
• Error Analysis
• Difference Scheme
• Neural Network

Devre Teorisinde Ortaya Çıkan Hiperbolik Telgraf Denklemlerinin Yaklaşık Çözümleri

Abstract

Bu çalışmada, telgraf hattı boyunca sinyallerin iletimi gibi belirli dalga benzeri olayların davranışını tanımlayan hiperbolik telgraf diferansiyel denklemi ele alınmıştır. Bir telgraf denkleminin sayısal çözümleri çeşitli operatör farkı şemaları ve fizik bilgili sinir ağları kullanılarak hesaplanmıştır. Hata analizi yapılmış ve sonuçlar karşılaştırılmıştır.

Keywords

• Hiperbolik Denklemler
• Sayısal Çözüm
• Hata Analizi
• Fark Şeması
• Sinir Ağı

References

1. [1] A. Ashyralyev, M. E. Koksal, A Numerical Solution of Wave Equation Arising in Non-Homogeneous Cylindrical Shells, Turkish Journal of Mathematics, 32 (4) 407-419, 2008
2. [2] M. E. Koksal, An Operator-Difference Method for Telegraph Equations Arising in Transmission Lines, Discrete Dynamics in Nature and Society, 1-17, 2011.
3. [3] M. E. Koksal, Time and frequency responses of non-integer order RLC circuits, AIMS Mathematics, 4 (1) 61-75, 2019.
4. [4] G. S. Krein, Linear Differential Equations in a Banach Space, Birkhauser, 1966.
5. [5] O. H. Fattorini, Second Order Linear Differential Equations in Banach Space, Mathematics Studies, 107, 1985.
6. [6] A. Ashyralyev, M. Modanli, An operator method for telegraph partial differential and difference equations, Boundary Value Problems, Artical ID;41, 1-17, 2015.
7. [7] A. Ashyralyev, M. E. Koksal, K. T. Turkcan. Numerical solutions of telegraph equations with the dirichlet boundary condition, International Conference on Analysis and Applied Mathematics, 1759;1, 1-6, 2016.
8. [8] J. Schmidhuber, ‘‘Deep learning in neural networks: An overview’’ Neural Networks, 61, 85–117, 2015.

9. [9] Y. LeCun, Y. Bengio & G. Hinton, Deep learning, nature, 521(7553), 436-444, 2015.
10. [10] V. J. da Cunha Farias, M. B. Siqueira, Physics-Informed Machine Learning for Numerical Solution of Hyperbolic Partial Differential Equations: An Application to the Second Order One Dimensional Linear and Nonlinear Telegraph Equation, https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4778353, 1-15, 2024.
11. [11] M. Raissi, P. Perdikaris, & G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations Journal of Computational Physics, 378, 686–707, 2019.