Browsing by Author "Algan, Ali"

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Bir boyutlu dalga denkleminin yapay sinir ağları ile çözümü
(2026) Altınöz, Nuray; Algan, Ali; Düşünceli, Faruk
Artificial neural networks (ANNs) have become an effective method widely used in recent years for solving complex problems due to their multi-layered structures. In this study, an effective approach has been developed to obtain the numerical solution of the one-dimensional wave equation using ANNs. Before proceeding to the numerical solution, analytical solutions for three different forms of the one-dimensional wave equation were considered and used for comparison. Numerical solutions of wave equations are more complex than analytical solutions and require high computational costs. The ANN method, however, enables numerical solutions to be obtained in a shorter time and with low error rates, independently of these processes. Furthermore, the ANN approach also allows for the visualisation of solutions. In the ANN method, activation functions are used to increase the learning ability of the model and to systematise the training process. The original aspect of this study is the choice of Fibonacci polynomials as the activation function. During the training of the network, the gradient descent algorithm was used, and the optimisation process was supported by the Hessian matrix to achieve the best solution. The continuous and complex structure of Fibonacci polynomials makes them suitable for use as activation functions. Using this method, the numerical solution of the one-dimensional wave equation with specific initial and boundary conditions has been successfully obtained. It has been observed that the numerical results obtained converge to the analytical solutions with very low error margins. Consequently, it has been demonstrated that Fibonacci polynomials can serve as an effective alternative activation function in the solution of partial differential equations using artificial neural networks.