Abstract
Recently, the development of neural network method for solving differential equations has made a remarkable progress for solving fractional differential equations. In this paper, a neural network method is employed to solve time-fractional telegraph equation. The loss function containing initial/boundary conditions with adjustable parameters (weights and biases) is constructed. Also, in this paper, a time-fractional telegraph equation was formulated as an optimization problem. Numerical examples with known analytic solutions including numerical results, their graphs, weights, and biases were also discussed to confirm the accuracy of the method used. Also, the graphical and tabular results were analyzed thoroughly. The mean square errors for different choices of neurons and epochs have been presented in tables along with graphical presentations.
Highlights
Fractional differential equations can be used to model many real-life problems
The study conducted in [4, 5] has emphasized on the property of the solution of fractional differential equations like its stability and existence
To the researchers’ knowledge, there has been a little study on solving fractional partial differential equations with the neural network approach
Summary
Fractional differential equations can be used to model many real-life problems. Recently, fractional partial differential equations have received much attention of the researchers due to their wide applications in the area of biological sciences and medicine [1,2,3]. Zhang and Meerschaert and Tadjeran [17, 18] employed a finite-difference approach for a solution of fractional partial differential equation. Solving fractional differential equations by the neural network method has become an active research area. A neural network is a type of machine learning algorithm which has amazing ability to solve large-scale problems It is based on the idea of minimizing the loss function that best approximates the solution to mathematical problems. This study focuses on solving fractional telegraph equation with the neural network method. Is paper extends this idea to solving time-fractional telegraph equation. To the researchers’ knowledge, there has been a little study on solving fractional partial differential equations with the neural network approach. E main contribution of this paper is to discuss the artificial neural network algorithm for solving time-fractional telegraph equations.
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