Abstract
In this paper, numerical methods for solving fractional differential equations by using a triangle neural network are proposed. The fractional derivative is considered Caputo type. The fractional derivative of the triangle neural network is analyzed first. Then, based on the technique of minimizing the loss function of the neural network, the proposed numerical methods reduce the fractional differential equation into a gradient descent problem or the quadratic optimization problem. By using the gradient descent process or the quadratic optimization process, the numerical solution to the FDEs can be obtained. The efficiency and accuracy of the presented methods are shown by some numerical examples. Numerical tests show that this approach is easy to implement and accurate when applied to many types of FDEs.
Highlights
Fractional differential equations (FDEs) have been a hot topic in many scientific fields, such as dynamical system control theory, fluid flow, modelling in rheology, dynamic process of self-similar porous structure, diffusion transport similar to diffusion, electric network, and probability statistics [1,2,3,4,5,6,7,8,9]
The neural network is trained to satisfy the differential equation at many selected points. The question for this method is that it is difficult to construct the first part of the trial solution and this method cannot be applied to fractional partial differential equations
The neural network method is a promising approach for solving fractional differential equations
Summary
Fractional differential equations (FDEs) have been a hot topic in many scientific fields, such as dynamical system control theory, fluid flow, modelling in rheology, dynamic process of self-similar porous structure, diffusion transport similar to diffusion, electric network, and probability statistics [1,2,3,4,5,6,7,8,9]. The neural network is trained to satisfy the differential equation at many selected points The question for this method is that it is difficult to construct the first part of the trial solution and this method cannot be applied to fractional partial differential equations. The two mentioned neural network techniques motivate us to develop more neural network methods to solve FDEs, but how to Journal of Function Spaces get the fractional order of the neural network is a difficult problem To overcome this difficulty, in this work, we use a triangle base neural network as basis function to propose an alternative method called triangle neural network methods.
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