Bilinear neural network method for obtaining the exact analytical solutions to nonlinear evolution equations and its application to KdV equation

Document Type : Original Article

Authors

Muhammad Abubakar Isah ¹

Asif Yokus ¹

Dogan Kaya ²

¹ Department of Mathematics‎, ‎Faculty of Science‎, ‎Firat University‎, ‎Elazig‎, ‎23100‎, ‎Turkey

² Department of Mathematics‎, ‎Istanbul Commerce University‎, ‎34445 Istanbul‎, ‎Turkey

10.22034/kjm.2024.396918.2865

Abstract

Most conventional test functions for solving nonlinear partial differential equations can be built using a single and double hidden layer neural network model via the bilinear neural network method, as is well known. The neural network test function model for the KdV equation is extended to the "2-3-1" and "2-2-2-1" models in this research. To find analytical solutions to the KdV equation, some new test functions are developed by giving some unique activation functions. By choosing some parameters, lump wave, bright, kink soliton solutions and various soliton solutions are generated. The relationship between the frequency velocity of the wave and the wave intensity as well as the stability analysis of the obtained solutions are studied which provide us a fresh perspective on how to explore the model in detail. The dynamical features of these waves are shown using two-dimensional graphs, three-dimensional graphs and contour plots.

Keywords

The bilinear neural network method

the KdV equation

traveling wave solution

Hirota bilinear form

Subjects

• 35 Partial differential equations

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