| Citation: |
Yunjie Yang, Jianping Shi, Hui Fang. NOVEL EXACT ANALYTICAL SOLUTIONS OF THE (3+1)-DIMENSIONAL POTENTIAL YU-TODA-SASA-FUKUYAMA EQUATION VIA LIE SYMMETRY ANALYSIS AND NEURAL NETWORK MODELS[J]. Journal of Applied Analysis & Computation, 2027, 17(1): 500-523. doi: 10.11948/20250287
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NOVEL EXACT ANALYTICAL SOLUTIONS OF THE (3+1)-DIMENSIONAL POTENTIAL YU-TODA-SASA-FUKUYAMA EQUATION VIA LIE SYMMETRY ANALYSIS AND NEURAL NETWORK MODELS
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Abstract
A novel method combining Lie symmetry analysis and neural network model is proposed for solving the (3+1)-dimensional potential-Yu-Toda-Sasa-Fukuyama (pYTSF) equation in this paper. This method has three processes: Firstly, the dimension of the given equation are reduced and two reduction equations are derived via Lie symmetry analysis; secondly, different test functions are constructed based on neural network models and various activation functions to yield solutions of the reduction equations; finally, a series of exact solutions of the original equation is achieved depending on the expression of group-invariant solutions and the transformation of variables. 12 novel exact analytical solutions of the (3+1)-dimensional pYTSF equation are obtained by the novel method. Furthermore, the dynamical characteristics of all solutions are illustrated by choosing appropriate parameters. The results validate the feasibility of this new method and show that the novel method may be applied in more nonlinear partial differential equations.
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References
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Yunjie Yang, Jianping Shi, Hui Fang. NOVEL EXACT ANALYTICAL SOLUTIONS OF THE (3+1)-DIMENSIONAL POTENTIAL YU-TODA-SASA-FUKUYAMA EQUATION VIA LIE SYMMETRY ANALYSIS AND NEURAL NETWORK MODELS[J]. Journal of Applied Analysis & Computation, 2027, 17(1): 500-523. doi: 10.11948/20250287
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Figure 1.
The framework of Lie-symmetry neural network method.
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Figure 2.
Six neural network models.
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Figure 3.
The profiles of solution
$ u_{1} $ $ z=1, t=1, b_3=2, w_{1,f}=1, w_{2,f}=1, w_{\eta,1}=1, w_{\tau,1}=1, w_{\chi,2}=1, w_{\tau,2}=1 $ -
Figure 4.
The profiles of solution
$ u_{2} $ $ z=1, t=1, b_4=2, w_{1,f}=1, w_{2,f}=1, w_{3,f}=10, w_{\eta,1}=1, w_{\tau,1}=1, w_{\chi,2}=1, w_{\chi,3}=1, w_{\tau,3}=1 $ -
Figure 5.
The real part profiles of solution
$ u_3 $ $ n $ $ t=1, z=1, b_2=2, w_{2,f}=1, w_{1,2}=1, w_{\tau,1}=1 $ -
Figure 6.
The real part evolution plots of solution
$ u_4 $ $ w_{1,3} $ $ w_{2,3} $ $ z=1, t=1, b_3=1, w_{3,f}=1, w_{\tau,1}=1, w_{\eta,2}=1 $ -
Figure 7.
The profiles of solution
$ u_{5} $ $ z=1, t=1, b_5=2, w_{3,f}=1, w_{4,f}=100, w_{1,3}=1, w_{2,3}=1, w_{2,4}=1, w_{\eta,1}=1, w_{\tau,1}=1, w_{\chi,2}=1, w_{\tau,2}=1 $ -
Figure 8.
The profiles of solution
$ u_{6} $ $ z=1, t=1, b_7=2, w_{4,f}=1, w_{5,f}=1, w_{6,f}=1, w_{\eta,1}=1, w_{\tau,1}=1, w_{\chi,3}=1, w_{\eta,2}=1 $ -
Figure 9.
The profiles of solution
$ u_{7} $ $ z=1, t=1, b_3=2, w_{1,f}=1, w_{2,f}=1, w_{\eta,1}=1, w_{\eta,2}=1 $ -
Figure 10.
The profiles of solution
$ u_{8} $ $ z=1, t=1, b_4=2, w_{1,f}=1, w_{2,f}=1, w_{3,f}=1, w_{\chi,1}=1, w_{\chi,2}=1, w_{\chi,3}=1, w_{\eta,1}=1, w_{\eta,3}=1 $ -
Figure 11.
The profiles of solution
$ u_{9} $ $ z=1, t=1, b_3=2, w_{2,f}=1, w_{1,2}=1, w_{\eta,1}=1 $ -
Figure 12.
The profiles of solution
$ u_{10} $ $ z=1, t=1, b_4=2, w_{3,f}=1, w_{1,3}=1, w_{2,3}=1, w_{\eta,1}=1, w_{\eta,2}=1 $ -
Figure 13.
The profiles of solution
$ u_{11} $ $ z=1, t=1, b_{3}=2, b_{5}=2, w_{3,f}=1, w_{4,f}=1, w_{1,3}=1, w_{2,3}=1, w_{2,4}=1, w_{\eta,1}=1, w_{\eta,2}=1 $ -
Figure 14.
The profiles of solution
$ u_{12} $ $ z=1, t=1, b_{7}=2 $