Search

In this paper, with the aid of symbolic computation system Python, and based on the Deep Neural Network, Automatic differentiation and Limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) optimization algorithms, the model parameters of modified Kortewegde Vries (mkdv) equation Kdv–Burger equation and Huxley equation are obtained. We added different amounts of noise to the clean data in experiment and found that with the addition of trace noise, the parameters of the differential equation can also be accurately found. The result indicates that the algorithm has little effect on trace noise and shows better robustness to data noise. The method in this paper has demonstrated the powerful mathematical and physical ability of deep learning to flexibly simulate the physical dynamic state represented by differential equations, which opens the way for us to understand more physical phenomena later and the algorithm may be suitable for the data in practical application. [ABSTRACT FROM AUTHOR]

With the development of computers and neural networks, the traditional methods of solving differential equations have been greatly developed. Typical examples are the differential equations of population, finance, infectious disease and traffic problems solved by neural network method. Recently, the popular physics-informed neural network (PINN) method has been proved to be able to solve the numerical solution of PDEs. Based on the PINN method, this paper proposes an improved PINN method (IPINN) that is to introduce local adaptive activation function of neurons into PINN network to improve the performance of neural network and successfully applies the IPINN method to the Ivancevic option pricing model and Black–Scholes model in finance. The rogue wave solution and soliton solution of the Ivancevic option pricing model, and the numerical solution of the Black–Scholes model are solved, respectively. At the same time, it can be shown that the IPINN method has the characteristics of faster convergence, more stability and higher accuracy than the PINN method by the results of numerical experiments. [ABSTRACT FROM AUTHOR]

Although many effective methods for solving partial differential equations (PDEs) have been proposed, there is no universal method that can solve all PDEs. Therefore, solving partial differential equations has always been a difficult problem in mathematics, such as deep neural network (DNN). In recent years, a method of embedding some basic physical laws into traditional neural networks has been proposed to reveal the dynamic behavior of equations directly from space-time data [i.e., physics-informed neural network (PINN)]. Based on the above, an improved deep learning method to recover the new soliton solution of Huxley equation has been proposed in this paper. As far as we know, this is the first time that we have used an improved method to study the numerical solution of the Huxley equation. In order to illustrate the advantages of the improved method, we use the same network depth, the same hidden layer and neurons contained in the hidden layer, and the same training sample points. We analyze the dynamic behavior and error of Huxley's exact solution and the new soliton solution and give vivid graphs and detailed analysis. Numerical results show that the improved algorithm can use fewer sample points to reconstruct the exact solution of the Huxley equation with faster convergence speed and better simulation effect. [ABSTRACT FROM AUTHOR]

This paper extends a method, called bilinear neural network method (BNNM), to solve exact solutions to nonlinear partial differential equation. New, test functions are constructed by using this method. These test functions are composed of specific activation functions of single-layer model, specific activation functions of "2-2" model and arbitrary functions of "2-2-3" model. By means of the BNNM, nineteen sets of exact analytical solutions and twenty-four arbitrary function solutions of the dimensionally reduced p-gBKP equation are obtained via symbolic computation with the help of Maple. The fractal solitons waves are obtained by choosing appropriate values and the self-similar characteristics of these waves are observed by reducing the observation range and amplifying the partial picture. By giving a specific activation function in the single layer neural network model, exact periodic waves and breathers are obtained. Via various three-dimensional plots, contour plots and density plots, the evolution characteristic of these waves are exhibited. [ABSTRACT FROM AUTHOR]

Let n be a positive integer, and 프(n)=F[x]/(xpn), the divided power algebra over an algebraically closed field F of prime characteristic p > 2. Let π(n) be the tensor product of 프(n) and the Grassmann superalgebra ∧(1) in one variable. The Zassenhaus superalgebra 풵(n) is defined to be the Lie superalgebra of the special super derivations of the superalgebra π(n). In this paper we study simple modules over the Zassenhaus superalgebra 풵(n) with p-characters of height 0. We give a complete classification of the isomorphism classes of such simple modules and determine their dimensions. A sufficient and necessary condition for the irreducibility of Kac modules is obtained. [ABSTRACT FROM AUTHOR]

In this paper, the exact solutions of a reduced extended (3 + 1)-dimensional Jimbo-Miwa equation are investigated with the help of its bilinear representation and symbolic computation. Firstly, a kind of bright-dark lump wave solutions is directly obtained by taking the solution F in bilinear equation as a quadratic function. Furthermore, the interaction solutions between one lump wave and one stripe wave are also presented by taking F as a combination of quadratic function and exponential function. Finally, by taking F as a combination of quadratic function and hyperbolic cosine function, the rogue wave which aroused by the interaction between lump soliton and a pair of stripe solitons are obtained. The dynamic properties of the above three kinds of exact solutions are displayed vividly by figures. [ABSTRACT FROM AUTHOR]

In this paper, based on the Hirota bilinear method, a kind of lump solutions and two classes of interaction solutions are discussed to the (2+1)-dimensional generalized KdV equation with the aid of symbolic computation system Mathematica. Analyticity is naturally guaranteed by taking special choices of the involved parameters to achieve a positive constant term. Particularly, these solutions with special values of the included parameters are plotted, as illustrative example. [ABSTRACT FROM AUTHOR]

For the boundary value problem (BVP) of a second-order partial differential equation on a plane triangle area, we propose a new algorithm based on the Adomian decomposition method (ADM) combined with a segmented technique. In addition, we present a new theorem that ensures the convergence of the algorithm. By this algorithm, the model for the effect of regional recharge on the plane triangle groundwater flow region is solved, from which we obtain the segmented exact solution of the problem, which satisfies the governing equation and all of the specified boundary conditions. Then, by the algorithm combined with Taylor's formula, the heterogeneous aquifer model on the plane triangle groundwater flow region is considered, from which we obtain the segmented high-precision approximate solution of the problem. [ABSTRACT FROM AUTHOR]