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This paper introduces a novel neural network approach based on Lie groups to effectively solve initial value problems of differential equations for nonlinear dynamical systems. Our method utilizes a priori knowledge inherent in the system, i.e. Lie group expressions, and employs a single-layer network structure with the essence of a multilayer perceptron (MLP). To validate the effectiveness of our approach, we conducted an extensive empirical study using various examples representing complex nonlinear dynamical systems. The research results demonstrate the outstanding performance and efficacy of our method, outperforming Neural Ordinary Differential Equations in terms of accuracy, convergence speed, and stability. [ABSTRACT FROM AUTHOR]

The study of non-linear partial differential equations is a complex task requiring sophisticated methods and techniques. In this context, we propose a neural network approach based on Lie series in Lie groups of differential equations (symmetry) for solving Burgers–Huxley nonlinear partial differential equations, considering initial or boundary value terms in the loss functions. The proposed technique yields closed analytic solutions that possess excellent generalization properties. Our approach differs from existing deep neural networks in that it employs only shallow neural networks. This choice significantly reduces the parameter cost while retaining the dynamic behavior and accuracy of the solution. A thorough comparison with its exact solution was carried out to validate the practicality and effectiveness of our proposed method, using vivid graphics and detailed analysis to present the results. [ABSTRACT FROM AUTHOR]

The study of wave-like propagation of information in nonlinear and dispersive media is a complex phenomenon. In this paper, we provide a new approach to studying this phenomenon, paying special attention to the nonlinear solitary wave problem of the Korteweg–De Vries (KdV) equation. Our proposed algorithm is based on the traveling wave transformation of the KdV equation, which reduces the dimensionality of the system, enabling us to obtain a highly accurate solution with fewer data. The proposed algorithm uses a Lie-group-based neural network trained via the Broyden–Fletcher–Goldfarb–Shanno (BFGS) optimization method. Our experimental results demonstrate that the proposed Lie-group-based neural network algorithm can simulate the behavior of the KdV equation with high accuracy while using fewer data. The effectiveness of our method is proved by examples. [ABSTRACT FROM AUTHOR]

To combine a feedforward neural network (FNN) and Lie group (symmetry) theory of differential equations (DEs), an alternative artificial NN approach is proposed to solve the initial value problems (IVPs) of ordinary DEs (ODEs). Introducing the Lie group expressions of the solution, the trial solution of ODEs is split into two parts. The first part is a solution of other ODEs with initial values of original IVP. This is easily solved using the Lie group and known symbolic or numerical methods without any network parameters (weights and biases). The second part consists of an FNN with adjustable parameters. This is trained using the error back propagation method by minimizing an error (loss) function and updating the parameters. The method significantly reduces the number of the trainable parameters and can more quickly and accurately learn the real solution, compared to the existing similar methods. The numerical method is applied to several cases, including physical oscillation problems. The results have been graphically represented, and some conclusions have been made. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation is investigated. Lump-type solutions and lump solutions are obtained with aid of symbolic computation via Hirota bilinear method and the ansatz technique. By taking the function f in the Hirota bilinear form of the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation as the general quadratic polynomial function, a kind of lump-type solution which contains eleven parameters with six arbitrary independent parameters and two non-zero conditions is obtained. Lump solutions are found from the lump-type solutions via taking a special set of parameters, and the motion track of the lump is also described both theoretically and graphically. [ABSTRACT FROM AUTHOR]

Abstract: Based on Adomian decomposition method, a new algorithm for solving boundary value problem (BVP) of nonlinear partial differential equations on the rectangular area is proposed. The solutions obtained by the method precisely satisfy all boundary conditions, except the small pieces near the four corners of the rectangular area. A theorem on the boundary error is given. Hence, the Adomian decomposition method is more efficiently applied to BVPs for partial differential equations. Segmented and weighted analytical solutions with a high accuracy for the BVP of nonlinear groundwater equations on a rectangular area are obtained by the present algorithm. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, based on differential characteristic set theory and the associated algorithm (also called Wuʼs method), an algorithmic method is presented to decide on the existence of a nontrivial non-classical symmetry of a given partial differential equation without solving the corresponding nonlinear determining system. The theory and algorithm give a partial answer for the open problem posed by P.A. Clarkson and E.L. Mansfield in [21] on non-classical symmetries of partial differential equations. As applications of our algorithm, non-classical symmetries and corresponding invariant solutions are found for several evolution equations. [Copyright &y& Elsevier]

In this paper, the problem of viscous heating in plane Couette flow is considered by the homotopy perturbation method. The non-linear terms are expanded to Taylor series of the homotopy parameter. The obtained solutions are shown graphically and are compared with the exact solutions. The obtained results illustrate the efficiency and convenience of the method. [ABSTRACT FROM AUTHOR]

Abstract: We give a general result about the relation between approximate symmetry and approximate homotopy symmetry for perturbed PDEs. We show that the two couple equations derived by approximate symmetry method and approximate homotopy symmetry method are connected by a transformation. Consequently, acting the transformation on the known solutions constructed by approximate symmetry method, we directly obtain approximate homotopy series solutions whose accuracy can be controlled by adjusting the convergence-control parameter. Applications to the Cahn–Hilliard equation illustrate the effectiveness of the transformation. [Copyright &y& Elsevier]

Abstract: In this paper, we successfully construct the new exact traveling wave solutions of the coupled Schrödinger–Boussinesq equation by using the extended simplest equation method. The exact traveling wave solutions with double arbitrary parameters, are expressed by the hyperbolic functions, the trigonometric functions and the rational functions respectively. When the parameters are taken as special values, some solitary wave solutions are derived from the hyperbolic function solutions. [Copyright &y& Elsevier]

In this paper, we present the Adomian decomposition method and its modifications combined with convergence acceleration techniques, such as the diagonal Padé approximants and the iterated Shanks transforms, to solve nonlinear fractional ordinary differential equations. Two nonlinear numeric examples demonstrate that either the diagonal Padé approximants or the iterated Shanks transforms can efficiently extend the effective convergence region of the decomposition series solution. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we present the generalized Adomian–Rach theorem and the generalized Rach–Adomian–Meyers modified decomposition method for solving multi-order nonlinear fractional ordinary differential equations. We consider different classes of initial value problems for nonlinear fractional ordinary differential equations, including the case of real-valued orders and another case of rational-valued orders, which are solved by the present method. This method can treat any analytic nonlinearity. The coefficients of the solution in the form of a generalized power series are determined by a convenient recurrence scheme, which does not involve integration operations compared with the classic Adomian decomposition method. [Copyright &y& Elsevier]

Abstract: This paper proposes a symmetry–iteration hybrid algorithm for solving boundary value problems for partial differential equations. First, the multi-parameter symmetry is used to reduce the problem studied to a simpler initial value problem for ordinary differential equations. Then the variational iteration method is employed to obtain its solution. The results reveal that the proposed method is very effective and can be applied for other nonlinear problems. [Copyright &y& Elsevier]

This paper proposes a symmetry-homotopy hybrid algorithm to solve boundary value problems of partial differential equations. Firstly, the multi-parameter symmetry is used to reduce the studied problem to a simpler initial value problem of ordinary differential equations. Then the homotopy perturbation method is employed to obtain its solution. The results reveal that the proposed method is very effective and can be applied for other nonlinear problems. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, an extended simplest equation method is proposed to seek exact travelling wave solutions of nonlinear evolution equations. As applications, many new exact travelling wave solutions for several forms of the fifth-order KdV equation are obtained by using our method. The forms include the Lax, Sawada–Kotera, Sawada–Kotera–Parker–Dye, Caudrey–Dodd–Gibbon, Kaup–Kupershmidt, Kaup–Kupershmidt–Parker–Dye, and the Ito forms. [Copyright &y& Elsevier]

Cui et al. [M. Cui and F. Geng, Solving singular two point boundary value problems in reproducing kernel space, J. Comput. Appl. Math. 205 (2007), pp. 6-15; H. Yao and M. Cui, A new algorithm for a class of singular boundary value problems, Appl. Math. Comput. 186 (2007), pp. 1183-1191] presents an algorithm to solve a class of singular linear boundary value problems in the reproducing kernel space. In this paper, we will present three new algorithms to solve a class of singular weakly nonlinear boundary value problems in reproducing kernel space. The algorithms are efficiently applied to solving some model problems. It is demonstrated by the numerical examples that those algorithms are highly accurate. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with a new algorithm for giving the analytical and approximate solutions of a class of boundary value problems in the reproducing kernel space. The analytical solution and approximate solution are represented in terms of series. For any initial function , we prove , , . Two numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective. [Copyright &y& Elsevier]

In this paper, we present an application of Wu's method (differential characteristic set (dchar-set) algorithm) for computing the symmetry of (partial) differential equations (PDEs) that provides a direct and systematic procedure to obtain the classical and nonclassical symmetry of the differential equations. The fundamental theory and subalgorithms used in the proposed algorithm consist of a different version of the Lie criterion for the classical symmetry of PDEs and the zero decomposition algorithm of a differential polynomial (d-pol) system (DPS). The version of the Lie criterion yields determining equations (DTEs) of symmetries of differential equations, even those including a nonsolvable equation. The decomposition algorithm is used to solve the DTEs by decomposing the zero set of the DPS associated with the DTEs into a union of a series of zero sets of dchar-sets of the system, which leads to simplification of the computations. [ABSTRACT FROM AUTHOR]

The aim of this work is to obtain a numerical solution of a time-fractional telegraph equation by a modified reproducing kernel method. Two numerical examples are given to show that the present method overcomes the drawback of the traditional reproducing kernel method and it is an easy and effective method. [ABSTRACT FROM AUTHOR]

Reproducing kernel Hilbert space method (RKHSM) is an effective method. This paper, for the first time, uses the traditional RKHSM for solving the temperature field in two phase flows of multilayer water injection well. According to 2D oil-water temperature field mathematical model of two phase flows in cylindrical coordinates, selecting the properly initial and boundary conditions, by the process of Gram-Schmidt orthogonalization, the analytical solution was given by reproducing kernel functions in a series expansion form, and the approximate solution was expressed by n-term summation. The satisfied numerical results were carried out by Mathematica 7.0, showing that the larger the difference between injected water temperature and initial borehole temperature or water injection conditions, the more obvious the indication of water accepting zones. The numerical examples evidence the feasibility and effectiveness of the proposed method of the two phase flows in engineering. [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]

By means of symbolic computation with the help of Maple, diverse of exact solutions for the Jimbo–Miwa-like equation are successfully derived, based on the generalized bilinear method. These new solutions, named the periodic wave and cross-kink wave have greatly enriched the existing literature on the Jimbo–Miwa-like equation. Particularly, we have obtained the interaction solutions between the lump solution and an arbitrary function for the Jimbo–Miwa-like equation. As long as the researcher needs it, he can choose an arbitrary function to observe the collision phenomenon which he wants between the lump and the solitary waves. Via the three-dimensional images and density images with the help of Maple, the physical characteristics of these waves are described very well. That will be widely used to describe many interesting physical phenomena in the fields of gas, plasma, optics, acoustics, heat transfer, fluid dynamics, classical mechanics and so on. [ABSTRACT FROM AUTHOR]

In this paper, abundant lump solutions and two types of interaction solutions of the (3+1)-D Kadomtsev-Petviashvili equation are obtained by the Hirota bilinear method. Some contour plots with different determinant values are sequentially given to show that the corresponding lump solution tends to zero when the determinant approaches to zero. The interaction solutions with special parameters are plotted to elucidate the solution properties. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we propose a new modified recursion scheme for the resolution of boundary value problems (BVPs) for second-order nonlinear ordinary differential equations with Robin boundary conditions by the Adomian decomposition method (ADM). Our modified recursion scheme does not incorporate any undetermined coefficients. We also develop the multistage ADM for BVPs encompassing more general boundary conditions, including Neumann boundary conditions. [Copyright &y& Elsevier]

Abstract: In this paper, the nonlinear differential equation for temperature distribution of convective straight fins with temperature-dependent thermal conductivity is solved by using a new modified decomposition method (MDM) for boundary value problems. In the new MDM the recursion scheme of the solution components does not involve any undetermined coefficients. Using the new method, the temperature distribution and the efficiency of the fin can be expressed analytically as functions containing two fin parameters without any undetermined coefficients, which greatly facilitates parameter analysis. [Copyright &y& Elsevier]