Results Involving Partial Differential Equations and Their Solution by Certain Integral Transform

In this study,we aimto investigate certain triple integral transformand its application to a class of partial differentialequations.We discuss various properties of the new transformincluding inversion, linearity, existence, scaling andshifting, etc. Then,we derive several results enfolding partial derivatives and establish amulti-convolution theorem.Further, we apply the aforementioned transform to some classical functions and many types of partial differentialequations involving heat equations,wave equations, Laplace equations, and Poisson equations aswell.Moreover,wedraw some figures to illustrate 3-D contour plots for exact solutions of some selected examples involving differentvalues in their variables.

Adaptive multi-step piecewise interpolation reproducing kernel method for solving the nonlinear time-fractional partial differential equation arising from financial economics

This paper is aimed at solving the nonlinear time-fractional partial differential equation with two small parameters arising from option pricing model in financial economics.The traditional reproducing kernel(RK)method which deals with this problem is very troublesome.This paper proposes a new method by adaptive multi-step piecewise interpolation reproducing kernel(AMPIRK)method for the first time.This method has three obvious advantages which are as follows.Firstly,the piecewise number is reduced.Secondly,the calculation accuracy is improved.Finally,the waste time caused by too many fragments is avoided.Then four numerical examples show that this new method has a higher precision and it is a more timesaving numerical method than the others.The research in this paper provides a powerful mathematical tool for solving time-fractional option pricing model which will play an important role in financial economics.

Machine learning of partial differential equations from noise data

Machine learning of partial differential equations(PDEs)from data is a potential breakthrough for addressing the lack of physical equations in complex dynamic systems.Recently,sparse regression has emerged as an attractive approach.However,noise presents the biggest challenge in sparse regression for identifying equations,as it relies on local derivative evaluations of noisy data.This study proposes a simple and general approach that significantly improves noise robustness by projecting the evaluated time derivative and partial differential term into a subspace with less noise.This method enables accurate reconstruction of PDEs involving high-order derivatives,even from data with considerable noise.Additionally,we discuss and compare the effects of the proposed method based on Fourier subspace and POD(proper orthogonal decomposition)subspace.Generally,the latter yields better results since it preserves the maximum amount of information.

LaNets:Hybrid Lagrange Neural Networks for Solving Partial Differential Equations

We propose new hybrid Lagrange neural networks called LaNets to predict the numerical solutions of partial differential equations.That is,we embed Lagrange interpolation and small sample learning into deep neural network frameworks.Concretely,we first perform Lagrange interpolation in front of the deep feedforward neural network.The Lagrange basis function has a neat structure and a strong expression ability,which is suitable to be a preprocessing tool for pre-fitting and feature extraction.Second,we introduce small sample learning into training,which is beneficial to guide themodel to be corrected quickly.Taking advantages of the theoretical support of traditional numerical method and the efficient allocation of modern machine learning,LaNets achieve higher predictive accuracy compared to the state-of-the-artwork.The stability and accuracy of the proposed algorithmare demonstrated through a series of classical numerical examples,including one-dimensional Burgers equation,onedimensional carburizing diffusion equations,two-dimensional Helmholtz equation and two-dimensional Burgers equation.Experimental results validate the robustness,effectiveness and flexibility of the proposed algorithm.

The Fractional Investigation of Some Nonlinear Partial Differential Equations by Using an Efficient Procedure

The nonlinearity inmany problems occurs because of the complexity of the given physical phenomena.The present paper investigates the non-linear fractional partial differential equations’solutions using the Caputo operator with Laplace residual power seriesmethod.It is found that the present technique has a direct and simple implementation to solve the targeted problems.The comparison of the obtained solutions has been done with actual solutions to the problems.The fractional-order solutions are presented and considered to be the focal point of this research article.The results of the proposed technique are highly accurate and provide useful information about the actual dynamics of each problem.Because of the simple implementation,the present technique can be extended to solve other important fractional order problems.

On Fuzzy Conformable Double Laplace Transform with Applications to Partial Differential Equations

The Laplace transformation is a very important integral transform,and it is extensively used in solving ordinary differential equations,partial differential equations,and several types of integro-differential equations.Our purpose in this study is to introduce the notion of fuzzy double Laplace transform,fuzzy conformable double Laplace transform(FCDLT).We discuss some basic properties of FCDLT.We obtain the solutions of fuzzy partial differential equations(both one-dimensional and two-dimensional cases)through the double Laplace approach.We demonstrate through numerical examples that our proposed method is very successful and convenient for resolving partial differential equations.

Numerical Solution of Parabolic in Partial Differential Equations (PDEs) in One and Two Space Variable

In this paper, we shall be concerned with the numerical solution of parabolic equations in one space variable and the time variable t. We expand Taylor series to derive a higher-order approximation for Ut. We begin with the simplest model problem, for heat conduction in a uniform medium. For this model problem, an explicit difference method is very straightforward in use, and the analysis of its error is easily accomplished by the use of a maximum principle. As we shall show, however, the numerical solution becomes unstable unless the time step is severely restricted, so we shall go on to consider other, more elaborate, numerical methods which can avoid such a restriction. The additional complication in the numerical calculation is more than offset by the smaller number of time steps needed. We then extend the methods to problems with more general boundary conditions, then to more general linear parabolic equations. Finally, we shall discuss the more difficult problem of the solution of nonlinear equations.

基于高维PDE投影恢复的低剂量CT重建方法

目的提出一种基于高维偏微分方程(PDE)投影恢复的低剂量CT重建方法。方法先将原始的投影数据映射到高维空间中,构造投影数据的高维表示,通过移动高维空间中的点来对高维表示进行更新,再使用偏微分方程对投影数据进行滤波,最后将恢复后的数据使用FBP算法重建出最终CT图像。结果在Shepp-Logan体模实验中,与FBP,PWLS-QM和TGV-WLS方法相比,新方法在相对均方根误差指标上分别降低了68.87%、50.15%和27.36%,结构相似性上分别提高了23.50%,8.83%和1.62%,特征相似性上分别提高了17.30%、2.71%和2.82%。在腹部临床数据实验中,与FBP,PWLS-QM和TGV-WLS方法相比,新方法在相对均方根误差中分别降低了42.09%、31.04%和21.93%,结构相似性上分别提高了18.33%、13.45%和4.63%,特征相似性上分别提高了3.13%、1.46%和1.10%。结论本研究提出的新方法在有效去除低剂量CT图像中的条形伪影和噪声的同时,可以保持图像的空间分辨率。

基于自适应神经网络的PDEs求解研究

针对当前基于神经网络的PDEs求解方法效率和精度均不够理想的缺陷,研究提出一种基于改进BP神经网络(BP neural network,BPNN)的PDEs求解模型。首先,参照自适应网格法来改进神经网络结构,构建自适应神经网络,改进模型的输出精度;其次,提出一种引入Levy飞行机制和鲸鱼优化算法(Whale optimization algorithm,WOA)的改进海鸥优化算法(Improved Seagull Optimization Algorithm,ISOA)来优化BPNN,寻找BPNN的最佳参数,提高模型的性能;基于上述内容,构建基于ISOA-BPNNPDEs智能求解模型。结果显示,该模型的F1值为95.74%,准确率达到97.96%,输出误差为0.021,优于当前最先进的两种PDEs求解模型。上述内容表明,研究构建的基于ISOA-BPNNPDEs智能求解模型能够高效、准确地实现PDEs求解,为PDEs求解研究提供了新的路径。

Finite-time consensus of multi-agent systems driven by hyperbolic partial differential equations via boundary control

The leaderless and leader-following finite-time consensus problems for multiagent systems(MASs)described by first-order linear hyperbolic partial differential equations(PDEs)are studied.The Lyapunov theorem and the unique solvability result for the first-order linear hyperbolic PDE are used to obtain some sufficient conditions for ensuring the finite-time consensus of the leaderless and leader-following MASs driven by first-order linear hyperbolic PDEs.Finally,two numerical examples are provided to verify the effectiveness of the proposed methods.

基于FPDE-SIFT的声呐干涉图像配准方法

图像配准是声呐进行高精度干涉测量的保障,该文针对水下目标的声呐图像配准,提出了一种基于4阶偏微分方程尺度不变特征变换的声呐干涉图像配准方法。该方法聚焦声呐图像配准的难点,首先基于4阶偏微分方程构建尺度空间,在保持图像细节的前提下滤除噪声,提高特征提取的准确度;对于残余噪声造成的特征点误检,借助特征点的相位一致性信息加以筛选,精简特征点样本集;最后对特征点匹配策略进行优化,提出改进的快速样本一致性匹配策略剔除特征点的误匹配。算法增加了匹配点对的数量,提高了匹配点对的准确度,实现了声呐干涉图像的精确配准。水池实验和外场试验表明,该文所提出的算法相较现有算法对声呐图像有着更好的适用性,配准后的均方根误差与留一法均方根均小于1像素,达到了亚像素配准精度。

An Extension of Mapping Deformation Method and New Exact Solution for Three Coupled Nonlinear Partial Differential Equations

In this paper, we extend the mapping deformation method proposed by Lou. It is used to find new exacttravelling wave solutions of nonlinear partial differential equation or coupled nonlinear partial differential equations(PDEs). Based on the idea of the homogeneous balance method, we construct the general mapping relation betweenthe solutions of the PDEs and those of the cubic nonlinear Klein-Gordon (NKG) equation. By using this relation andthe abundant solutions of the cubic NKG equation, many explicit and exact travelling wave solutions of three systemsof coupled PDEs, which contain solitary wave solutions, trigonometric function solutions, Jacobian elliptic functionsolutions, and rational solutions, are obtained.

REDUCTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATION AND EXACT SOLUTIONS

Nonlinear partial differetial equation(NLPDE) is converted into ordinary differential equation(ODE) via a new ansatz.Using undetermined function method,the ODE obtained above is replaced by a set of algebraic equations which are solved out with the aid of Mathematica.The exact solutions and solitary solutions of NLPDE are obtained.

Method of Lines for Third Order Partial Differential Equations

The method of lines is applied to the boundary-value problem for third order partial differential equation. Explicit expression and order of convergence for the approximate solution are obtained.

A lattice Boltzmann model with an amending function for simulating nonlinear partial differential equations

This paper proposes a lattice Boltzmann model with an amending function for one-dimensional nonlinear partial differential equations （NPDEs） in the form ut ＋αuux ＋βu^nuz ＋γuxx ＋δuzxx ＋ζxxxx = 0. This model is different from existing models because it lets the time step be equivalent to the square of the space step and derives higher accuracy and nonlinear terms in NPDEs. With the Chapman-Enskog expansion, the governing evolution equation is recovered correctly from the continuous Boltzmann equation. The numerical results agree well with the analytical solutions.

DISTRIBUTED PARAMETER NEURAL NETWORKS FOR SOLVING PARTIAL DIFFERENTIAL EQUATIONS

Novel distributed parameter neural networks are proposed for solving partial differential equations, and their dynamic performances are studied in Hilbert space. The locally connected neural networks are obtained by separating distributed parameter neural networks. Two simulations are also given. Both theoretical and computed results illustrate that the distributed parameter neural networks are effective and efficient for solving partial differential equation problems.

Oscillation for Solutions of Systems of High Order Partial Differential Equations of Neutral Type

In this paper,some sufficient conditions are obtained for the oscillation for solutions of systems of highd order partial differential equations of neutral type.

New Implementation of Legendre Polynomials for Solving Partial Differential Equations

In this paper we present a proposal using Legendre polynomials approximation for the solution of the second order linear partial differential equations. Our approach consists of reducing the problem to a set of linear equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The performance of presented method has been compared with other methods, namely Sinc-Galerkin, quadratic spline collocation and LiuLin method. Numerical examples show better accuracy of the proposed method. Moreover, the computation cost decreases at least by a factor of 6 in this method.

On Current Developments in Partial Differential Equations

The first part of this article is an overview on some recent major developments in the field of analysis and partial different equations.It is a brief presentation given by the author at a round table discussion.The second part is a supplement of various details provided by several outstanding researchers on subjects.

Mixture of a New Integral Transform and Homotopy Perturbation Method for Solving Nonlinear Partial Differential Equations

In this paper, we present a new method, a mixture of homotopy perturbation method and a new integral transform to solve some nonlinear partial differential equations. The proposed method introduces also He’s polynomials [1]. The analytical results of examples are calculated in terms of convergent series with easily computed components [2].