THE EXACT MEROMORPHIC SOLUTIONS OF SOME NONLINEAR DIFFERENTIAL EQUATIONS

We find the exact forms of meromorphic solutions of the nonlinear differential equations■,n≥3,k≥1,where q,Q are nonzero polynomials,Q■Const.,and p_(1),p_(2),α_(1),α_(2)are nonzero constants withα_(1)≠α_(2).Compared with previous results on the equation p(z)f^(3)+q(z)f"=-sinα(z)with polynomial coefficients,our results show that the coefficient of the term f^((k))perturbed by multiplying an exponential function will affect the structure of its solutions.

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.

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.

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.

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.

Data-driven and physical-based identification of partial differential equations for multivariable system

Data-driven partial differential equation identification is a potential breakthrough to solve the lack of physical equations in complex dynamic systems.However,existing equation identification methods still cannot effectively identify equations from multivariable complex systems.In this work,we combine physical constraints such as dimension and direction of equation with data-driven method,and successfully identify the Navier-Stocks equations from the flow field data of Karman vortex street.This method provides an effective approach to identify partial differential equations of multivariable complex systems.

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.

A SOLVING METHOD FOR A SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS WITH AN APPLICATION TO THE BENDING PROBLEM OF A THICK PLATE

A theorem of solving a system of linear non-homogeneous differential equations through integrating and adding its basic solutions is put forward and proved, the mathematical role and physical nature of the theorem is interpreted briefly. As an example, the theorem is applied to solve the problem of thermo-force bending of a thick plate.

FORCED OSCILLATIONS OF BOUNDARY VALUE PROBLEMS OF HIGHER ORDER FUNCTIONAL PARTIAL DIFFERENTIAL EQUATIONS

In this paper we study the forced oscillations of boundary value problems of a class of higher order functional partial differential equations.The principal tool is an everaging techniqe which enables one to establish oscillation in terms of related functional differential inequallities.

A SYMBOLIC COMPUTATION METHOD TO DECIDE THE COMPLETENESS OF THE SOLUTIONS TO THE SYSTEM OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS

A symbolic computation method to decide whether the solutions to the system Of linear partial differential equation is complete via using differential algebra and characteristic set is presented. This is a mechanization method, and it can be carried out on the computer in the Maple environment.

SINGULARLY PERTURBED METHODS IN THE THEORY OF OPTIMAL CONTROL OF SYSTEMS GOVERNED BY PARTIAL DIFFERENTIAL EQUATIONS

In this paper, the various problems associaled with the optimal control of systemsgoverned by partial differential equations are introduced by using singularly perturbedmethods for analysis based on stale equations, or the cost funtction and also stateequations defined in perturbed domains.

Numerical Solution of Nonlinear System of Partial Differential Equations by the Laplace Decomposition Method and the Pade Approximation

In this paper, Laplace decomposition method (LDM) and Pade approximant are employed to find approximate solutions for the Whitham-Broer-Kaup shallow water model, the coupled nonlinear reaction diffusion equations and the system of Hirota-Satsuma coupled KdV. In addition, the results obtained from Laplace decomposition method (LDM) and Pade approximant are compared with corresponding exact analytical solutions.

Remarks on Oscillation for Systems of High Order Partial Differential Equations of Neutral Type

在这份报纸，一些足够的条件为摆动被获得因为高度的系统的答案订中立类型的部分微分方程。

THE BASIC THEORY OF SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS WITH BOUNDED DELAY IN THE SPACE OF INTEGRABLE FUNCTIONS

在可积函数空间中讨论有界滞量泛函微分方程，建立其解的基本理论，包括解的存在性，唯一性及延展性。

On the partial stability of nonlinear impulsive Caputo fractional systems

In this work,stability with respect to part of the variables of nonlinear impulsive Caputo fractional differential equations is investigated.Some effective sufficient conditions of stability,uniform stability,asymptotic uniform stability and Mittag Leffler stability.The approach presented is based on the specially introduced piecewise continuous Lyapunov functions.Furthermore,some numerical examples are given to show the effectiveness of our obtained theoretical results.

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.

SOME RESULTS REGARDING PARTIAL DIFFERENTIAL POLYNOMIALS AND THE UNIQUENESS OF MEROMORPHIC FUNCTIONS IN SEVERAL VARIABLES

In this paper,we mainly investigate the value distribution of meromorphic functions in Cmwith its partial differential and uniqueness problem on meromorphic functions in Cmand with its k-th total derivative sharing small functions.As an application of the value distribution result,we study the defect relation of a nonconstant solution to the partial differential equation.In particular,we give a connection between the Picard type theorem of Milliox-Hayman and the characterization of entire solutions of a partial differential equation.

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.

The Existence of Meromorphic Solutions to Non-Linear Delay Differential Equations

In this paper, we study the existence of the transcendental meromorphic solution of the delay differential equations , where a(z) is a rational function, and are polynomials in w(z) with rational coefficients, k is a positive integer. Under the assumption when above equations own transcendental meromorphic solutions with minimal hyper-type, we derive the concrete conditions on the degree of the right side of them. Specially, when w(z)=0 is a root of , its multiplicity is at most k. Some examples are given here to illustrate that our results are accurate.

THE GROWTH OF SOLUTIONS OF SYSTEMS OF COMPLEX NONLINEAR ALGEBRAIC DIFFERENTIAL EQUATIONS

We investigate the problem of growth order of solutions of a type of systems of non-linear algebraic differential equations, and extend some results of the growth order of solutions of algebraic differential equations to systems of algebraic differential equations.