Double Elzaki Transform Decomposition Method for Solving Non-Linear Partial Differential Equations

In this paper, we discuss a new method employed to tackle non-linear partial differential equations, namely Double Elzaki Transform Decomposition Method (DETDM). This method is a combination of the Double ELzaki Transform and Adomian Decomposition Method. This technique is hereafter provided and supported with necessary illustrations, together with some attached examples. The results reveal that the new method is very efficient, simple and can be applied to other non-linear problems.

ON THE HOLOMORPHIC SOLUTION OF NON-LINEAR TOTALLY CHARACTERISTIC EQUATIONS WITH SEVERAL SPACE VARIABLES

In this paper the authors study a class of non-linear singular partial differential equation in complex domain C-t x C-x(n). Under certain assumptions, they prove the existence and uniqueness of holomorphic solution near origin of C-t x C-x(n).

Linear Quadratic Optimal Control for Systems Governed by First-Order Hyperbolic Partial Differential Equations

This paper focuses on linear-quadratic(LQ)optimal control for a class of systems governed by first-order hyperbolic partial differential equations(PDEs).Different from most of the previous works,an approach of discretization-then-continuousization is proposed in this paper to cope with the infinite-dimensional nature of PDE systems.The contributions of this paper consist of the following aspects:(1)The differential Riccati equations and the solvability condition of the LQ optimal control problems are obtained via the discretization-then-continuousization method.(2)A numerical calculation way of the differential Riccati equations and a practical design way of the optimal controller are proposed.Meanwhile,the relationship between the optimal costate and the optimal state is established by solving a set of forward and backward partial difference equations(FBPDEs).(3)The correctness of the method used in this paper is verified by a complementary continuous method and the comparative analysis with the existing operator results is presented.It is shown that the proposed results not only contain the classic results of the standard LQ control problem of systems governed by ordinary differential equations as a special case,but also support the existing operator results and give a more convenient form of computation.

The Jaffa Transform for Hessian Matrix Systems and the Laplace Equation

Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.

Interaction of Conormal Waves With Strong and Weak Singularities For Semi－Linear Equations

We first define a kind of new function space, called the space of twice conormal distributions. With some estimates on these function spaces, we can prove that if there exist two characteristic hypersurfaces bearing strong and weak singularities respectively intersect transversally, then some new singularities will take place anti their strength will be no more than the original weak one.

An Alternative Method for Solving Lagrange＇s First-Order Partial Differential Equation with Linear Function Coefficients

An alternative method of solving Lagrange＇s first-order partial differential equation of the form（a1x ＋b1y＋C1z）p＋ （a2x ＋b2y＋c2z）q =a3x ＋b3y＋c3z,where p = Эz/Эx, q = Эz/Эy and ai, bi, ci （i = 1,2,3） are all real numbers has been presented here.

Solving Inhomogeneous Linear Partial Differential Equations

Lagrange＇s variation-of-constants method for solving linear inhomogeneous ordinary differential equations （ode＇s） is replaced by a method based on the Loewy decomposition of the corresponding homogeneous equation. It uses only properties of the equations and not of its solutions. As a consequence it has the advantage that it may be generalized for partial differential equations （pde＇s）. It is applied to equations of second order in two independent variables, and to a certain system of third-order pde＇s. Therewith all possible linear inhomogeneous pde＇s are covered that may occur when third-order linear homogeneous pde＇s in two independent variables are solved.

On a Linear Partial Differential Equation of the Higher Order in Two Variables with Initial Condition Whose Coefficients are Real-valued Simple Step Functions

By using the method developed in the paper [Georg. Inter. J. Sci. Tech., Volume 3, Issue 1 （2011）, 107-129], it is obtained a representation in an explicit form of the weak solution of a linear partial differential equation of the higher order in two variables with initial condition whose coefficients are real-valued simple step functions.

New Kamenev-type Oscillation Criteria for Half-linear Partial Differential Equations

We establish new Kamenev-type oscillation criteria for the half-linear partial differential equation with damping div（A（x）｜｜ u｜｜^p-2 u）＋〈b（x）,｜｜ u｜｜^p-2 u〉＋c（x）｜u｜^p-2u=0（E）under quite general conditions. These results are extensions of the recent results developed by Sun [Y.C. Sun, New Kamenev-type oscillation criteria of second order nonlinear differential equations with damping, J. Math. Anal. Appl. 291 （2004） 341-351] for second order ordinary differential equations in a natural way, and improve some existing results in the literature. As applications, we illustrate our main results using two different types of half-linear partial differential equations.

Mild Solutions of a Class of Conformable Fractional Differential Equations with Nonlocal Conditions

This paper deals with the existence,uniqueness and continuous dependence of mild solutions for a class of conformable fractional differential equations with nonlocal initial conditions.The results are obtained by means of the classical fixed point theorems combined with the theory of cosine family of linear operators.

ODE-Based Multistep Schemes for Backward Stochastic Differential Equations

In this paper,we explore a new approach to design and analyze numerical schemes for backward stochastic differential equations(BSDEs).By the nonlinear Feynman-Kac formula,we reformulate the BSDE into a pair of reference ordinary differential equations(ODEs),which can be directly discretized by many standard ODE solvers,yielding the corresponding numerical schemes for BSDEs.In particular,by applying strong stability preserving(SSP)time discretizations to the reference ODEs,we can propose new SSP multistep schemes for BSDEs.Theoretical analyses are rigorously performed to prove the consistency,stability and convergency of the proposed SSP multistep schemes.Numerical experiments are further carried out to verify our theoretical results and the capacity of the proposed SSP multistep schemes for solving complex associated problems.

A Novel Nonlinear Algorithm for Typhoon Cloud Image Enhancement

A novel nonlinear gray transform method is proposed to enhance the contrast of a typhoon cloud image.Generally,the typhoon cloud image obtained by a satellite cannot be directly used to make an accurate prediction of the typhoon＇s center or intensity because the contrast of the received typhoon cloud image may be bad.Our aim is to extrude the typhoon＇s eye in the typhoon cloud image.A normalized arc-tangent transformation operation is designed to enhance global contrast of the typhoon cloud image.Differential evolution algorithm is used to choose the optimal nonlinear transform parameter.Finally,geodesic activity contour model is used to extract the typhoon＇s eye to verify the performance of the proposed method.Experimental results show that the proposed method can efficiently enhance the global contrast of the typhoon cloud image while greatly extruding the typhoon＇s eye.

变分量子线性求解算法在高速飞行器定常绕流数值模拟中的应用

CFD方法在用于三维大规模复杂流动的空气动力学数值模拟时,面临着计算成本高、模拟时间长等瓶颈问题。近年来,量子计算为航空航天CFD领域带来了新的解决思路,通过利用量子比特的叠加态和纠缠特性,理论上相比于经典计算机能够实现对数级的存储缩减和指数级的效率加速,在处理大规模空气动力学模拟等任务上具有巨大的潜力。聚焦于量子计算技术在航空航天CFD领域的应用探索,针对高速定常流动问题,采用变分量子线性求解器(variational quantum linear solver,VQLS)辅助求解CFD时间离散环节得到的高维线性方程组,进而发展了可以在含噪声中等规模量子(noisy intermediate-scale quantum,NISQ)器件上实现的VQLS-CFD耦合方法。通过三维双椭球模型和探测器火星科学实验室模型的超声速数值模拟测试,验证了VQLS-CFD耦合方法可以实现大规模复杂流动的鲁棒及准确模拟,并产生与试验数据吻合度较高的计算结果,最终获得高可信度的气动预测结果。然而,介绍的量子计算流体力学(quantum computational fluid dynamics,QCFD)方法在计算效率方面尚未达到真正意义上的加速效果,这一现象的突破将依赖量子计算机硬件的持续发展与QCFD算法的不断优化与成熟。

动边界上Burgers方程的非线性边值一初值问题

借助应用齐次平衡原则导出的B¨acklund变换和Cole-Hopf变换将动边界上给定流量型条件的Burgers方程的非线性边值—初值问题化为线性方程的边值—初值问题。当动边界是时间t的线性函数时,得出了问题的精确解析解。

单相有源电力滤波器非线性统一控制策略

目前有源电力滤波器控制策略都采用电流内环和电压外环的双环控制.本文提出了一种基于精确反馈线性化的单相有源电力滤波器统一控制策略.在单相有源电力滤波器仿射非线性模型基础上,通过求解偏微分方程得到一个包含补偿电流变量和直流侧电压变量的输出函数,并推导出了其状态反馈精确线性化非线性控制律,将原非线性系统转换成微分同胚的二阶线性系统.选取适当的反馈系数设计控制器使输出函数渐近跟踪指令参考值,从而实现了电流和电压的统一控制.最后通过仿真分析和实验验证了所提方案的优越性和可行性.

基于PDE的图像去噪和反差增强同步算法

针对反差较低且包含噪声污染的灰度图像,设计分段线性拉伸函数,引入TV下降流,建立新的偏微分方程(PDE)数学模型。该模型通过设置参数λ灵活控制去噪和反差增强的程度,实现2种灰度图像处理手段的同步进行。对比实验表明,该方法可有效缓解传统处理方法存在的问题,在抑制噪声的同时增强图像的反差。

齐次平衡法与Burgers-KdV方程的精确解

对齐次平衡法进行适当修改。

双重介质分形油藏非线性流动分析

在非线性偏微分方程中 ,根据弱可压缩液体的假设 ,忽略了二次梯度项 ,这在混气石油和低渗透储层的计算中将产生误差。对于双重分形介质流动系统及其物质平衡方程 ,在保留非线性偏微分方程中所有项的前提下建立了新的双孔模型。采用Douglas Jones方法 ,获得了定产生产和定压生产时分形介质双孔模型的数值解。结果表明 ,对于定产生产 ,油藏参数cLD对非线性影响更敏感 ,线性解和非线性解的偏差约为 10 %。对于定压生产 ,cLD的影响可忽略不计。讨论了液体压缩系数和双重分形介质参数变化时的压力变化规律 ,给出了典型压力曲线。

精确反馈线性化中虚拟输出函数的求解与应用

为使精确反馈线性化方法得到更广泛的应用,给出虚拟输出函数的简便求解方法.首先将虚拟输出函数的求解转化为偏微分方程组的求解,然后针对这类偏微分方程组的特点提出仿照求解齐次线性方程组的方法可以简便的求出虚拟输出函数.利用该方法为非线性舵鳍联合系统实现了状态反馈线性化,并基于闭环增益成形算法设计了非线性鲁棒控制器,仿真表明该控制器对原非线性系统能起到良好的控制作用.该方法简单、实用,对于状态反馈线性化的实现具有一定的指导意义.

一类无散射偏微分方程组的变换和精确解

将 3个非线性物理方程组通过近似约化 ,转化为一类无散射的偏微分方程组 ,然后通过变换 ,化为一阶拟线性双曲型方程。利用拟线性双曲型方程的解 ,得到无散射偏微分方程组的精确解。参