A stochastic log-Laplace equation
where b, c, e are real functions on R (c, e ≥ 0), W , B1, B2, . . . are independent (standard) Brownian motions and ηti is the position of the ith particle at time t. Let MF (R) denote the set of all finite Borel measures on R. It is established by Skoulakis and Adler [18] that the high-density limit Xt of this system is the unique MF (R)-valued solution to the following martingale
existing theory of SPDE.
A STOCHASTIC LOG-LAPLACE EQUATION
3
1.2. Main results. First we study the existence and uniqueness for the solution to (1.3). We also establish its particle system representation in the spirit of Kurtz and Xiong [15].
of the LLE:
(1.3)
t
yt(x) = f (x) + (b(x)∂xyr(x) + a(x)∂x2yr(x) − yr(x)2) dr
0
t
+ c(x)∂xyr(x) dWr.
0
The stochastic partial differential equation (SPDE) is an important field
this case, the derivative of the solution is not involved in the noise term.
To the best of our knowledge, the LLE (1.3) does not fit into the setups of
This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Probability, 2004, Vol. 32, No. 3B, 2362–2388. This reprint differs from the original in pagination and typographic detail.
arXiv:math/0410164v1 [math.PR] 6 Oct 2004
The Annals of Probability 2004, Vol. 32, No. 3B, 2362–2388 DOI: 10.1214/009117904000000540 c Institute of Mathematical Statistics, 2004
To begin with, we introduce some notation needed in this paper. Let H0 = L2(R) be the set of all square integrable functions on R, and let H0+ consist of all the nonnegative functions in H0. Let Hm = {φ ∈ H0 : φ′, . . . , φ(m) ∈ H0}. Define the Sobolev norm on Hm by
1. Introduction and main results.
1.1. Introduction. We study the behavior of a branching interacting particle system in a random environment. For simplicity of notation, we assume that the particles move in the one-dimensional space R. The branching is critical binary; that is, at independent exponential times, each particle will die or split into two with equal probabilities. Between branchings, the motion of the ith particle is governed by an individual Brownian motion Bi(t) and a common Brownian motion W (t) which applies to all particles in the system:
challenge.
In this paper, we study the LLE (1.2). The main result is Theorem 1.2 in
which we prove that the log-Laplace transform of Xt is indeed given by the solution to (1.2). For simplicity of notation, we consider the forward version
A STOCHASTIC LOG-LAPLACE EQUATION1
By Jie Xiong
University of Tennessee
We study a nonlinear stochastic partial differential equation whose solution is the conditional log-Laplace functional of a superprocess in a random environment. We establish its existence and uniqueness by smoothing out the nonlinear term and making use of the particle system representation developed by Kurtz and Xiong [Stochastic Process. Appl. 83 (1999) 103–126]. We also derive the Wong–Zakai type approximation for this equation. As an application, we give a direct proof of the moment formulas of Skoulakis and Adler [Ann. Appl. Probab. 11 (2001) 488–543].
in
[18].
A
re-
lated model is studied by Wang [19] and Dawson, Li and Wang [4].
The log-Laplace equation has been used by many authors in deriving
various properties for superprocesses (cf. [2, 5]). It is natural, as indicated in
s
+ t c(x)∂xyr,t(x) dˆWr,
s
where f is the test function for the Laplace transform [cf. (1.8)], ∂x, ∂x2 are the first and second partial derivatives with respect to x and the last integral
of current research. We refer the reader to [1, 11] and [17] for an introduction
toied linear SPDEs. Here we only mention two
0
is a continuous martingale with quadratic variation process
t
M (φ) t = ( Xs, φ2 + | Xs, cφ′ |2) ds,
0
where
a(x)
=
1 2
(e(x)2
+
c(x)2).
Moment
formulas
are
derived
Received May 2002; revised June 2003. 1Supported in part by the NSA. AMS 2000 subject classifications. Primary 60G57, 60H15; secondary 60J80. Key words and phrases. Superprocess, random environment, Wong–Zakai approximation, particle system representation, stochastic partial differential equation.
两个与高阶积分有关的laplace变换公式的证明
两个与高阶积分有关的laplace变换公式的证明一、Laplace变换公式证明1. Laplace变换Laplace变换是一种线性、无限可微的变换，可将时域函数$f(t)$映射为复平面上的函数$F(s)$，其表达式为：$$F(s)=\int_0^\infty e^{-st}f(t)dt$$2. 常用Laplace变换公式(1) 指数函数：$\mathcal{L}(e^{at})=\frac{1}{s-a}$(2) 正弦函数：$\mathcal{L}(\sin at)=\frac{a}{s^2+a^2}$(3) 余弦函数：$\mathcal{L}(\cos at)=\frac{s}{s^2+a^2}$二、高阶积分和Laplace变换公式的证明1. 高阶积分高阶积分是指对一个已经被求过积分的函数再次进行积分的过程，形如：$$\int\dots\int f(x)dx^{n}$$其中$n$为积分次数。
2. Laplace变换的性质(1) 线性性：$\mathcal{L}\{af(t)+bg(t)\}=a\mathcal{L}\{f(t)\}+b\mathcal{L}\{g(t)\}$(2) 积分性质：$\mathcal{L}\{f^{(n)}(t)\}=s^nF(s)-s^{n-1}f(0)-s^{n-2}f'(0)-\dots-f^{(n-1)}(0)$3. 高阶积分的Laplace变换对于高阶积分$\int\dots\int f(x)dx^{n}$，我们可以通过反复应用积分性质，将其转化为一阶积分的Laplace变换形式：$$\mathcal{L}\{\int\dots\intf(x)dx^{n}\}=\frac{1}{s^{n+1}}\mathcal{L}\{f(t)\}$$这个公式可以用来简化一些高阶积分的计算。
4. 积分上限的Laplace变换当我们要计算形如$\int_{0}^{t}f(x)dx$这样的积分的Laplace变换时，可以使用下列公式：$$\mathcal{L}\{\int_{0}^{t}f(x)dx\}=\frac{F(s)}{s}$$其中$F(s)$为$f(t)$的Laplace变换。
p-laplace方程 博士文
p-laplace方程博士文英文回答：The p-Laplace equation is a generalization of the Laplace equation, which is a second-order partial differential equation commonly used in physics and engineering to describe the behavior of scalar fields. The p-Laplace equation is given by:Δ_pu = div(|∇u|^(p-2) ∇u) = 0。
where Δ_pu denotes the p-Laplacian operator, ∇u is the gradient of u, |∇u| is the magnitude of the gradient, and p is a positive constant. The p-Laplace equation arises in various areas of mathematics and physics, including elasticity, fluid mechanics, image processing, and geometric analysis.One important property of the p-Laplace equation isthat it exhibits a nonlinear behavior, unlike the Laplaceequation. This nonlinearity arises from the power p in the equation, which affects the growth and decay of the solution. For example, when p = 2, the p-Laplace equation reduces to the Laplace equation, and the solution behaves linearly. However, for p ≠ 2, the solution can exhibit different types of nonlinear behavior, such as concentration or diffusion effects.中文回答：p-Laplace方程是Laplace方程的一种推广形式，Laplace方程是一种常见的二阶偏微分方程，常用于描述物理和工程中标量场的行为。
一类分数阶p-Laplace方程解的对称性与单调性
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通讯作者 ：瞿 萌，ｑｕｍｅｎｇ＠ｍａｉｌ．ａｈｎｕ．ｅｄｕ．ｃｎ。收稿 日期 ：２０１８—１—１５ 基金项 目：国家 自然科学基金（１１４７１０３３）。
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——Βιβλιοθήκη ————
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吴 家 彦 ，杨 柳 ，瞿 萌
（安 徽 师 范 大 学 数 学 与统 计 学 院，安徽 芜 湖 ，２４１００２）
摘要 ：用移动平 面法 讨论一类单位球上非线性项 带奇性权函数的分数阶 ｐ－Ｌａｐｌａｃｅ 方程经典解 的对称性和单
调性 得到方程 的解是关于原 点径 向对称和单调递减 的。
关键词 ：分数 阶ｐ．Ｌａｐｌａｃｉａｎ；移动平面法；径 向对称；单调递减
Ａｂｓｔｒａｃｔ：Ｂｙ ｔｈｅ ｍ ｏｖｉｎｇ ｐｌａｎｅ ｍ ｅｔｈｏｄ，ｉｔ ｉｓ ｏｂｔａｉｎｅｄ ｔｈａｔ ｔｈｅ ｓｏｌｕｔｉｏｎ ｔｏ ａ ｃｌａｓｓ ｏｆ ｆｒａｃｔｉｏｎａｌ Ｐ。Ｌａｐｌａｃｅ ｅｑｕａｔｉｏｎ ｗｉｔｈ ｓｉｎｇｕｌａｒ ｗｅｉｇｈｔ ｎｏｎｌｉｎｅａｒ ｔｅｒｍ ｉｓ ｒａｄｉａｌｌｙ ｓｙｍｍｅｔｒｉｃ ａｎｄ ｍｏｎｏｔｏｎｅ ｄｅｃｒｅａｓｉｎｇ ａｂｏｕｔ ｔｈｅ ｏｒｉｇｉｎ． Ｋ ｅｙ ｗｏｒｄｓ：ｆｒａｃｔｉｏｎａｌ ｐ—Ｌａｐｌａｃｉａｎ；ｍｏｖｉｎｇ ｐｌａｎｅ ｍｅｔｈｏｄ；ｒａｄｉａｌ ｓｙｍ ｍｅｔｒｙ；ｍｏｎｏｔｏｎｉｃｉｔｙ
p-Laplace方程解的存在性
硕士学位论文（高校教师）p-Laplace方程解的存在性 )(xEXISTENCE OF SOLUTIONS OF )p-LAPLACE EQUATION(x房维维哈尔滨工业大学2009年6月中图分类号:O175.2 学校代码：10213 UDC：517.9 密级：公开理学硕士学位论文（高校教师）(xp-Laplace方程解的存在性)硕士研究生: 房维维导 师: 付永强教授申 请 学 位: 理学硕士学 科: 基础数学所 在 单 位: 哈尔滨师范大学阿城学院答 辩 日 期: 2009年6月授予学位单位: 哈尔滨工业大学Classified Index: O175.2U.D.C: 517.9Dissertation for the Master Degree in ScienceEXISTENCE OF SOLUTIONS OFp-LAPLACE EQUATION(x)Candidate:Weiwei FangSupervisor:Prof. Yongqiang FuAcademic Degree Applied for:Master of ScienceSpecialty:Fundamental Mathematics Affiliation: Harbin Normal University AchengCollegeDate of Defence:June, 2009Degree-Conferring-Institution:Harbin Institute of Technology摘要对非标准增长条件的-Laplace 方程问题的研究是近年来发展起来的一个新的研究课题。
由于Laplace 方程和)(x p p -Laplace 方程的研究方法已经不再适用于-Laplace 方程， 所以目前对-Laplace 方程的研究只有很少的成果出现， 因此对这类问题的研究具有广泛的理论与实际意义。
对-Laplace 方程的研究， 有很多不同的方法。
自动控制理论词汇表
Glossary for Feedback Control of Dynamic Systems自动控制理论词汇表Chapter 1thermostat n.恒温器predictive control 预测控制power generation plant 发电厂micron n.微米cell phone 移动电话jumbo jet 巨型喷气式客机block diagram 方框图actuator 执行机构process n.过程feedback n.反馈plant n.被控对象mph=mile per houropen-loop 开环closed-loop 闭环throttle n.油门gain n.增益orifice n.孔、小孔controlled variable 被控变量error n.误差incubator n.孵化器flue n.烟道chronicle n.编年史、年代记录conical a.圆锥体的mill wright 技工、造水车工匠inertia n.惯性、惯量oscillate about…在….周围振荡reference input 参考输入prescribed direction 预定的方向actual direction 实际方向flyball n.飞球governor n.控制器、调节器、总督、省长equilibrium n平衡点differencial equation 微分方程characteristicequation 特征方程third-order 三阶polynomial n.多项式state variable 状态变量distortion n.畸变complex variable 复变量methdology n.方法论proportional a. 比例的、成比例的integral a.积分的derivative a.微分的stochastic a.随机的servomechanism n.伺服机构calculus n.微积分ubiquitous a.到处存在的、普遍存在的radar-tracking 雷达跟踪SISO systems 单输入单输出系统Laplace transform 拉普拉斯变换pole n.极点zero n.零点transfer function传递函数trajectory optimization 轨迹优化root locus 根轨迹specifications n.指标、规格、规范discrete-data 离散数据sampled-data 采样数据performance n.性能Chapter 2desired reference variable 期望参考变量prototype n.原型system identification 系统辨识time response 时域响应step input 阶跃输入defer n.推迟、延期vector n.向量、矢量slug n.斯（勒格），质量单位（32.2磅）impart n.赋予、传授、告知heavy line 粗实线dashed line 虚线coordinate n.坐标numerator n.分子denominator n.分母suspension n.悬架、悬挂deflection n.偏移、偏转、偏差displacement n.位移shock absorber 减震器、缓冲器dashpot n.缓冲器bump n. vt.颠簸bounce n. vt.反弹、跳跃moment of inertia 转动惯量、惯性矩attitude n.姿态、姿势antenna n.天线perpendicular n.垂直线 a.垂直的asymmetry a.不对称的torque n.转矩resonant n.谐振、共振damper n.阻尼器prudent a.谨慎的，有远见的，精打细算的anti-alias 抗混频operational amplifer 运算放大器passive circuit 无源电路Kirchhoff’s current law 基尔霍夫电流定律algebraic sum 代数和summer n.加法器integrator n.积分器tesla n.特斯拉(磁通量单位)louderspeaker n.扬声器bobbin线轴，线筒stator n.定子rotor n.转子back emf 反电势maze n.曲径，迷宫specific heat 比热spatially ad.空间地hydraulic a.液压的、水力学的gimbal n.平衡环，万向接头nozzle n.喷嘴grooming n.修饰，美容piston n.活塞porous a.可渗透的，多孔的laminar a.多层的、层流的n.层流turbulent a.湍流的Chapter 3linear time-invariant systems 线性时不变（定常）系统signal flow graph 信号流图simulation n.仿真frequency-response 频率响应superposition n.叠加convolution n.卷积inpulse-response 脉冲响应unit step function 单位阶跃函数root-locus 根轨迹stability properties 稳定性特性principles linear algebra 线性代数原理state variable methods 状态变量法matrix n..矩阵nonlinear n.非线性mathematical mode 数学模型trivial a.琐碎的、不重要的linearize vt.线性化operating point 工作点state-space 状态空间partial differential equations 偏微分方程equilibrium n.平衡点complex frequency variables 复频率变量zero initial conditions 零初始条件steady-state 稳态ramp input 斜坡输入dc gain 直流增益inverse Laplace transform 逆拉氏变换partial fraction expansion 部分分式展开rantional a.有理的residue n.余式unilateral a.单边的convergence n.收敛final-value theorem 终值定理homogeneous differential equation 齐次微分方程ordinary differential equation 常微分方程overall transfer function 总的传递函数“loading” effect 负载效应cascade blocks 方框串联（级联）to reduce 化简eliminating 消去equivalent a.等效的simplification n.化简、简化integrodifferential a.积分-微分的time constant 时间常数imaginary axis 虚轴damping ratio n.阻尼比natural undamped frequency 自然无阻尼频率overdamped a.过阻尼critically damped n.临界阻尼rectangular coordinate 直角坐标oscillatory a.振荡的transient response 瞬态响应overshoot n.超调量delay time 延迟时间peak time 峰值时间rise time 上升时间settling time 调节时间steady state 稳态characteristic equation 特征方程RHP(Right Half-Plane) 右半平面elevator n.飞机升降舵，飞机升降仪，电梯nonminimum-phase 非最小相位diverge v.发散、分歧asymptotically stable 渐进稳定stability n.稳定性absolute stability 绝对稳定性relative stability 相对稳定性stability criterion 稳定性判据equilibrium state 平衡状态product n.乘积coefficient n..系数nagtive feedback 负反馈positive feedback 正反馈unity feedback system 单位负反馈系统reduction n.化简simultaneous a.联立的common factor 公因子expedient a.权宜的，有用的attenuate v.变弱，衰减，变细，变薄，稀释cofactor n.公因子Routh stability criterion 劳斯稳定性判据determinant n. 行列式tune v.调节retune v.再调节pseudorandom-noise 伪随机噪声signal-to-noise ratio 信-噪比Mason Gain Formula 梅森（增益）公式term n.术语signal flow graph 信号流图nodepathenvelope n.包络线dominant root 主导极点Chapter 4steady-state 稳态with respect to 关于….deviation n.偏离steady-state error 稳态误差load torque 负载转矩viscous friction 粘性摩擦repeater n.中继器drift v.漂移fidelity n.准确性，忠实，忠诚parabolic antenna 抛物线天线position error constant. 位置误差常数velocity error constant 速度误差常数robust property 鲁棒性shaft n.轴tachometer n.转速计inductance n.电感sampled v.采样quantized v.量化extrapolate v.预测，推测trapezoid n.梯形，不等边四边形vertices n.顶点order n.阶次，数量级proportional control 比例控制derivative control 微分控制sinusoidal a.正弦parameter n.参数Chapter 5root-locus method 根轨迹法monic a.首一的feedforward a.前向的denominator n.分母numerator n.分子quadratic n.二次项branch n.分支factored a.分解的asymptote vt.渐进n.渐进线division n.除法vantage point 有利地位，观点imaginary part 虚部breakaway point 分离点common denominator 公分母conjugate pairs 共扼对multiplicity n.多重，多数trial and error 凑试（法）spirule n.螺旋尺intersection n.交汇symmetrical a.对称的magnitude condition 幅值条件angle condition 相角条件phase condition 相角条件origin n.起始点terminus n.终点angle of departure 分离角、出发角angle of arrival汇合角、到达角cubic a.三阶的、立方的quartic a.四次的remainder n.留数、余数remainder theorem 留数定理taking the limit 取…极限synthetic division 综合除法dominant root 主导根compensator n.补偿器azimuth n.地位角、地平经度inertial guidance 惯性导航constant term 常数项symmetrical with respect to 关于…对称trial point 试验点terminate vt.终止于first differentiate 一阶微分real parts实部imaginary part 虚部lag compensator 滞后补偿器lead compensator 超前补偿器spill over 溢出，无法容纳autopilot n.自动导航trim v.n.使整齐，微调trim tab 平衡调整片margin n.裕量iteration n.重复、循环、迭代intact a.完好无缺的，原封不动的Chapter 6frequency response 频率响应rendered v.使成为，提供，报答，着色; 执行ratio of the magnitudes 幅值比bandwidth n.带宽resonant peak 谐振峰值low-pass filter低通滤波器sanity n.神智健全，头脑清楚，健全tangent n.正切、切线reciprocal a.互补的，相互的，互惠的phase difference 相角差transport lag 传输延迟irrational factor 非有理因子phase shift 相位移动moduli n.模(复数)poke vt.戳、刺、捅drudgery n.苦工、单调乏味的工作logarithmic coordinate 对数坐标semilog n.半对数decibel n.分贝decade n.十倍量程octave n.八倍频程、八度、八阶asymptotic behavior 渐进行为dotted n.虚线break frequency 转折频率corner frequency 转折频率slope n.斜率20dB/decade 20分贝/十倍频程superimpose vt.迭加polar plot 极坐标图pass function 旁路函数servomotor-amplifier 伺服电机-放大器angular velocity 角速度minimum phase 最小相位tilt angle 倾斜角lateral force 侧面力、横向力perceived velocity 可察觉的速度croseover frequency 穿越频率appendage n.附件、备件Nyquist criterion 奈奎斯特判据Semi-graphical 半图形Nyquist plot 奈奎斯特图Bode diagram 伯德图positive real part 正实部necessary and sufficient condition 充分必要条件left half of the s-plane s平面左半平面formidable a. 可怕的、令人生畏的determinant 行列式pole-zero cancellation 零极点相消rational functions 有理函数quotient n. 商、份额multi-loop control system 多环控制系统encircled vt. 环绕enclosed vt. 包围closed path 闭合路径counterclockwise a.逆时针的clockwise a.顺时针的encirclement n. 环绕enclosure n. 包围contour n.围线，轮廓线argument principle 幅角原理complex variable 复变量single-valued rational function 单值有理函数analytic a.解析的Nyquist path 奈奎斯特路径singularity n.奇异（点、值）semicircle n.半圆artifice n.技巧、技能gain margin 增益裕量phase margin 相角裕量vicinity n.邻近compromise n.折中，妥协trapezoidal a.梯形的iterate v.重复、循环、迭代bracket v.放在括号内，归入一类，包含octave n.八个一组的事物，八度enumerate v.数，点detrimental a.有害的，不利的threshold n.阈值Chapter 8sampling n.采样sample period 采样周期aliasing n.混频，别名inherent a.内在的z transform Z变换radar tracking system 雷达跟踪系统discrete period 离散周期discrete equivalent 离散等效digitization n.数字化recursive a.递归的，循环的difference equation 差分方程sample rate 采样速率sampler 采样器zero-order holder 零阶保持器inverse Z transform 反Z变换、逆Z变换long division 长除法unit circle 单位圆overlap n.重叠rephrase v.重新措辞，改述extrapolate v.预测，推测alleviate v.减轻，使 ... 缓和judicious a.明智的，贤明的，审慎的fictitious a.假想的，虚伪的impulse transfer function 脉冲传递函数piecewise-continuous 分段连续的pseudo-continuous-time 准连续时间Pade approximation Pade 近似Fourier analysis 傅立叶分析modulation n.调制Fourier transform 傅立叶变换spurious a.寄生的、伪的、假的ideal sampler 理想采样器impulse train 脉冲列、脉冲串transcendental a.超自然的、超常的rational function 有理函数closed form 封闭形式degree n.阶denominator n 分母numerator n 分子initial value 初始值identical a.相等的starred a.打星号的impulse response transfer function 脉冲响应传递函数uniformly spaced 均匀分布map into 影射、映射circles of radius 圆弧multiple-sheeted surfaceRiemann surface 黎曼曲面radial ray 射线by virtue of 借助、凭借、依靠….（的力量）logarithmic spiral 对数螺旋intersection n.相交power series 幂级数sampling instant 采样时刻natural logarithm 自然对数rationalizing 有理化cascading property 串联（级联）特性attenuation factor 衰减因子warp vt.使弯曲、使变形tune vt.调节、调整cross-hatched vt.用交叉线画出（图画上）阴影performance specification 性能指标trial-and-error approach 试凑法bilinear n.双线性Chapter 9equilibrium point 平衡点neighborhood n.邻域saturate n.饱和robotic n.机器人学heuristic a.启发式的，搜索式的sinusoidal a.正弦的sinusoid n.正弦harmonic a.谐波的describing-function 描述函数static nonlinearity 静态非线性dynamic nonlinearity. 动态非线性periodic response 周期响应phase-plane 相平面catastrophe n.灾难、浩劫shaky a.不稳定的，不可靠的scalar function 标量函数Liapunov function 李亚普诺夫函数linearization n.线性化inverse nonlinearity 可逆非线性perturbation n.摄动operating point 工作点eigenvalue n.特征值bearing n.轴承levitate v.浮动，使漂浮，使悬浮turbo n.汽轮机deviation n.偏差rigid link 刚性连接regime n.情形，体制dead-zone 死区viscous friction 粘性摩擦coulomb friction 库仑摩擦relay n.继电（特性）limit cycle 极限环deflect v.使偏，使歪windup n.终结，结束akin a.同类的，相似的odd function 奇函数backlash n.齿轮间隙magnetic hysteresis 磁滞coincident a.重合的，一致的on/off system 通断（控制）系统superposition n. 迭加sub-harmonic a.谐波的magnetic flux 磁通iron-cored coil 铁芯线圈stiction n. 静摩擦力autonomous a.自治的hypersphere n. 超球stability in the sense of Liapunov 李亚普诺夫意义下的稳定性asymptotically stable 渐进稳定monotonically stable 单调稳定origin n. 原点globally stable 全局稳定locally stable 局部稳定electronic oscillator 电子谐振器Van der Pol’s differential equation 范德波尔微分方程nonsinusoidal waveform 非正弦波形rated voltage 额定电压phase variable 相变量phase portrait 相图perpendicular a. 垂直的、正交的、成直角的Taylor series 泰勒级数increment n.增量Euler method 欧拉法singular point 奇异点。
拉普拉斯变换相关书籍英文
Laplace Transform: A Comprehensive StudyIntroductionLaplace Transform is a powerful mathematical tool used in various fields such as engineering, physics, and mathematics. It provides a way to transform differential equations into algebraic equations, making them easier to solve. In this article, we will explore the concept of Laplace Transform in detail and discuss its applications.Basic Definition and Properties1.The Laplace Transform is defined for a function f(t) defined for t≥ 0 by the equation: > F(s) = L{f(t)} = ∫[0, ∞] e^(-st) f(t) dtHere, s is a complex variable and F(s) is the Laplace Transform of f(t).2.Linearity Property: For any constants a and b, and functions f(t)and g(t), we have: > L{af(t) + bg(t)} = aF(s) + bG(s)This property allows us to apply Laplace Transform to linearcombinations of functions.3.Shifting Property: If F(s) is the Laplace Transform of f(t),then: > L{e^(-at)f(t)} = F(s + a)This property enables us to manipulate the time domain bymultiplying the function with an exponential term.4.Differentiation Property: If F(s) is the Laplace Transform of f(t),then: > L{df(t)/dt} = sF(s) - f(0)This property allows us to transform derivatives in the timedomain into algebraic expressions in the Laplace domain.Inverse Laplace Transform1.The Inverse Laplace Transform of F(s) is denoted by L^{-1}{F(s)}and is defined as: > f(t) = L^{-1}{F(s)} = (1/2πi) ∫[-i∞, i∞]e^(st) F(s) dsThis transformation allows us to retrieve the original functionf(t) from its Laplace Transform F(s).mon Pairs: There are several common Laplace Transform pairsthat are frequently used for solving differential equations. Some examples include:–L{1} = 1/s, L^{-1}{1/s} = 1–L{t^n} = n!/(s^(n+1)), L{-1}{n!/(s(n+1))} = t^nThese pairs provide a shortcut for transforming common functionsin the time domain.Applications of Laplace Transform1.Solving Ordinary Differential Equations (ODEs): Laplace Transformis extensively used to solve linear ordinary differentialequations with constant coefficients. By applying LaplaceTransform to both sides of the equation, we can convert theequation into an algebraic equation that is easier to solve. Once solved, the inverse Laplace Transform is applied to obtain thesolution in the time domain.2.Circuit Analysis: Laplace Transform is widely used in electricalcircuit analysis. By transforming the circuit equations into theLaplace domain, we can analyze the behavior of the circuits interms of complex impedance and transfer functions. This simplifies the analysis of complex circuits and facilitates the design ofcircuits for specific applications.3.Control Systems: The Laplace Transform plays a crucial role in theanalysis and design of control systems. By transforming theequations governing the dynamics of the system, we can analyzestability, transient response, and steady-state behavior. Laplace Transform also allows us to design controllers and compensators to achieve desired system performance.4.Signal Processing: Laplace Transform is utilized in signalprocessing to analyze and manipulate signals. By transformingsignals into the frequency domain, we can analyze their spectralcharacteristics and apply various filters or modifications.Laplace Transform is particularly useful for analyzing continuous-time signals and systems.ConclusionIn conclusion, Laplace Transform is a fundamental mathematical tool that has numerous applications in various fields. Its ability to transform differential equations into algebraic equations simplifies problem-solving and analysis. By understanding the basic properties and applications of Laplace Transform, we can effectively solve complex problems in engineering, physics, and mathematics.。
拉普拉斯变换英语作文
拉普拉斯变换英语作文Laplace Transform。
The Laplace transform is a powerful tool in the field of mathematics and engineering. It is used to solve differential equations and is particularly useful in the analysis of linear time-invariant systems. The Laplace transform is named after Pierre-Simon Laplace, a French mathematician who made significant contributions to the field of mathematics.The Laplace transform of a function f(t) is defined as the integral of the function multiplied by the exponential function e^(-st), where s is a complex number. The Laplace transform of f(t) is denoted as F(s) and is defined as:F(s) = ∫[0 to ∞] f(t) e^(-st) dt。
The Laplace transform has many important properties that make it a valuable tool in the analysis of linearsystems. One of the most important properties of the Laplace transform is its linearity. This means that the Laplace transform of a linear combination of functions is equal to the linear combination of their individual Laplace transforms. Mathematically, this property can be expressed as:L{af(t) + bg(t)} = aF(s) + bG(s)。
拉普拉斯和拉格朗日函数的关系
拉普拉斯和拉格朗日函数的关系Lagrange and Laplace functions are two important mathematical concepts that have significant connections and applications in various areas of science and engineering. While they both deal with optimization problems, they have distinct characteristics that set them apart from each other. The Lagrange function, named after the Italian mathematician Joseph-Louis Lagrange, is commonly used in the field of calculus of variations to find extrema of a function subject to constraints. On the other hand, the Laplace function, named after the French mathematician Pierre-Simon Laplace, is used in probability theory and statistics to analyze random variables and stochastic processes.The relationship between the Lagrange and Laplace functions lies in the concept of optimization and constraint satisfaction. In optimization problems, the Lagrange function is used to formulate the constrained optimization problem, where the objective function is maximized or minimized subject to a set of constraints. The Lagrange multiplier, which is a key component of the Lagrange function, helps in incorporating the constraints into the optimizationproblem and finding the extremal points. The Laplace function, on the other hand, is used in probability theory to analyze the probability distribution of random variables and stochastic processes. It provides a way to model and study the behavior of random phenomena and make predictions based on statistical analysis.Despite their differences in application and methodology, the Lagrange and Laplace functions share common principles in mathematical optimization and analysis. Both functions aim to find the optimal solution to a given problem by considering various constraints and parameters that influence the outcome. They also play a crucial role in modeling real-life scenarios and making informed decisions based on mathematical principles. The Lagrange function, for example, is widely used in economics, physics, and engineering to optimize resource allocation, design efficient systems, and solve complex problems with multiple constraints. Similarly, the Laplace function is essential in statistics, finance, and insurance to analyze risks, predict outcomes, and make probabilistic assessments based on historical data.In addition to their practical applications, the Lagrange and Laplace functions have theoretical significance in mathematics and contribute to the development of mathematical theories and frameworks. The Lagrange function, for instance, is a fundamental concept in the calculus of variations, a branch of mathematics that deals with functionals and extremal problems. It provides a powerful tool for solving variational problems and studying optimization in continuous spaces. The Laplace function, on the other hand, is central to probability theory and stochastic processes, which are essential for understanding random phenomena and modeling uncertainties in real-world situations. By combining theoretical principles with practical applications, the Lagrange and Laplace functions offer a comprehensive approach to problem-solving and decision-making in various domains.Overall, the relationship between the Lagrange and Laplace functions highlights the interdisciplinary nature of mathematics and its applications in different fields. By understanding the connections and differences between these functions, mathematicians, scientists, and engineers can leverage their unique properties to solve complex problems, optimize systems, and make informed decisions. Whetherit is maximizing the efficiency of a manufacturing process using the Lagrange function or analyzing the risk factors in a financial portfolio using the Laplace function, the integration of mathematical principles and practical applications is essential for advancing knowledge and driving innovation in modern society. As we continue to explore the depths of mathematics and its applications, the synergy between the Lagrange and Laplace functions will continue to play a crucial role in shaping our understanding of optimization, probability, and uncertainty in a rapidly changing world.。
Laplace方程Cauchy问题求解的一种正则化方法
收稿日期:2023-04-14;修订日期:2023-05-26作者简介:祝志栋(1982 ),男,硕士,讲师,主要从事数学物理方程反问题研究㊂基金项目:2021年福建省中青年教师教育科研项目(科技类)(JAT210498)㊂第41卷㊀第5期2023年10月江㊀㊀西㊀㊀科㊀㊀学JIANGXI㊀SCIENCEVol.41No.5Oct.2023㊀㊀doi :10.13990/j.issn1001-3679.2023.05.002Laplace 方程Cauchy 问题求解的一种正则化方法祝志栋,石红岩,时秀娟(仰恩大学数学系,362014,福建,泉州)摘要:利用Tikhonov 正则化方法求解圆环域上Laplace 方程Cauchy 问题,提出了利用Morozov 相容性原理确定正则化参数的改进模型函数方法,并给出了迭代算法㊂数值算例验证了该方法的有效性㊂关键词:Laplace ;正则化参数;模型函数;Morozov 方程中图分类号:O175.25㊀㊀㊀㊀文献标识码:A㊀㊀㊀㊀文章编号:1001-3679(2023)05-825-06A Regularization Method for Solving the Cauchy Problemof Laplace EquationZHU Zhidong,SHI Hongyan,SHI Xiujuan(Department of Mathematics,Yang en University,362014,Quanzhou,Fujian,PRC)Abstract :The Tikhonov regularization method was utilized to solve the Cauchy problem of Laplace e-quation on annular domain.An improved model function method based on Morozov discrepancy prin-ciple was proposed to determine the regularization parameter ,and an iterative algorithm was presen-ted.Numerical examples verify the effectiveness of the proposed method.Key words :Laplace;regularization parameter;model function;Morozov equation0㊀引言Laplace 方程Cauchy 问题是一类在地球物理㊁无损探伤㊁医学成像[1]等多个领域有着广泛应用的数学物理问题㊂这一类问题大都是不适定的,求解该类问题的本质困难在于解的不稳定性,观测数据的微小误差将会引起未知解的急剧变化[2],因而难以使用古典的数值方法求解[3]㊂对该类问题的求解国内外目前已有大量的工作,包括基本解方法[4]㊁拟逆方法[5-6]㊁磨光化方法[7-9]㊁Fourier 方法[10]㊁中值差分法[11]等㊂其中应用比较广泛的为Tikhonov 正则化方法,而正则化参数的选取对于该方法的有效性非常重要[3]㊂文献[12-14]提出了确定正则化参数的模型函数方法,并验证了其在处理不适定问题中的有效性㊂本文将该方法加以改进,以期得到比较理想的正则化参数,并给出算例㊂为了方便检验该方法的有效性,选定在圆环区域上求解该问题㊂1㊀圆环域上Laplace 方程Cauchy 问题的正则化方法1.1㊀解的存在性考虑一类特殊区域,即圆环域上的问题:1ρ∂∂ρ(ρ∂u ∂ρ)+1ρ2∂2u∂θ2=0,㊀㊀(ρ,θ)ɪ(a ,b )ˑ(0,2π)(1)u (b ,θ)=f (θ),θɪ(0,2π)(2)∂u∂ρ(b,θ)=g(θ),θɪ(0,2π)(3)该问题由容易测量的外边界数据反演不易测量的内边界函数值,这是一个不适定的问题[3]㊂通过引进边界数据u(a,θ)=h(θ),θɪ(0,2π)(4)代替式(3),则得到一个适定的问题,其解为: u(ρ,θ)=ʏ2π0[ln b-lnρ2π(ln b-ln a)+ð n=1 a-nρ-n-a-n b-2nρnπ(a-2n-b-2n)cos n(τ-θ)]h(τ)dτ+ʏ2π0[lnρ-ln a2π(ln b-ln a)+ð n=1a-2n b-nρn-b-nρ-nπ(a-2n-b-2n) cos n(τ-θ)]f(τ)dτ,(5)在式(5)两边关于ρ求导后再取ρ=b,由Cauchy 数据(3)可得Kh=L(f,g),(6)其中:L(f,g)(θ):=g(θ)-ʏ2π0[12πb(ln b-ln a)+ð n=1nb-1a-2n+nb-2n-1π(a-2n-b-2n)cos n(τ-θ)]f(τ)dτ, (Kh)(θ):=ʏ2π0[12πb(ln a-ln b)+ð n=1 2nb-n-1a-nπ(b-2n-a-2n)cos n(τ-θ)]h(τ)dτ㊂原Cauchy问题(1)~(3)有解的必要条件为Cauchy数据(f,g)满足L(f,g)ɪRange(K)㊂1.2㊀Tikhonov正则化方法若外边界上的数据为扰动数据fδ,gδ,则方程(6)的右端项变为L(fδ,gδ)㊂定义:k^(τ,θ):=12πb(ln b-ln a)+ð n=1 nb-1a-2n+nb-2n-1π(a-2n-b-2n)cos n(τ-θ)㊂假定外边界r=b上的有误差的Cauchy数据由下面的方式给定fδ(θ)=f(θ)+1πδsinθ,gδ(θ)=g(θ)+ 1πδcosθ,(7)则它在整个[0,2π]上的误差为fδ(㊃)-f(㊃) 2L2[0,2π]=δ2π2ʏ2π0sin2θdθ δ2, gδ(㊃)-g(㊃) 2L2[0,2π] δ2㊂注意到L(fδ,gδ)(θ)-L(f,g)(θ)=gδ(θ)-g(θ)-ʏ2π0k^(τ,θ)[fδ(τ)-f(τ)]dτ,从而方程Kh=L(f,g)的右端数据的误差是(δ∗)2:= L(fδ,gδ)-L(f,g) 2L2[0,2π]=ʏ2π0 [L(fδ,gδ)(θ)-L(f,g)(θ)]2dθ 2 gδ-g 2L2[0,2π]+2ʏ2π0[ʏ2π0k^(τ,θ)[fδ(τ)-f(τ)]dτ]2dθ 2 gδ-g 2L2[0,2π]+2 fδ-f 2L2[0,2π] k^ 2L2([0,2π]ˑ[0,2π])=C20δ2㊂下面由此来求h(τ)的近似值hδ(τ)㊂利用Tikhonov正则化方法构造K-1的正则化解算子Rα,即求hδ(τ),使得下面Tikhonov泛函值极小:J^α(h)=12 Khδα-L(fδ,gδ) 2L2[0,2π]+α2 hδα 2L2[0,2π],(8)其中α称为正则化参数㊂关于Tikhonov泛函有如下结论[3]:引理1:设K是Hilbert空间XңY的有界线性算子,则1)J^α(h)在X上存在唯一的极小元hδα;2)hδα满足αhδα+K∗Khδα=K∗L(fδ,gδ)(9)其中α>0,K∗为K的伴随算子㊂由于K是<L2,L2>在L2内积下的自伴算子,从而式(9)可写为αhδα+K2hδα=KL(fδ,gδ)㊂(10)在该方法中,正则化解算子Rα和正则化解hδα分别为:Rα=(αI+K2)-1K,hδα=RαL(fδ,gδ)= (αI+K2)-1KL(fδ,gδ)㊂由于δ∗= L(fδ,gδ)-L(f,g) L2[0,2π] C0δ,则可得到正则化解和精确解的误差为hδα-h = RαL(fδ,gδ)-h RαL(fδ, gδ)-RαL(f,g) + RαL(f,g)-h Rα L(fδ,gδ)-L(f,g) + RαKh-h δ∗ Rα + RαKh-h ㊂因此,问题的关键在于选取合适的正则化参数㊃628㊃江㊀西㊀科㊀学2023年第41卷α,使得δ∗R α + R αKh -h 极小,而正则化参数的选取是否合适是决定正则化解和精确解之间的误差水平的关键因素㊂下面利用Morozov 相容性原理确定正则化参数㊂该原理通过求解方程(9)的最小模解来确定α[3]㊂正则化参数α(δ)由Kh α(θ)-L (f δ,g δ) 2L 2[0,2π]=(δ∗)2(11)确定,其中α,h α(θ)满足方程(9),从而式(11)为求解α的隐函数方程㊂可以证明式(11)有解[3]:引理2:如果L (f δ,g δ)⊈Ker (K ∗),J (0)<12(δ∗)2<J (1),则式(11)在αɪ(0,1]存在唯一的根α∗ɪ(0,1]㊂1.3㊀正则化参数选取的模型函数方法(拟)Newton 迭代法为解决该问题的常用方法,但该方法只有选取合适的迭代初值才能得到比较理想的结果,不易于数值实现且计算量比较大㊂模型函数方法基于Morozov 相容性原理构造变量为正则化参数的模型函数,来近似逼近正则化泛函的极小值,通过求解模型函数所满足方程来确定正则化参数[12]㊂模型函数方法不仅给出了Morozov 方程的解法,并且由此得到的正则化参数在某种条件下可以收敛到某个最优的正则化参数[3]㊂下面利用该方法解决本文的不适定问题㊂对Kh α=L (f δ,g δ),不妨记L (f δ,g δ)=L δ㊂由式(11)可得到J (α)=12Kh α(θ)-L δ 2L 2[0,2π]+α2h α(θ) 2L 2[0,2π]=12Kh α(θ) 2L 2[0,2π]-R (Kh α(θ),L δ)+12 L δ 2L 2[0,2π]+α2 h α(θ) 2L 2[0,2π]㊂另一方面,在式(9)两边用h α(θ)作内积得:(Kh α(θ),L δ)= Kh α(θ) 2L 2[0,2π]+α h α(θ)2L 2[0,2π]=R (Kh α(θ),L δ),从而有J (α)=12 L δ 2L 2[0,2π]-12Kh α(θ) 2L 2[0,2π]-α2 h α 2L 2[0,2π]㊂(12)在式(8)的两边关于α求导,并利用式(9)得到:J ᶄ(α)=(Kh α(θ)-L δ,K (h α(θ))')+α(h α(θ),(h α(θ))')+12h α 2L 2[0,2π]=12h α 2L 2[0,2π],将上式代入式(12),得到:2J (α)+2αJ ᶄ(α)+ Kh α(θ) 2L 2[0,2π]= L δ 2L 2[0,2π]㊂(13)下面通过引进近似的模型函数来转化 Kh α(θ) 2L 2[0,2π]㊂对于α>0,存在C ~(α)>0,使得下式成立(Kh α(θ),Kh α(θ))=C ~(α)(h α(θ),h α(θ))㊂不妨令㊀(Kh α(θ),Kh α(θ))≃T (h α(θ),h α(θ))=2TJ ᶄ(α),其中T 为常数㊂将此意义下的模型函数记为m (α),于是由(13)知m (α)满足αm ᶄ(α)+m (α)+Tm ᶄ(α)=12L δ 2,(14)求解(14)可得到含有参数C ,T 的m (α)表达式m (α)=12 L δ 2+C T +α,(15)在不同的α的邻域内以及C ,T ,J (α)可由m (α)近似得到㊂下面给出确定2个参数C ,T 的模型函数方法,即:给定α0>0和ε>0,令k =0:1)解式(9)得到h αk ,计算J (αk )和J '(αk ),然后由表达式(15),更新T k 和C k :m k (αk )=12 L δ 2+C k T k +αk=J (αk ),(16)m 'k(αk )=-C k(T k +αk )2=J ᶄ(αk ),(17)由上式可以确定m (α)中的T k 和C k :T k =Kh αk2h αk2,C k =-( Kh αk 2+αk h αk 2)22 h αk2㊂(18)2)设第k 次的模型函数:m k (α)=12 L δ 2+C k T k +α,(19)在Morozov 方程中求解αk +1:㊃728㊃第5期㊀㊀㊀㊀㊀㊀祝志栋等:Laplace 方程Cauchy 问题求解的一种正则化方法G k (α):=m k (α)-αm 'k (α)=12(δ∗)2㊂(20)3)如果|αk +1-αk | ε,则停止;否则令k :=k +1,回到1)㊂显然,对α>0:m 'k(α)=-C k(T k +α)2>0.m ᵡk(α)=2C k(T k +α)3<0,(21)由式(20)得到:G 'k (α)>0㊂由式(15)可得T k 为引进模型函数时的近似关系(Kh α,Kh α)≃T (h α,h α)㊂这表明,由此迭代更新参数C ,T 最终得到的m (α)应该是J (α)在上式意义下的一个近似㊂关于式(20)解的存在性有如下结论:定理1:如果L δ⊈Ker (K ∗),J (0)<12(δ∗)2<J (1)㊂假定G k (αk )>12(δ∗)2且确定G k (α)的αk 充分小,则式(20)存在唯一解αk +1[3]㊂该定理给出了Morozov 方程有解的一个必要条件㊂只有选择合适的迭代初值,才能使该迭代算法收敛到Morozov 方程的准确解㊂这说明该迭代算法有其局限性,需要做出改进才能得到比较理想的结果㊂1.4㊀模型函数方法的改进对函数G k (α)做如下改进:G ^k (α):=G k (α)+λk (G k (α)-G k (αk )),αɪ[0,αk ],(22)其中λk 是合适的松弛常数㊂由于G 'k (α)>0,其中λk (G k (α)-G k (αk ))的符号由λk 决定(与λk 异号),再用方程G ^k (α)=12(δ∗)2(23)来代替式(20),其中λk 的选取需满足一定的要求,即使得式(23)存在唯一解㊂由于G ^k (αk )=G k (αk )>12(δ∗)2,故只要G ^k (0)<12(δ∗)2且G ^k (α)单调上升即可得到解的唯一性㊂松弛常数λk 可由下式给出:G ^k (0)=G k (0)+λk (G k (0)-G k (αk ))=-λ^(δ∗)2,其中^λɪ(0,1/2)是任意给定的常数,即满足^Gk (0)<12(δ∗)2㊂由此解得λk =G k (0)+^λ(δ∗)2G k (αk )-G k (0),(24)进一步可得1+λk =G k (αk )+^λ(δ∗)2G k (αk )-G k (0)>0㊂对式(22)两边关于α求导,得到^Gᶄk (α)=(1+λk )Gᶄk (α)>0,即^G k (α)单调上升㊂由此可得改进的模型函数方法,其迭代算法如下:取迭代初值α0>0和误差水平ε>0,k =0:1)解式(9)得到h αk ,计算J (αk )和J '(αk )㊂由式(15),更新:m k (αk )=12 L δ 2+C k T k +αk=J (αk ),mᶄk (αk )=-C k(T k +αk )2=J ᶄ(αk ),由上式可确定m (α)中的T k 和C k :T k = Kh αk 2h αk 2,C k =-( Kh αk 2+αk h αk 2)22 h αk2㊂2)设第k 次的模型函数为:m k (α)=12 L δ 2+C k T k +α,在Morozov 方程中求解αk +1:^Gk (α):=G k (α)+λk (G k (α)-G k (αk ))=12(δ∗)2,确定新的近似零点,它也是式(20)的近似零点,其中λk 由式(24)确定㊂3)若^Gk (αk ) 12(δ∗)2或|αk +1-αk | ε,则停止迭代;否则令k :=k +1,回到1)㊂此处αk 可以不用很小,因为松弛参数λk 保证了^Gk (α)<12(δ∗)2㊂2㊀数值实现下面给出具体的例子验证用正则化方法确定内边界数据h (θ)的有效性㊂例1:不妨取模型问题的解为u (ρ,θ)=(3ρ2+4ρ-2)cos2θ,1 ρ 2,0 θ㊃828㊃江㊀西㊀科㊀学2023年第41卷2π,则易得外边界的Cauchy 数据f (θ)=13cos2θ,g (θ)=11cos2θ,内边界精确的Dirichlet 数据为h (θ)=7cos2θ㊂取扰动输入数据为f δ=f +δπsin θ,g δ=g +δπcos θ.对于给定的扰动水平δ,可以由下式得到δ∗:(δ∗)2=ʏ2π0[L (f δ,g δ)(θ)-L (f ,g )(θ)]2dθ≃2πm ðml =0[L (f δ,g δ)(θl )-L (f ,g )(θl )]2=δ22mπðm l =0[cos θl -2πm ðm j =0k ^(θj ,θl )sin θj ]2㊂(25)下面用改进的模型函数方法来确定正则化参数,进而求h (θ)的数值解㊂对式(9)进行离散化,由给定的正则化参数α0>0,可解出h α0(τk ),k =0,㊃㊃㊃,m ㊂给定ε>0,l =0:1)不妨设h αℓ(τk )=h ℓ(τk ),k =0, ,m ,,再如下计算J (αℓ),Jᶄ(αℓ):J (αℓ):=12 Kh -L δ 2L 2[0,2π]+αℓ2h 2L 2[0,2π]=12ʏ2π0[ʏ2π0k ~(τ,θ)h (θ)dθ-g δ(τ)+ʏ2π0^k(τ,θ)f δ(θ)dθ]2dτ+αℓ2ʏ2π0h 2(θ)dθ≃πm ðmj =0a j [2πm ðmi =0a ik ~(τj ,τi )h ℓ(τi )-g δ(τj )+2πmðmi =0a i ^k(τj ,τi )f δ(τi )]2+παℓmðmi =0a i h 2ℓ(τi ),(26)Jᶄ(αℓ)=12 h 2L 2[0,2π]≃πm ðmi =0a i h 2ℓ(τi ),(27)并且有T ℓ=Kh ℓ 2L 2[0,2π]h ℓ2L 2[0,2π]=ʏ2π0[ʏ2π0k ~(τ,θ)h ℓ(θ)dθ]2dτʏ2π0h 2ℓ(θ)dθ≃ðmj =0a j [2πm ðmi =0a i k ~(τj ,τi )h ℓ(τi )]2ðmi =0a i h 2ℓ(τi ),C ℓ=-(T ℓ+αℓ)2J ᶄ(αℓ)≃-πm(T ℓ+αℓ)2ðmi =0a i h 2ℓ(τi )㊂2)构造模型函数m ℓ(α):m ℓ(α)=12 L δ 2+C ℓT ℓ+α≃πmðmj =0a j [g δ(τj )-2πm ðm i =0a i ^k (τj ,τi )f δ(τi )]2+C ℓT ℓ+α(28)mᶄℓ(α)=-C ℓ(T ℓ+α)2,(29)取^λ=14,计算:G ℓ(0)=12L δ 2+C ℓT ℓ≃πmðmj =0a j [g δ(τj )-2πm ðm i =0a i ^k (τj ,τi )f δ(τi )]2+C ℓT ℓ,(30)G ℓ(αℓ)=J (αℓ)-αℓJᶄ(αℓ),(31)λℓ=G ℓ(0)+14(δ∗)2G ℓ(αℓ)-G ℓ(0),(32)3)由式(28)㊁(29)构造如下的改进模型函数方程:^Gℓ(α):=G ℓ(α)+λℓ(G ℓ(α)-G ℓ(αℓ))=12(δ∗)2,(33)由式(20)㊁(31)㊁(32)可以将该式化简为C ℓT ℓ(T ℓ+α)2=(δ∗)2+2λℓG ℓ(αℓ)2(1+λℓ)-12L δ 2,(34)由上式可以解得αℓ+1㊂4)如果|αℓ+1-αℓ| ε或者^Gℓ(αℓ) 12(δ∗)2,迭代停止;否则令ℓ=ℓ+1,返回1)㊂取初始的正则化参数α0=0.4,误差水平δ=0.01,δ∗由式(25)得到,迭代停止水平ε=10-8,m =100㊂经过计算可得,经过3次迭代,得到正则化参数α=0.0002,其对应的正则化解与精确解的曲线见图1㊂例2:取u (ρ,θ)=ρ2cos2θ+ρ-2sin2θ,1 ρ 2,0 θ 2π,则可得到边界数据f (θ)=4cos2θ+sin2θ4,g (θ)=4cos2θ-sin2θ4㊂精确解为h (θ)=㊃928㊃第5期㊀㊀㊀㊀㊀㊀祝志栋等:Laplace 方程Cauchy 问题求解的一种正则化方法-8-6-4-20246801234567内边界解hα0=0.4,δ=0.01θ([0,2π])图1㊀精确解与数值解cos2θ+sin2θ㊂扰动数据为f δ=f +δπsin θ,g δ=g+δπcos θ㊂内边界解h 取模型函数方法中初始的正则化参数为α0=0.55,误差水平δ=0.01,迭代停止水平ε=10-8,m =100㊂经过计算可以得到,经过5次迭代,得到正则化参数α=0.0001,其对应的正则化解与精确解的曲线见图2㊂1234567α0=0.55,δ=0.01θ([0,2π])-1.5-1-0.500.511.5图2㊀精确解与数值解3㊀结论1)通过引进内边界数据u (a ,θ)=h (θ)构造一个适定的Dirichlet 问题,从而将不适定的La-place 方程Cauchy 问题转化为第一类积分方程的求解,并给出了解存在的必要条件㊂2)给出了利用Tikhonov 正则化方法求解的正则化策略,提出了利用Morozov 原理确定正则化参数的改进模型函数方法,并给出了迭代算法㊂数值算例表明了该算法的有效性㊂参考文献:[1]㊀程晋,刘继军,张波.偏微分方程反问题:模型㊁算法和应用[J].中国科学:数学,2019,49(4):643-666.[2]芮秀杏,叶智群,邱淑芳,等.一类数值求导方法及其正则化参数的后验选取[J].江西科学,2021,39(5):782-789.[3]刘继军.不适定问题的正则化方法及应用[M].北京:科学出版社,2005.[4]WEI T,CHEN Y G,LIU J C.A variational -typemethod of fundamental solutions for a Cauchy problemof Laplace s equation[J].Applied Mathematical Mod-elling,2013,37(3):1039-1053.[5]BOURGEOIS L.Convergence rates for the quasi -re-versibility method to solve the Cauchy problem for La-place s equation [J].Inverse Problems,2006,22(2):413-430.[6]熊向团,宋菲菲.三维Laplace 方程Cauchy 问题的拟逆正则化方法及误差估计[J].西北师范大学学报(自然科学版),2022,58(5):12-16.[7]LI Z P,FU C L.A mollification method for a Cauchyproblem for the Laplace equation [J].Applied Mathe-matics and Computation,2011,217(22):9209-9218.[8]许涵,冯立新.求解Laplace 方程Cauchy 问题的磨光化方法[J].黑龙江大学自然科学学报,2022,39(4):379-387.[9]丁凤霞,程浩.椭圆方程柯西问题磨光正则化参数的后验选取[J].山东大学学报(理学版),2018,53(2):18-24.[10]FU C L,LI H F,QIAN Z,et al.Fourier 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拉普拉斯方程、水平集方法等
拉普拉斯方程（Laplace's equation），又名调和方程、位势方程，是一种。
定义三维情况下，拉普拉斯方程可由下面的形式描述，问题归结为求解对实自变量x、y、z二阶的实函数φ ：上面的方程常常简写作：或其中div表示的（结果是一个），grad表示标量场的（结果是一个），或者简写作：Δφ = 0其中Δ称为.拉普拉斯方程的解称为调和函数。
如果等号右边是一个给定的函数f(x, y, z)，即：则该方程称为泊松方程。
拉普拉斯方程和泊松方程是最简单的。
偏微分算子或Δ（可以在任意维空间中定义这样的算子）称为拉普拉斯算子，英文是Laplace operator或简称作Laplacian。
拉普拉斯方程的可归结为求解在区域D内定义的函数φ，使得φ在D的边界上等于某给定的函数。
为方便叙述，以下采用拉普拉斯算子应用的其中一个例子——作为背景进行介绍：固定区域边界上的温度（是边界上各点位置坐标的函数），直到区域内部热传导使温度分布达到稳定，这个温度分布场就是相应的狄利克雷问题的解。
拉普拉斯方程的不直接给出区域D边界处的温度函数φ本身，而是φ沿D的边界法向的。
从物理的角度看，这种边界条件给出的是矢量场的势分布在区域边界处的已知效果（对热传导问题而言，这种效果便是边界热流密度）。
拉普拉斯方程的解称为调和函数，此函数在方程成立的区域内是。
任意两个函数，如果它们都满足拉普拉斯方程（或任意线性微分方程），这两个函数之和（或任意形式的线性组合）同样满足前述方程。
这种非常有用的性质称为。
可以根据该原理将复杂问题的已知简单组合起来，构造适用面更广的。
二维拉普拉斯方程（u(r=2)=0、u(r=4)=4sin(5*θ)）下的拉普拉斯方程（r=2、R=4）图形两个自变量的拉普拉斯方程具有以下形式：解析函数解析函数的实部和虚部均满足拉普拉斯方程。
换言之，若z = x+ iy，并且那么f(z)是解析函数的是u(x,y)，v(x,y)可微，且满足下列柯西-黎曼方程：上述方程继续求导就得到所以u满足拉普拉斯方程。