偏微分方程-1

Basic partial differential equations

First order partial differential equations, linear and quasi-linear PDE, Wave equations: initial condition and boundary condition, well-poseness, Sturn-Liouville eigen-value problem, energy functional method, uniqueness and stability of solutions  Heat equations: initial conditions, maximal principle and uniqueness and stability Potential equations: Green functions and existence of solutions of Dirichlet problem, harmonic functions, Hopf’s maximal principle and existence of solutions of Neumann’s problem, weak solutions, eigen-value problem of the Laplace operator Generalized functions and fundamental solutions of PDE

1， 偏微分方程学科的发展、数学物理方程的导出、第一边值问题、第二边值问题、Dirichlet问题、第三边值问题。

2， Cauchy问题、Cauchy-Kovalevskaya定理、强函数、Cauchy-Kovalevskaya定理的证明、广义Cauchy问题。

3， 特征流形、特征方程、Holmgren定理、Carleman定理、化二阶线性偏微分方程为标准型。

4， 二阶线性偏微分方程标准型的存在性、二阶线性偏微分方程的分类、偏微分方程问题提法的适定性、反射法、依赖区域、决定区域、影响区域、

特征锥、能量不等式、波动方程Cauchy问题解的唯一性。

5， 球面平均法、Kirchhoff公式、Poisson公式、d’Alembert公式、降维法、波动方程Cauchy问题解的稳定性、波的弥散、依赖集合、Duhamel原理、波动方程的边值问题与混合问题、Goursat问题。

6， 波动方程混合问题解的唯一性、波动方程混合问题解的稳定性、Holder不等式、Friedrichs不等式。

7， 磨光函数、单位分解定理、广义导数、广义导数的唯一性、Sobolev空间、Sobolev空间的基本性质、Meyers-Serrin定理。

8， 光滑函数的局部逼近定理、光滑函数的大范围逼近定理、延拓定理、Sobolev空间中函数的迹、迹定理、零迹函数定理、H_0^1{\Omega}空间上的函数的迹的连续依赖性。Gagliardo-Nirenberg—Sobolev 不等式。

9， Morrey不等式、Sobolev不等式、Rellich-Kondrachov定理、Poincare不等式、广义解、基本解。

10， Laplace方程的基本解、调和函数、广义调和函数、Green公式、热流定理、球面平均值定理、极值原理、Hopf-Oleinik定理、Laplace方程的Dirichlet问题解的唯一性、Dirichlet原理。

11， Lax-Milgram定理、能量估计、椭圆方程边值问题广义解的存在性定理、能量等式、Sturm-Liouville问题、本征值、本征函数、Green函数。

12， 将Sturm-Liouville问题归结为积分算子本征函数问题、双曲方程混合问题解的存在性、Laplace方程第一边值问题的Green函数、Green函数的对称性、Poisson公式、Harnack不等式。

13， 伴随微分算子与伴随边值问题、最小位能原理、正算自与算子方程、正定算子。

偏微分方程-2

1， Laplace算子的本征值与本征函数、Laplace方程边值问题解的唯一性与连续依赖性。

2， 导数的先验估计、调和函数的解析性、解析延拓定理、Liouville定理、Phragmen-Lindelof定理。

3， Dirichlet外问题、Dirichlet内问题、Neumann外问题、Neumann内问题、可去奇点定理、调和函数在无穷远邻域中的性质、广义调和函数与调和函数的关系、Weyl引理。

4， Laplace方程Cauchy问题可解性的充要条件、调和函数族的紧性定理、Newton势、单层势、双层势、对数势、亚椭圆算子、Newton势的密度、Lyapunov曲面。

5， 双层势的间断、双层势的法向导数的间断、一维波动方程的分离变量法。

6， 固有振动、热传导方程的Green公式、热传导方程的基本解、热势、热传导方程解的分析性质、热传导方程的边值问题、热传导方程的Cauchy问题、用分离变量法解矩形区域的热传导方程。

7， 热传导方程在有界区域与无界区域中的极值原理、严格极值原理、热传导方程边值问题解的先验估计、热传导方程第一与第二边值问题解的唯一性、热传导方程Cauchy问题解的唯一性、热传导方程边值问题解的连续依赖性、热传导方程Cauchy问题解的连续依赖性、二阶抛物型方程的广义解。

8， 二阶抛物型方程的Galerkin方法、二阶抛物型方程广义解的存在性、二阶抛物型方程广义解的正则性、二阶双曲型方程广义解。

9， 二阶双曲型方程的Galerkin方法、二阶双曲型方程广义解的存在性、二阶双曲型方程广义解的正则性、二阶线性方程的弱间断解、弱间断面。

10，弱间断解与特征曲面的关系、方程组的弱间断线、方程组的特征理论、方程组的分类、双曲型方程组的标准型、Godunov可对称化条件、对称双曲型方程组。

11， 对称双曲型方程Cauchy问题解的唯一性、对称双曲型方程Cauchy问题解的能量不等式、Sobolev嵌入定理、常系数对称双曲型方程Cauchy问题解的存在性、常系数对称双曲型方程Cauchy问题的求解。

12， 振荡积分、振荡积分的磨光化、用振荡积分定义广义函数的光滑性、Hadamard引理、Fourier积分算子、Fourier积分算子的核、算子相位函数、伪微分算子。

13， 逆紧支伪微分算子、逆紧支伪微分算子的符号、逆紧支伪微分算子的符号的展开、平移算子的符号、对偶符号、复合公式、古典符号与伪微分算子、奇异积分算子。

(Linear) Partial Differential Equations

1. 《Basic Partial Differential Equations》, D. Bleecker, G. Csordas 著, 李俊杰 译，高等教育出版社，2008.

1. 《数学物理方法》，柯朗、希尔伯特著。

Evans, “Partial differential equations”

L. Hormander, “Linear Partial Differential Operators”

Aleksei.A.Dezin, “Partial differential equations”

Jeffrey Rauch, “Partial Differential Equations”

David Gilbarg, “Elliptic Partial Differential Equations of Second Order”

陈祖墀《偏微分方程》中国科技大学出版社

《偏微分方程教程》华中师范大学

姜礼尚，《数学物理方程讲义》，高教版
谷超豪，《数学物理方程》，高教版

北大，二階偏微分

118《常微分方程与偏微分方程》 管志成，李俊杰编

【习题集】

119《偏微分方程习题集》沙玛耶夫主编

【提高】

120《Handbook of Linear Partial Differential Equations for Engineers and Scientists》

（《线性偏微分方程手册：工程师和科学家必备》英文版）Andrei D. Polyanin编著

九、“数学物理方程”和“数学物理方法”

     一般是物理专业、力学、信息等专业的课程。其内容是基本上是“偏微分方程”加上“复变函数”整合而成的一本综合课程。“数学物理方法”相当于“工程数学”的三本（即复变函数，积分变换，场论初步）。

【教材】

122《特殊函数概论》王竹溪,郭敦仁编著

123《广义函数与数学物理方程》齐民友著

126《数学物理方法》梁昆淼著

【习题集】

129《数学物理方程习题集》弗拉基米洛夫编  

【提高】

130《矢算场论札记》梁洪昌著

结合《数学物理方程》一起使用，会对自身水平有很大帮助。

131《数学物理方程及其应用》吴小庆编著

132《数学物理方程》 张渭滨

133《数学物理方程与特殊函数》 杨奇林

134《数学物理方法》 郭玉翠 

135《数学物理方程–方法导引》陈恕行,秦铁虎

136《The Boudary Value Problems of Mathematical Physics》O A. Ladyzhenskaya

137《物理学与偏微分方程》李大潜,秦铁虎著

138《积分方程》李星编著

139《积分方程论》(修订版) 路见可, 钟寿国编著 

Partial differential equations -1

Basic partial differential equations

First order partial differential equations, linear and quasi-linear PDE, Wave equations: initial condition and boundary condition, well-poseness, Sturn-Liouville eigen-value problem, energy functional method, uniqueness and stability of solutions Heat equations: initial conditions, maximal principle and uniqueness and stability Potential equations: Green functions and existence of solutions of Dirichlet problem, harmonic functions, Hopf’s maximal principle and existence of solutions of Neumann’s problem, weak solutions, eigen-value problem of the Laplace operator Generalized functions and fundamental solutions of PDE

1 , Partial differential equations disciplinary development, export equations of mathematical physics, the first boundary value problem, the second boundary value problem,Dirichlet problem, the third boundary value problem.

2 , Cauchy problem,Cauchy-Kovalevskaya theorem and strong function, theCauchy-Kovalevskaya theorem proved and generalized Cauchy problem.

3 , Characteristic manifold, characteristic equation,Holmgren theorem,Carleman theorem, second-order linear partial differential equation into standard form.

4 , The existence of standard type of second-order linear partial differential equation, the classification of second-order linear partial differential equations, partial differential equations, reflected the well-posedness of the problem method and rely on regional, regional, regional,

Characteristic cones and energy inequalities, equations Cauchy The uniqueness of the solution.

5 , Spherical means law,Kirchhoff equation,Poisson formula,d ‘ Alembert dimension reduction method, formulas, equationsCauchy Problem of stability, wave dispersion, dependency collection,Duhamel principle, boundary value problem of wave equation and mixed issues,Goursat problem.

6 , The only solution for the mixed problem of wave equation, wave equation mixed problems of stability,Holderinequality,Friedrichs inequality.

7 , Smoothing function, decomposition theorem and generalized derivatives, the generalized derivatives of uniqueness, andSobolev spaces,Sobolev space, basic properties,Meyers-Serrin theorem.

8 , Local approximation of smooth functions theorems, wide range of smooth function approximation theorem, extension theorem, theSobolev space of functions in trace, trace theorems and zero-tracking function theorem,H_0^1{\Omega}function on the space of continuous dependence of the trace. Gagliardo-Nirenberg-Sobolev inequalities.

9 , Morrey inequality,Sobolev inequality,Rellich-Kondrachov theorem,Poincare inequality, the generalized solution, basic solutions.

10 , Laplace equation solutions, harmonic functions, generalized harmonic function,Green formula, heat flux theorem, the spherical mean value theorem, the maximum principle,Hopf-Oleinik theorem,Laplace Equation Dirichlet Problem of uniqueness of solution, Dirichlet Principle.

11 , Lax-Milgram theorem, the energy and the existence of generalized solution of boundary value problems for elliptic equation theorem, the energy equation,Sturm-Liouville problems, intrinsic value, intrinsic functions,Greenfunctions.

12 , Sturm-Liouville integral operator eigenfunctions problems boil down to issues, the existence of solution for the mixed problem for hyperbolic equations and theLaplace equation boundary value problems of first Green functions, Greenfunction of symmetry, andPoisson equations,Harnack inequalities.

13 , Adjoint differential operators and with boundary value problems, principle of minimum potential energy, work, positive definite operator equations and operator.

Partial differential equations -2

1 , Laplace operators eigenvalues and eigenfunctions, andLaplace equations of boundary value problems of uniqueness and continuous dependence.

2 , Derivative of prior estimates, harmonic functions are analytic, and analytic continuation theorem,Liouvilletheorem, thePhragmen-Lindelof theorem.

3 , Dirichlet problem and theDirichlet problem,Neumann external problem,Neumann problem, removable Singularity theorems, nature of harmonic functions at infinity in the neighborhood, Generalized harmonic function and harmonic function relationships,Weyl ‘s lemma.

4 , Laplace equations Cauchy problem solvability if and only if, harmonic functions of the compactness theorem,Newtonpotential, potential of single layer, double layer potential, logarithmic potentials, and elliptic operators,NewtonPotential density,Lyapunov surfaces.

5 , Double layer potential of discontinuous and the normal derivative of double layer potential interruption, one dimensional wave equation method of separation of variables.

6 , Vibration, heat conduction equation of Green formula, the fundamental solutions of the heat equation, heat potential, analysis of heat conduction equations, heat conduction equations of boundary value problems, heat conduction equations Cauchy problem, using separation of variables method for heat conduction equation of the rectangular region.

7 , Hot conduction equation in has territories regional and no territories regional in the of extreme principle, and strictly extreme principle, and hot conduction equation side value problem solutions of prior estimated, and hot conduction equation first and second side value problem solutions of only sex, and hot conduction equation Cauchyproblem solutions of only sex, and hot conduction equation side value problem solutions of continuous dependence, and hot conduction equation Cauchy Solution of continuous dependence, generalized solutions of second-order parabolic equations.

8 , Second-order parabolic equations of Galerkin methods, the existence of solutions of second order parabolic generalized and second order parabolic generalized solutions of regularity and second order generalized solutions of hyperbolic type.

9 , Second-order hyperbolic equations of Galerkin methods, the existence of solutions of second order uniqueness of hyperbolic type and second order hyperbolic generalized solutions of regularity, weakly discontinuous solutions of second-order linear equations, weak surface.

10 , Weakly discontinuous solutions and feature relations, equation of the surface of weak continuous line, the character theory of equations, classification of equations, hyperbolic equations in standard, Godunov Condition of symmetric and symmetric hyperbolic equations.

11 , Symmetric hyperbolic equations Cauchy problem of uniqueness of solution, symmetric hyperbolic equations Cauchyproblem of energy inequality and theSobolev embedding theorem, symmetric hyperbolic equations with constant coefficients Cauchy Existence of solutions to problems, constant coefficients symmetric hyperbolic equations Cauchyproblem solving.

12 , Oscillatory integral, oscillating integrals polished smooth, oscillatory integral definition of generalized functions, andHadamard ‘s lemma, andFourier integral operators,Fourier integral operators nuclear phase functions, pseudo-differential operator, operator.

13 , Tight inverse pseudo differential operators, tight inverse pseudo differential operator symbol symbols, tight inverse pseudo differential operators, translation operator symbols, symbol of duality, the compound formula, classical symbols and pseudo differential operators and singular integral operators.

(Linear)Partial Differential Equations

1. 《 Basic Partial Differential Equations 》 , D. Bleecker, G. Csordas The , Lee Chun kit Translation, higher education press,2008.

2. Of the methods of mathematical physics, r.Courant, Hilbert with.

Evans, “Partial differential equations”

L. Hormander, “Linear Partial Differential Operators”

Aleksei.A.Dezin, “Partial differential equations”

Jeffrey Rauch, “Partial Differential Equations”

David Gilbarg, “Elliptic Partial Differential Equations of Second Order”

Chen zuchi of the partial differential equations of the University of science and technology of China press

The partial differential equations course in central China Normal University

Jiang lishang, lectures on the equations of mathematical physics, Education Edition
Gu chaohao, of the equations of mathematical physics, Education Edition

North, nikai second floor partial derivatives

118 Of the ordinary differential equations and partial differential equations Guan zhicheng, Lee Chun kit series

“Onward”

119 Shamayefu editor of the partial differential equation problem sets

“Increase”

120 《 Handbook of Linear Partial Differential Equations for Engineers and Scientists 》

(The Handbook of linear partial differential equations: engineers and scientists must have the English version) Andrei D. Polyanin Authoring

Nine, “mathematical physics” and “methods of mathematical physics”

     General Physics, mechanics, information and other professional courses. Its contents are essentially “partial differential equations” with “complex functions” integrated into a comprehensive curriculum. “Mathematical physical methods” equivalent to “Engineering Mathematics” of the three (that is, functions of a complex variable and integral transform, preliminary field theory).

“Textbook”

122 Wang Zhuxi, an introduction to special functions , Written by Guo d r

123 The generalized function with friends with all equations of mathematical physics

126 Liang Kunmiao the methods of mathematical physics with

“Onward”

129 The fulajimiluofu series of equations of mathematical physics problem set 

“Increase”

130 The Vector calculus theory notes Liang Hongchang the

Used in conjunction with the equations of mathematical physics, is of great help for their level.

131 Written by Wu Xiaoqing of the equations of mathematical physics and its applications

132 Of the equations of mathematical physics Zhang Weibin

133 Of the equations of mathematical physics and specific functions Yang Qilin

134 Of the methods of mathematical physics Guo Yu-Cui

135 The equations of mathematical physics — Chen shuxing method guidance , Qin tiehu

136 《 The Boudary Value Problems of Mathematical Physics 》 O A. Ladyzhenskaya

137 Li Daqian the physics and partial differential equations , Qin tiehu with

138 Written by Li Xing of the integral equation

139 The theory of integral equations ( Revised edition ) Can be on the road , Written by Zhong Shouguo