2934阅读
0回复

[资料共享]PARTIAL DIFFERENTIAL EQUATIONS ---Part II

离线aoso

<a href="http://b

UID ：4796

注册：2007-09-04
登录：2025-06-23
发帖：592
等级：八级仿真大师

0楼发表于: 2008-09-09 20:38:18

PARTIAL DIFFERENTIAL EQUATIONS ---Part II
Sergei Borisenok
This second in the series of three volumes builds upon the basic theory of linear PDE given in volume 1, and pursues more advanced topics. Analytical tools introduced here include pseudodifferential operators, the functional analysis of self-adjoint operators, and Wiener measure. The book also develops basic differential geometrical concepts, centred about curvature. Topics covered include spectral theory of elliptic differential operators, the theory of scattering of waves by obstacles, index theory for Dirac operators, and Brownian motion and diffusion.
The aim of the second part of the course on PDE is to present the basic methods and to achieve an adequate treatment of the most important aspects of the theory of partial differential equations.
1. Second order PDEs (Part II)
1.1. Boundary value problems. Dirichlet, Neumann and mixed (Robin etc.) conditions. Interior Dirichlet problem for a circle. Poisson integral solution for a circle. Exterior Dirichlet problem for a circle.
1.2. Cylindrical and spherical coordinates. Diffusion equation in cylindrical coordinates. Diffusion equation in spherical coordinates. Laplace equation in cylindrical coordinates. Laplace equation in spherical coordinates.
2. Orthogonal expansions
2.0. Orthogonality. Inner product. Orthonormal set of functions. Orthogonal polynomials. Weighed inner product. Families of orthogonal polynomials: Chebyshev, Hermite, Jacobi, Laguerre, Legendre.
2.1. Series of orthogonal functions. Generalized Fourier series. Ortogonal expansions of PDE solutions. Gibbs phenomenon.
2.2. Eigenfunction expansion.
3. Integral transforms
3.0. Integral transforms: Fourier cosine transform, Fourier sine transform, Fourier complex transform, Laplace transform, Mellin transform, Hankel transform.
3.1. Laplace transform. Basic properties. Inversion theorem. Application of Laplace transform to PDEs.
3.2. Fourier transform. Basic properties. Convolution theorems. Application of Fourier transform to PDEs.
4. Green’s functions
4.1. Basic definitions. Distributions. Regular and singular distributions. Dirac distribution and Dirac delta-function. Heaviside function. Green’s function.
4.2. Green’s function for parabolic PDEs. Green’s function for homogeneous diffusion operator. Schroedinger equation.
4.3. Green’s function for elliptic PDEs. Laplace and Helmholtz operators.
4.4. Green’s function for hyperbolic PDEs.
5. Perturbation methods
5.1. Taylor series expansions.
5.2. Successive approximations.
6. Non-linear dynamical equations of high orders
Dynamical PDEs. Self-similar solutions. KdV equation. Burgers’ equation. Solitons.

Literature:
1. P. R. Garabedian. Partial Differential Equations. New York: John Wiley & Sons, 1964.
2. D. G. Duffy. Solutions of Partial Differential Equations. Delhi: CBS, 1988.
3. G. L. Lamb. Introductory Applications of Partial Differential Equations. New York: John Wiley & Sons, 1995.
4. P. K. Kythe, P. Puri, M. R. Schaferkotter. Partial Differential Equations and Mathematica. New York: CRC Press, 1997.
5. G. B. Folland. Introduction to Partial Differential Equations. New D ..

未注册仅能浏览部分内容，查看全部内容及附件请先登录或注册

发帖回复