丘赛计算与应用数学考试大纲

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Computational Mathematics

Interpolation and approximation

Polynomial interpolation and least square approximation; trigonometric inter
polation and approximation, fast Fourier transform; approximations by ration
al functions; splines.

Nonlinear equation solvers

Convergence of iterative methods (bisection, secant method, Newton method, o
ther iterative methods) for both scalar equations and systems; finding roots
of polynomials.

Linear systems and eigenvalue problems

Direct solvers (Gauss elimination, LU decomposition, pivoting, operation cou
nt, banded matrices, round-off error accumulation); iterative solvers (Jacob
i, Gauss-Seidel, successive over-relaxation, conjugate gradient method, mult
i-grid method, Krylov methods); numerical solutions for eigenvalues and eige
nvectors

Numerical solutions of ordinary differential equations

One step methods (Taylor series method and Runge-Kutta method); stability, a
ccuracy and convergence; absolute stability, long time behavior; multi-step
methods

Numerical solutions of partial differential equations

Finite difference method; stability, accuracy and convergence, Lax equivalen
ce theorem; finite element method, boundary value problems

References:

[1] C. de Boor and S.D. Conte, Elementary Numerical Analysis, an algorithmic
approach, McGraw-Hill, 2000.

[2] G.H. Golub and C.F. van Loan, Matrix Computations, third edition, Johns
Hopkins University Press, 1996.

[3] E. Hairer, P. Syvert and G. Wanner, Solving Ordinary Differential Equati
ons, Springer, 1993.

[4] B. Gustafsson, H.-O. Kreiss and J. Oliger, Time Dependent Problems and D
ifference Methods, John Wiley Sons, 1995.

[5] G. Strang and G. Fix, An Analysis of the Finite Element Method, second e
dition, Wellesley-Cambridge Press, 2008.


Applied Mathematics

ODE with constant coefficients; Nonlinear ODE: critical points, phase space
& stability analysis; Hamiltonian, gradient, conservative ODE's.

Calculus of Variations: Euler-Lagrange Equations; Boundary Conditions, param
etric formulation; optimal control and Hamiltonian, Pontryagin maximum princ
iple.

First order partial differential equations (PDE) and method of characteristi
cs; Heat, wave, and Laplace's equation; Separation of variables and eigen-fu
nction expansions; Stationary phase method; Homogenization method for ellipt
ic and linear hyperbolic PDEs; Homogenization and front propagation of Hamil
ton-Jacobi equations; Geometric optics for dispersive wave equations.

References:

W.D. Boyce and R.C. DiPrima, Elementary Differential Equations, Wiley, 2009

F.Y.M. Wan, Introduction to Calculus of Variations and Its Applications, Cha
pman & Hall, 1995

G. Whitham, "Linear and Nonlinear Waves", John-Wiley and Sons, 1974.

J. Keener, "Principles of Applied Mathematics", Addison-Wesley, 1988.

A. Benssousan, P-L Lions, G. Papanicolaou, "Asymptotic Analysis for Periodic
Structures", North-Holland Publishing Co, 1978.

V. Jikov, S. Kozlov, O. Oleinik, "Homogenization of differential operators a
nd integral functions", Springer, 1994.

J. Xin, "An Introduction to Fronts in Random Media", Surveys and Tutorials i
n Applied Math Sciences, No. 5, Springer, 2009