Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2013
Lecture notes on Variational and Approximate Methods in Applied Mathematics  A Peirce UBC
1
Lecture 7: Singular Integrals, Open Quadrature rules, and
Gauss Quadrature
(Compiled 18 September 2012)
In
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2014
1
Math 406 E: Ass. 1: Due Wednesday 17 Sept.
1. Least squares tting  mth degree polynomial through N points.
(a) Find a system of equations for the coecients of an mth degree
polynomial that ts a fun
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2014
1
Lecture notes on Variational and Approximate Methods in Applied Mathematics
Lecture 3: The Runge Phenomenon and Piecewise
Polynomial Interpolation
(Compiled 3 September 2014)
In this lecture we cons
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2014
Introduction to variational methods and nite elements
1.2.3. Variational formulations of BVP:
ax b
Problem: Sove ax = b
x=
x
b
a
Reformulate the problem:
Consider E = 1 ax2 + bx
2
Find x : E(x ) = min
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2014
1
Math 406 E: Ass. 1: Due Wednesday 17 Sept.
1. Least squares tting  mth degree polynomial through N points.
(a) Find a system of equations for the coecients of an mth degree
polynomial that ts a fun
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2014
1
Mathematics 406, Assignment 314 Due Oct 10 th
1. Repeated Richardson extrapolation applied to the Trapezium Rule is known as Romberg
integration. Use the following asymptotic expansion for the error
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2014
1
Mathematics 406, Assignment 4 Due Friday October 24 th
1. (a) Show that f (x)(x a) = f (a)(x a)
(b) Show that x 0 (x) = (x).
(c) Find the generalized derivative of f (x) = H(x 1) sin x.
(d) Write (s
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2014
1
Mathematics 406, Assignment 314 Due Oct 10 th
1. Repeated Richardson extrapolation applied to the Trapezium Rule is known as Romberg
integration. Use the following asymptotic expansion for the error
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2014
1
Mathematics 406, Assignment 5 Due Friday November 31 st
1. Consider the boundary value problem
Lu = u00 + u = f (x); 0 < x < l
u0 (0) = 0 and u(l) = 0
(a) Find the Greens function and express the so
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2014
MATH 406, HWK 7, Due 19 November 2014
1. Consider the following boundary value problem
Lu = u00 + k2 u (x) = f (x) ,
u (0) = ,
u0 (1) =
(1)
(a) Use integration by parts to obtain the weak statement o
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2014
MATH 406, HWK 6, Due 7 November 2014
1. Alter the program poisson.m (a copy of poisson.m, V.m, and VN.m
can be found on the course web site) to be able to solve the following
cracklike mixed boundary
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2014
MATH 406, HWK 7
1. Consider the following boundary value problem
Lu = u00 + k2 u (x) = f (x) ,
u0 (1) =
u (0) = ,
(1)
(a) Use integration by parts to obtain the weak statement of the BVP.
(b) If f is
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2014
MATH 406, HWK 8, Due 26 November 2014
1. Consider determining the eigenvalues for Laplaces equation:
2
u 2u
+ 2 = u
u =
x2
y
on the region subject to Neumann boundary conditions:
u
=0
n
(1)
(2)
St
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2014
1
Lecture notes on Variational and Approximate Methods in Applied Mathematics
Lecture 5: Numerical Integration
(Compiled 3 September 2014)
In this lecture we introduce techniques for numerical integra
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2014
Lecture notes on Variational and Approximate Methods in Applied Mathematics
1
Lecture 6: Singular Integrals, Open Quadrature rules, and
Gauss Quadrature
(Compiled 3 September 2014)
In this lecture we
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2014
1
Lecture notes on Variational and Approximate Methods in Applied Mathematics
Lecture 1: Interpolation and approximation
(Compiled 3 September 2014)
In this lecture we introduce the concept of approxi
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2013
1
Lecture notes on Variational and Approximate Methods in Applied Mathematics  A Peirce UBC
Lecture 5: Numerical Integration
(Compiled 15 September 2012)
In this lecture we introduce techniques for n
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2013
Lecture notes on Variational and Approximate Methods in Applied Mathematics  A Peirce UBC
1
Lecture 4: Piecewise Cubic Interpolation
(Compiled 15 September 2012)
In this lecture we consider piecewise
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2013
Lecture notes on Variational and Approximate Methods in Applied Mathematics  A Peirce UBC
1
Lecture 2: Error in polynomial approximation and
interpolation with equally spaced sample points
(Compiled
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2013
Lecture notes on Variational and Approximate Methods in Applied Mathematics  A Peirce UBC
1
Lecture 3: The Runge Phenomenon and Piecewise
Polynomial Interpolation
(Compiled 22 September 2013)
In this
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2013
Math 406 E: Ass. 2: Due Friday 18 Oct 2013
1. GaussChebyshev Quadrature: GaussChebyshev quadrature with m 3 integration
points evaluates integrals of the form
1
;
"1
fx
dx
1/2
1 " x 2
exactly if f
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2013
MATH 406, HWK 5, Solutions
1. Consider the following boundary value problem
Lu u k 2 ux fx , u0 ,
u 1
#
a. Use integration by parts to obtain the weak statement of the BVP.
b. If f is sufficiently di
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2013
1
Lecture notes on Variational and Approximate Methods in Applied Mathematics  A Peirce UBC
Lecture 1: Interpolation and approximation
(Compiled 1 September 2012)
In this lecture we introduce the con