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 this lecture we discuss the evaluation of singular inte
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 function f at given data points (xk , f (xk ), k =
1, . .
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 consider the dangers of high degree polynomial interpolatio
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 E(x)
x
x
1. The Calculus of Variations
Consider a dier
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 function f at given data points (xk , f (xk ), k =
1, . .
Variational and Approximate Methods in Applied Mathematics
MATH 406

Fall 2014
1
Mathematics 406, Assignment 214 Due Sept 26 th
1. Numerical Dierentiation
(a) Simplify the formula for the numerical dierentiation
1
1
hDyn ' ( + 2 + 3 )yn
2
3
by expanding the operators explicitly.
(b) Use the result in (a) to nd rst derivative of the
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 in the Trepezium rule:
I(0) I(hs ) =
X
ci h2i
s
i=1
to
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 (sin x) as a sum of functions.
2. Consider the following
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 in the Trepezium rule:
I(0) I(hs ) =
X
ci h2i
s
i=1
to
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 solution in terms of it.
(b) Find the values of l for whi
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 of the BVP.
(b) If f is suciently dierentiable show that
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 value problem for Laplaces equation on a
semicircular
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 suciently dierentiable show that the strong and weak f
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)
Starting with the weighted residual statement of this bou
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 integration, which are primarily based on integrating interpol
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 discuss the evaluation of singular integrals using soc
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 approximation of functions by a linear combination of a nite n
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 numerical integration, which are primarily based on inte
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 cubic interpolation in which a cubic polynomial approx
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 6 September 2012)
In this lecture we discuss the error
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 lecture we consider the dangers of high degree polynom
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 fx is a polynomial of degree 2m " 1 5. Use this fact to
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 differentiable show that the strong and weak formulations
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 concept of approximation of functions by a linear combinat