Topics for Final Exam and some useful information
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Important Information:
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Time & Location: 12:00PM in BUCH A101, Wednesday, August 19, 2015
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You MUST bring your UBC card to take the final exam.
I. The (singular) power series solutions to an ODE:
– what are the ordinary points and singular points
– Classify a singular point to be regular or irregular
– Find the power series solution near an ordinary point and a lower bound of the radius of convergence of
this power series.
– Find the singular power series solution near a regular point (Case I, II and III) and a lower bound of the
radius of convergence of this singular power series.
II. The heat equation:
– Separation
of
variables
for
the
heat
equation
without
source
with
homogeneous
Dirich-
nonhomogeneous
Dirich-
let/Neumann/Robin/periodic boundary conditions on a finite interval.
– Eigenfunction
expansion
for
the
heat
equation
with
source
with
let/Neumann/Robin/periodic boundary conditions on a finite interval.
Be careful that how to
find w(x, t)
III. The wave equation:
– The D’Alembert solution for the wave equation on the real line
– Understand the domain of dependence and the region of influence, and use them to compute the expression
of u(·, t0 ) at a given time t0 .
– Separation of variables for the wave equation without external force with homogeneous Dirichlet/Neumann/Robin/periodic boundary conditions on a finite interval.
– Eigenfunction expansion for the wave equation with external force with nonhomogeneous Dirichlet/Neumann/Robin/periodic boundary conditions on a finite interval. Be careful that how to find
w(x, t)
IV. The Laplace equation:
– Separation of variables for the Laplace equation with Dirichlet/Neumann/Robin/periodic boundary conditions on a rectangle. Pay more attention on the additional condition of the Neumann boundary condition.
In this case, only one boundary condition is inhomogeneous, other boundary conditions are homogeneous.
– Separation of variables for the Laplace equation with Dirichlet/Neumann/Robin/periodic boundary conditions on a circular type domain (pizza type, pizza-cut type, disk type, annulus type). In this case, only
one boundary condition is inhomogeneous, other boundary conditions are homogeneous.
1
2
V. The Fourier series:
– Compute the Fourier sine/cosine series on [0, L].
– Compute the full Fourier series on R.
– Find the even/odd 2L-periodic extension in R of a function f (x) on [0, L].
– Applications of the Theorem of Convergence of Fourier series.
– Applications of the Parseval’s identity.
VI. The finite difference method:
– Finite-difference approximations to first and second derivatives
– Write a finite difference scheme with notation ukn for an initial-boundary value problem for the
heat/wave/Laplace equations with source/external force and a Dirichlet/Neumann/Robin/periodic boundary condition on a finite interval or a rectangle. Can compute the ukn with M = 2 and N = 2, 3.
VII. The Sturm-Liouville eigenvalue Problem:
– Write a second order ODE into the Sturm-Liouville form.
– Find eigenvalues and the corresponding eigenfunctions for the Sturm-Liouville problem.
– Understand the Strum-Liouville theorem.
– Use the Sturm-Liouville theorem and the separation of variables to solve an initial-boundary value problem
for the heat/wave/Laplace equations without source/external force/ and a mixed boundary condition on a
finite interval or a rectangular or circular domain.
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You can save a lot of time if you can remember the eigenvalues and eigenfunctions
for the five common eigenvalue problems( You can use them without detailed work).
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Expectation of the distribution of topics in Final Exam:
Part I Part II, III, IV, V Part VI Part VII
≈15
•
≈65
≈10
≈10
The following topics will NOT be in Final Exam:
– The method of characteristic curves and characteristic coordinates.
– The relation between the Fourier series solution of the wave equation without external force with homogeneous Dirichlet/Neumann/Robin/periodic boundary conditions on a finite interval and the D’Alembert’s
solution.
– The method of eigenfunction expansions for the Laplace/Poisson equations.
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There are NO Bonus Problems in Final Exam.
There are NO hints for those five common eigenvalue problems in Final Exam.
Z
Z
• The integrals of
x sin(αx) dx and
x cos(αx) dx are NOT in Final Exam.
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