MATH3365 / MATH5366M Course Overview for Session 2010-11
This
course (15 credits at level 3 and 20
credits at level 5) has the following content. All supporting material,
i.e. examples sheets, printed handouts and Maple worksheets, can be
obtained here.
i.
Approximating roots
(5 lectures): regular
and singular algebraic equations; approximations by iteration and direct
expansion; distinct and repeated roots; fractional powers; degeneracy;
rescaling; dominant balance; general integer-power expansions; transcendental
equations.
ii.
Gauge functions and
asymptotics (5 lectures):
gauge functions; order symbols; asymptotic sequences; asymptotic expansions;
uniqueness; uniformity; scaling out non-uniformity; coupling distinguished
gauges with rescaling.
iii.
Boundary-value
problems (5 lectures):
regular and singular ODEs; failure of naive expansion; transcendentally small
terms; boundary layers; non-uniform convergence; inner and outer variables;
inner and outer expansions; matched asymptotic expansions; Prandtl's rule for
O(1) matching; Van Dyke's rule for higher-orders matching; principle of least
degeneracy; location and nature of boundary layers.
iv.
Initial-value
problems (3 lectures):
Linstedt-Poincaré technique; strained coordinates; multiple scales;
averaging methods.
v.
Summation of series
(3 lectures): Radius
of convergence; Domb-Sykes plots; multiplicative singularity extraction;
Padé approximants; Shanks' transformation; summation of slowly
convergent and divergent asymptotic expansions.
vi.
WKB theory (level 5 only, 4 lectures): Eikonal & transport equations;
approximation of eigenvalues; comparison with boundary-layer theory;
alternative WKB approximation; higher-order approximations; solution of IVPs;
turning points; Airy's equation; trig-exp connection across turning points;
large eigenvalues; Liouville-Green and Langer transformations; uniformly valid
solution spanning turning points.
vii.
Exponential integrals
(level 5 only, 3 lectures):
