I
Preliminary Material
a. Constant Coefficient Equations
b. Cauchy-Euler Equations
I. Boundary Value Problems
a. Direct Solution
b. Separation of Variables – Heat Equation
c. Solution of Eigenvalue Equations with Homogeneous BCs
II. Vector Spaces and Function Spaces
a. Bases/Eigenfunctions
b. Scalar Product [pic]
c. Orthogonal, Orthonormal, Normalization
d. Determination of Expansion Coefficients
III. Fourier Series
a. Trigonometric Fourier Series Expansion on [pic]
[pic]
b. Fourier Coefficients
[pic]
c. Orthogonality of [pic] on [pic], etc.
d. Half Range Expansions on [pic]
[pic]
[pic]
e. Fourier Sine and Cosine Series
[pic], [pic]
f. Even Periodic Extension, Odd Periodic Extension, Periodic Extension
IV. Sturm-Liouville Eigenvalue Problems
a. Converting Second Order Linear ODEs to Self-Adjoint Form, [pic] Leads to Sturm-Liouville Operator: [pic]
b. Types of Boundary Conditions:
i. Regular BCs [pic]
ii. Dirichlet: [pic]
iii. Neumann: [pic]
iv. Periodic: [pic]
c. Sturm-Liouville Eigenvalue Problems: [pic] plus BCs
d. Lagrange's Identity: [pic]
e. Green's Identity: [pic]
f. Adjoint Operators [pic]
g. Proof that eigenvalues are real and eigenfunctions are orthogonal for self-adjoint problems. – Use Green’s Identity
h. Eigenfunction Expansion Method:
Assume [pic] and find coefficients.
V. Special Functions
a. Classical Orthogonal Polynomials
b. Gram-Schmidt Orthogonalization Process
c. Legendre Polynomials
i. Rodrigues Formula [pic]
ii. Three Term Recursion Formlua: [pic]
iii. Generating Function: [pic], [pic]
iv. Binomial expansion: [pic]
d. Gamma Function: [pic]
e. Factorials and double factorials.
f. Be able to iterate recursion (finite difference) formulae, like [pic] given [pic]
g. Bessel Functions – General Properties
VI. Green's Functions
a. Variation of Parameters [pic] [pic][pic]
b. [pic]
c. Initial Value Green's Function
[pic] [pic]
d. Boundary Value Green's Function
[pic] [pic]
e. Properties of BVP Green's Functions
i. [pic]
ii. Satisfies Homogeneous BCs
iii. Symmetry: [pic]
iv. Continuity: [pic]
v. Jump Discontinuity of Derivative: [pic]
f. Dirac Delta Function
i. [pic] and [pic]
ii. [pic]
iii. For the case that a function has multiple simple roots, [pic] for [pic] one has that that [pic].
g. Green's Functions from Eigenfunction Expansions
[pic]
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