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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