Section 25.1

Boundary Value Problems

Unlike Initial Value Problems, BVPs specify conditions at two different points, often the boundaries of a physical object.

1

Introduction

An IVP specifies and .
A BVP typically specifies and .

Visualize a beam fixed at both ends, or a string held taught at and .

2

Types of Boundary Conditions

  • Dirichlet: Values are fixed. .
    (e.g., Temperature held constant at ends).
  • Neumann: Derivatives (flux) are fixed. .
    (e.g., Insulated ends, heat flux is 0).
  • Robin (Mixed): Combination .
    (e.g., Convection cooling).
3

Existence & Uniqueness

Warning: BVPs are not guaranteed to have solutions, or unique solutions!

1. No Solution
2. Unique Solution
3. Infinitely Many Solutions (Eigenfunctions!)
4

Worked Examples

Example 1: Unique Solution

.

.

.

.

. Unique.

Example 2: No Solution

.

.

.

Contradiction. No solution possible.

Example 3: Infinite Solutions

.

.

.

This is . can be anything.

. Infinite family of solutions.

5

Practice Quiz

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