# Boundary Value Problems

Welcome to An Introduction to Boundary Value Problems (Orthogonal Functions and Partial Differential Equations).

## For those interested in editing this course, some of thoughts on how this course is supposed to work.

This is an adaptive course, of course, it is a wiki. The structure of the educational materials shall be adapted based on the response from users, both instructors and students. At the start, the material will be sparse and will borrow from others, but over time it is expected that editing and modification by users will modify the content, and the course structure to generate a course that is adjusted to the learning style(s) of a variety of learners. Possibly there will be several mainstream approaches (structures). That is the hope. Please keep the content bounded (Boundary Value Problems).

Unlike a Wikipedia article, the content of an course must focus on educating the learner, thus the structure and content need to be directed at reaching well defined learning outcomes at each level, rather than providing a strictly encyclopedic description of a topic. When you are editing/adding content, please identify the learning objective of the current content and make sure your modifications align to that objective. When adding new content, determine a specific learning objective of the new content and provide the expected skills and knowledge to be gained by understanding this content.

Initially, a fixed sequence of lessons shall be provided to cover the material and verify content understanding. In the future, the course should provide multiple tree structures whereby student(s) may tailor the presentation to their needs. Thus the need for a "wikian" template to allow addition and clarification of material, modification to address different learning styles, and multiple layers of assessment.

A template for each Lesson currently consists of:

1. Title: Distinctive Name
2. Objective(s): Specifically the "conceptual or procedural knowledge" that is to be understood and retained by the learner.
3. Activities: Material directed at the learner to achieve learning objectives.
4. Assessment: Methods that will provide the learner and content designer with information as to what "level of success" was achieved in reaching the learning objective(s).

## Course Introduction

Objective is to educate the student such that they gain additional mathematical concepts and procedures that will help them develop mathematical models of classical/textbook physical processes and then use those models to simulate the process. The processes to be considered in the course are:

1. Heat equation: Diffusion of heat associated with the three geometries:
1. Rod shaped object (1-D);
2. Plate rectangular and disk (2-D);
3. Sphere.
2. Wave equation: For the motion of a point mass:
1. For a string (1-D);
2. On a disk (2-D);
3. and on a sphere (3-D).

For each of the above geometries the Heat and Wave Equations will be solved for a variety of simple boundary conditions.

Solutions to the boundary value problems will be based on orthogonal functions. This will include:

• Fourier series,
• Bessel functions
• Legendre Polynomials

These will be introduced as needed in the lessons. MAPLE 15 has been used to develop graphs and simulations included in this course. MAPLE is a copyrighted software package. .

## Prerequisites

1. A solid background in calculus, specifically elementary functions, derivative and integration for single and multivariable functions.
2. An understanding of 1st and 2nd order ordinary differential equations.
3. Some exposure to applied linear algebra.

## Outline

1. Introduction
2. Review of initial value problems.
3. Introduction to two point boundary value problems and their solution using orthogonal functions.
4. Diffusion equation
1. Basics of vector analysis
2. Flow generated by a vector field
5. Heat Equation
1. Heat Flow
2. 1-D Heat equation
3. 2-D Heat equation
4. Heat equation in polar coordinates
5. Heat equation cube
6. Heat equation sphere
6. Wave Equation
7. Electromagnetism
8. Continuum mechanics
9. Fluid Mechanics

Note: Solutions to quizzes and homework will be posted as they become available. And C is for under CONSTRUCTION.

## Introduction

• Introduction
• Lesson 1: What are the class of physical problems to be addressed through mathematical models.(C)

## Review of Initial Value Problems

• Lesson 2: Initial Value Problems Lesson 2: Review of concepts and skills associated with solving basic first order initial value problems (IVPs).(C)
• Lesson 3: Initial Value Problems Lesson 3: Review of concepts and skills associated with solving second order initial value problems (IVPs). (C)

## Introduction to Two Point Boundary Value Problems (BVPs) and orthogonal functions

• Lesson 4: Introduction to two point BVPs Lesson 4: Problems using ODEs with two conditions.
• Lesson 4.1: Fourier series and orthogonal functions Lesson 4.1. (C)

## Vectors, Vector Fields, and associated operation and operators

• Lesson : Background on vectors,gradients,scalar fields,vector fields,curl, and divergence.Lesson 5.1
• Lesson : Divergence Theorem Divergence Theorem. (C)

## Heat Equation

• Lesson : Derivation of Heat Equation Derivation of Heat Equation
• Lesson : 1-D Heat Equation Lesson 5.2$u_{xx}={\frac {1}{k}}u_{t}$ (C)
• Lesson : 2-D Potential Equation Lesson 7$\nabla ^{2}u=0$ (C)
• Lesson : 2-D Heat Lesson 8 $\nabla ^{2}u=u_{t}$ (C)
• Lesson : 3-D Heat Cube and Sphere

## Wave Equation

• Lesson : 1-D Wave equation Lesson 6$u_{xx}={\frac {1}{c^{2}}}u_{tt}$ (C)
• Lesson : 2-D Wave Lesson 9 $\nabla ^{2}u={\frac {1}{c^{2}}}u_{tt}$ (C)
• Lesson : 3-D Spherical 3-D Spherical
• Lesson: Spherical harmonics Lesson 10
• Lesson : Numerical Methods (C)

## Course Specific Resources

• Problem Database Problem Lookup (C): This is a table of BVPs: 1) Type of PDE; 2) Type of BCs; 3) Initial Conditions. This will allow learners to find an example for a specific type of problem they are interested in, or a more general class without searching through all of the course material. This would be a great location for persons to add BVPs and solutions.