Differential Equations
The Language of Continual Change
We need to start with some basic definitions. "The naming of things" is often the first step in wading into any new topic.
(NOTE: I am initially filling out the details of these posts, while studying PMTH339 Differential Equations, at UNE (Unit Coordinator: Wenjie Ni ). I'm not sure, yet, of the schedule for publishing these pages. I may keep them "in draft" for a while. If the content on this topic seems to abruptly end at some point, that may simply be as far as I have progressed on the subject myself, or as far as I have created content I am willing to publish.)
We need to define what Differential Equations are, then continue to name and define a few other related concepts. Some of them will not make sense straight away. The ones that do might seem to be without a purpose, or... even, so obvious that you wonder "why do we bother having a special name for that."
These questions are usually answered by further study - but they also often remain the important bedrock of what is most important about a topic (when compared to others) - and should be remembered, as you continue deeper into the topic's subject matter.
So let's start with the real basics:
What is a Differential Equation?
At its simplest, a Differential Equation is:
an equation that contains a reference to a function, and any of that functions derivatives.
It's as broad and simple as that.
These Differential Equations (DEs) are then be broken down into subcategories - so we can analyse the features of each type. But, it is important to remember just how broad that definition is - and to recognise how powerful that relationship (between a function, and its derivatives) is, in general.
What does it mean to "solve" a Differential Equation
To solve a Differential Equation means:
to find a function or a set of functions that satisfies the equation for all values in its domain.
Essentially, solving a Differential Equation involves determining the mathematical expression that accurately describes the relationship between the quantities that vary, within that equation, as they vary relative to each other. This solution must also exclude any values that are not possible, given the values (and the "constraints" - see below) as they each vary across their domain.
This solution can either be:
exact - expressed as a formula - or,
numerical, approximating values at specific points.
TODO: review this definition this is an amalgam of my understanding so far - but I suspect it can be much more clearly / simply defined for the beginner in this subject - in a way (maybe with examples) that gives someone unfamiliar with the subject more insight.
Types of Differential Equations
Type | Definition | Example |
Ordinary Differential Equation (ODE) | an equation that involves a function of one variable, and any of its derivatives | ` y'' + 4y = \cos{x} ` |
Partial Differential Equations (PDE) | an equation that involves a function of several variables and its partial derivatives | ` \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 ` |
The "Order" of a Differential Equation
Let n be the the highest order of a derivative that occurs in an Ordinary Differential Equations (ODE) - OR, the highest order of a partial derivative that occurs in a Partial Differential Equation (PDE). The "order" of that Differential Equation is then said to be n.
Examples
` y'' + 4y = \cos{x} `
is a second order ODE for the unknown function f(x), and:
` \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 `
is a second order PDE for the unknown function u(x, y).
This second example is a famous Differential Equation, known as "Laplace's Equations". We will see more of Laplace's Equation later.
TODO: make section on Laplace's Equations + Harmonic Functions - what they are and why they are important. Link to here.
System
If we have a collection of Differential Equations that a system satisfies simultaneously, we call that collection:
a System of Differential Equations
This is analogous to the use of the word "System", in relation to Algebraic Questions.
Constraints
Boundary conditions and initial conditions both specify constraints that help determine the solutions to Differential Equations, but they apply in different contexts and have distinct roles.
Initial Conditions
Within some systems, there will be further limitations on how solutions to a Differential Equation can be defined. Many systems require, or naturally impose, initial conditions, in order for the a specific solution to that Differential Equation to be found.
The idea of initial conditions in Differential Equations, is explored further on this page.
Boundary conditions
Boundary conditions specify the values that a solution to a Differential Equation must take on at the boundary of the domain in which the solution is defined. These conditions are are often used to solve differential equations that describe physical phenomena where the solution is constrained by the conditions at the edges or boundaries of a domain.
Types of Boundary Conditions
The three common forms of boundary conditions are:
Dirichlet boundary conditions: These specify the value of the function at the boundary.
Neumann boundary conditions: These specify the value of the derivative of the function at the boundary.
Robin boundary conditions: A combination of Dirichlet and Neumann, involving both the function and its derivative at the boundary.
There is also a concept of a Mixed boundary condition, and a Cauchy boundary condition.
(TODO: spell out these definitions in a separate section / page. For now, for more information, one good resource I found is here: https://www.simscale.com/docs/simwiki/numerics-background/what-are-boundary-conditions/)
Constraint Comparison
Dimensionality: Initial conditions are often given at a specific time point, defining a reference point in time from which the evolution of the system can be defined. Boundary conditions are often given at spatial boundaries and can be static or evolve over time.
Usage in Equations: Initial conditions are important in time-dependent problems to start the integration process, while boundary conditions are important in spatial problems, so that the solution can fit the physical or geometric constraints of the problem.
While both types of constraints help determine the solutions of a Differential Equation, general speaking boundary conditions deal with spatial limits and physical constraints at the edges, whereas initial conditions set the stage for the evolution of a system over time.
"Well-posed" and "Ill-posed" Differential Equations
The concept of "ill-posed" and "well-posed" Differential Equations help to determine whether a mathematical model and its solutions are practically useful and theoretically sound.
Well-posed Problem
A problem in Differential Equations is considered well-posed if it satisfies the following three criteria:
Existence: The problem has a solution.
Uniqueness: The solution is unique; no two different solutions can satisfy the problem under the same conditions.
Stability: The solution's behavior changes continuously with the initial conditions or parameters. This means small changes in the input lead to small changes in the output, ensuring that the model is sensitive and responsive in a controlled manner.
Well-posed problems are important in both theoretical and applied mathematics. The distinction is important in ensuring that the problem is meaningful and the solutions are dependable for modeling real-world phenomena.
(NOTE: I am currently fascinated by the interaction between "well-posed"/"ill-posed" problems and Chaos theory. I don't know enough of the details of either, yet - but it looks to me, at first glance, that many chaotic systems could be defined as "well-posed" in the sense that the solution is "continuous", in a strict sense - but that the changes (while "smooth" and "continuous" are still "too fast" to allow for practical application of the solution. This would seem to me to be the essential insight of Chaos Theory... but, as I say, I don't know enough about either yet, to be sure. TODO: come back - investigate this, and flesh out the thought if it seems to be worth it. As a start I threw this page together, on my findings related to "well-posed" vs "ill-posed" problems and Chaos Theory)
