In Chapter 3 we continue with our discussion of ODEs, including code examples in C++ and Python.
2.2 BACKGROUND AND PROBLEM STATEMENT
In this section we introduce the very first differential equation of this book. It is a scalar first-order linear ordinary differential equation (ODE), and we shall analyse it from several qualitative and quantitative viewpoints.
Consider a bounded interval where . This interval could represent time or distance, for example. In most cases we shall view this interval as representing time values. In the interval we define the initial value problem (IVP) for an ODE:
where is a first-order linear differential operator involving the derivative with respect to the time variable and is a strictly positive function in . The term is called the inhomogeneous forcing term, and it is independent of . Finally, the solution to the IVP must be specified at ; this is the so-called initial condition.
In general, the problem (2.1) has a unique solution given by:
(See Hochstadt (1964), where the so-called integration factor is used to determine a solution.)
A special case of (2.1) is when the right-hand term is zero and is constant; in this case the solution becomes a simple exponential term without any integrals, and this will be used later when we examine difference schemes to determine their feasibility. In particular, a scheme that behaves badly for the above special case will be unsuitable for more general or more complex problems unless some modifications are introduced.
2.2.1 Qualitative Properties of the Solution and Maximum Principle
Before we introduce difference schemes for (2.1), we discuss a number of results that allow us to describe how the solution behaves. First, we wish to conclude that if the initial value and inhomogeneous term are positive, then the solution should also be positive for any value in . This so-called positivity or monotonicity result should be reflected in our difference schemes (not all schemes possess this property). Second, we wish to know how the solution grows or decreases as a function of time. The following two results deal with these issues.
Lemma 2.1 (Positivity). Let the operator be defined in Equation (2.1), and let be a well-behaved function satisfying the inequalities:
Then the following result holds true:
Roughly speaking, this lemma states that you cannot get a negative solution from positive input.
You can verify it by examining Equation (2.2) because all terms are positive.
The following result gives bounds on the growth of .
