Mathematical Modeling with Multidisciplinary Applications

CHAPTER 7

INVERSE PROBLEMS IN ODEs

H. KUNZE¹ AND D. LA TORRE²

¹Department of Mathematics and Statistics, University of Guelph, Canada

²Department of Economics, Business and Statistics, University of Milan, Italy

Early undergraduate courses in differential equations typically focus on solution methods, mathematical modeling, and interpretation of solutions and models. Seeking the solution of a given differential equation or system is often called the “direct problem.” On the other hand, the “inverse problem” asks us to find an appropriate model (differential equation or system), the solution of which agrees well with some experimental or real-world observations.

For example, we might believe that a particular predator-prey system of differential equations models the interactions between the rabbits and foxes in a given region, and we can gather some population data over some period of time. From this data, the inverse problem might ask us to estimate the parameters (the coefficients) in the model. Many questions may come to mind. Some are:

1. Since the differential equations are nonlinear, we cannot solve them explicitly, and we can only solve them numerically if we pick values for the coefficients. So, how do we solve the inverse problem?

2. If we only gather population data every week, say, and there are surely measurement errors, our method for solving the inverse problem needs to be pretty robust. Is it possible to construct such a method?

3. The method should be mathematically ...