Linking modeling to ideas

One corner of the modeling triangle is deriving mathematical statements from experimental observations and physical laws. Another is using mathematical concepts--analytical, graphical, and numerical--to analyze the behavior predicted by the model. The third is interpreting that behavior in light of the original physical problem. Tables 3, 4, and 5 summarize the mathematical concepts that are associated with the analysis and interpretation of many of the models introduced in this text.

From a modeling point of view, the best approach to constructing a syllabus is to decide which ideas you regard as most important and which models are likely to be of most interest to you and your students. Use tables 3, 4, and 5 to strike the balance that is best for you--the left-hand columns of those tables list models, the right-hand columns list the ideas that are introduced using those differential equations.

The time you spend in your course with analysis and interpretation is time in part devoted to modeling and time in part devoted to mathematics. For many students, seeing mathematics explain physical phenomena is the most powerful motivation for its study. From that perspective, you might decide to choose the models you emphasize by the ideas that are exemplified in their analysis, that is, by selecting solely from the right-hand column of tables 3, 4, and 5.

 

Section Ideas used

Rock model: v' = -g 1.2 direction field, solution graph Euler's method, solution formula

3.2 direction field, solution graph

Simple: P' = kP 2.1 direction field, solution graph, solution formula

3.1 Euler's, Heun's methods

3.2 direction field, solution graph,

4.1 general solution

4.2-4.3 solution formula

Emigration: P' = kP - E 2.2.1 direction field, solution graph, steady state, stability

3.2 direction field, solution graph, phase line

3.3 steady state, stability

4.1 general solution

4.2, 4.4 solution formula

10.4-10.5 Laplace transform solution

Logistic: P' = aP - s P² 2.2.2; 3.2 direction field, solution graph, phase line

3.3 linear stability analysis

4.2 solution formula

4.6 uniqueness

$$T' = -(Ak/cm)(T - T_{\mathrm{out}})$$ 2.3 direction field, solution graph, steady state, stability, Euler's method

3.1 Euler's, Heun's methods

3.2 phase line

3.3 steady state, stability

4.1 general solution

 

 

Section Ideas used

Predator-prey (2.24-2.25), p. 67 2.4.1 steady state, stability, Euler's method

Epidemic (SIR, SIRS) 2.4.2 steady state, stability,

(2.29-2.30), p. 71 phase plane

7.2 nullclines, local linearization

Competition (2.35-2.36), p. 73 2.4.3 steady state, phase plane Spring-mass:

mx'' + px' + kx = F(t) 5.1 solution formula, phase plane, Euler's method

6.1 general solution

6.3-6.5 solution formulas (homogeneous: undamped, overdamped, etc.

6.6-6.7 solution formulas (nonhomogeneous: resonance, etc.)

$$L \theta'' + g \sin \theta = 0$$ 5.2 linearization, solution formula

6.1 steady state, linearization

6.5, 6.8 solution formula (linear)

6.8 steady state, linear stability

7.1 phase plane

7.2 nullclines, local linearization

8.3 linear stability analysis, phase plane van der Pol:

$$L i'' + \epsilon(i^2 - l)i' + i/C = 0$$ 7.3 limit cycle

 

Section Ideas used

-D(A c')' +(A V c)' = 0 9.2 solution formula

9.3 numerical approximation (finite differences)

11.2 solution formula

$$\partial T/\partial t = \kappa \partial T^2/\partial x^2$$ 9.5 eigenfunction solution

9.6 numerical approximation (method of lines), equilibria

11.5 spatial eigenfunctions (Bessel)

caption

[Ideas Used in Analyzing Higher-Order Models (cont'd)] Continuation from table 4 of the list of analytical, numerical and graphical tools used in analyzing higher-order models. The center column identifies the section in which the ideas listed in the right-hand column are used to analyze the given model. See tables 1 and 2 for the section in which each model is introduced.