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Math 308, Sections 301, 302, Summer 2008 Lecture 4. 06/4/2008 Chapter 3. Mathematical methods and numerical methods involving first order equations. Section 3.1 Mathematical modeling. Formulate the problem Here you must pose the problem in such a way that it can be ”answered” mathematically. Develop the model There are two things to be one here. First, you must decide which variables are important and which are not. The former are then classified as independent variables or dependent variables. The unimportant variables are those that have very little or no effect on the process. The independent variables are those whose effect is significant. The dependent variables are those that are affected by the independent variables and that are important to solving the problem. Second, you must determine or specify the relationships that exist among the relevant variables. Test the model The following questions should be answered: ◮ Are the assumptions reasonable? ◮ Are the equations dimensionally consistent? ◮ Is the model internally consistent in the sense that equations do not contradict one another? ◮ Does the relevant equation have solutions? ◮ Is the solution unique? ◮ How difficult is to obtain the solutions? ◮ Do the solutions provide an answer for the problem being studied? ◮ You should always keep in mind that a model is not a reality but only representation of reality. Section 3.2 Compartmental analysis Many complicated processes can be broken down into distinct stages and the entire system modeled by describing the interactions between the various stages. Such systems are called compartmental. The basic one-compartment system consists of a function x(t) that represents the amount of a substance in the compartment at time t, an input rate at which the substance enters the compartment, and an output rate at which the substance leaves the compartment INPUT RATE −→ x(t) −→ OUTPUT RATE Because the derivative of x with respect to t can be interpreted as the rate of change in the amount of the substance in the compartment with respect to time, the one-compartment system suggests dx = input rate − output rate dt as a mathematical model for the process. Mixing problem. Example 1. A brine solution of salt flows at a constant rate of 8L/min into a large tank tat initially held 100L of brine solution in which was dissolved 0.5kg of salt. The solution inside the tank is kept well stirred and flows out of the tank in the same rate. If the concentration of salt in the brine entering the tank is 0.05kg /L, determine the mass of salt in the tank after t min. When will the concentration of salt in the tank reach 0.02kg /L? Example 2. For the mixing problem described in Example 1, assume now that the brine leaves the tank at a rate of 6L/min, with all else being the same. Determine the mass of salt in the tank after t min. Population models Let p(t) be the population of bacteria at time t. In our model we assume that the growth rate is proportional to the population present. We also assume that the death rate is zero. The mathematical model for population of bacteria is dp = k1 p, dt p(0) = p0 , where k1 > 0 is the proportionality constant for the growth rate and p0 is the population at time t = 0. For human population the assumption that the death rate is zero is wrong! If we assume that the people die only of natural causes, we might expect the death rate also to be proportional to the size of the population. So, we can rewrite formula dp = k1 p − k2 p = (k1 − k2 )p = kp, dt where k = k1 − k2 and k2 is the proportionality constant for the death rate. Let’s assume that k1 > k2 so that k > 0. This gives the mathematical model dp = kp, dt p(0) = p0 , which is called the Malthusian or exponential, law of population growth. The solution to this initial value problem is p(t) = p0 ekt . Example 3. In 1980 the Department of Natural Resources released 1000 splake (a crossbreed of fish) into a lake. In 1987 the population of splake in the lake was estimated to be 3000. Using the Malthusian law, estimate the population of splake in the lake in the year 2010. What about premature death? We might assume that another component of the death is proportional to the number of two-party interactions. There are p(p − 1)/2 such possible interactions for a population of size p. Thus, if we combine the birth rate with the death rate and rearrange constants, we get the logistic model dp = −Ap(p − p1 ), p(0) = p0 , dt where A = k3 /2 and p1 = (2k1 /k3 ) + 1. This equation has two constant (equilibrium) solutions p(t) = p1 and p(t) = 0. The nonequilibrium solutions can be found by separating variables Z Z dp = −A dt p(p − p1 ) If p(0) = p0 , and c = 1 − p1 /p0 , then solving for p(t), we find p(t) = p1 p0 p1 = . −Ap t 1 1 − ce p0 + (p1 − p0 )e−Ap1 t The function p(t) is called the logistic function. Example 4. Assume that the rate of decay of a radioactive substance is proportional to the amount of the substance present. If initially there are 50 g of a radioactive substance and after 3 days there are only 10 g remaining, what percentage of the original amount remains after 4 days?