1. Process & Systems Modeling with Differential Equations
A Lecture in ENGIANA

2. Introduction
It is often desirable to describe the behavior of some real-life system or phenomenon (whether physical, sociological, or even economic) in mathematical terms.
The mathematical description of a system of phenomenon is called a mathematical model and is constructed with certain goals in mind.

3. Introduction For example, we may wish to understand
the mechanism of a certain ecosystem by studying the growth of animal populations in that system, or
we may wish to date fossils by analyzing the decay of radioactive substance either in the fossil or in the stratum in which it was discovered.

4. Introduction
Since the descriptions or assumptions made about a system frequently involve a rate of change of one or more of the variables involved, the mathematical depiction of all these assumptions may be one or more equations involving derivatives.
In other words, the mathematical model may be a differential equation or a system of differential equations.

5. Introduction
A mathematical model of a physical system will often involve the variable time t.
A solution of the model then gives the state of the system.
In other words, the values of the dependent variable (or variables) for appropriate values of t describe the system in the past, present and future.

6. Introduction We will examine the modeling of the following topics:
Population Dynamics
Logistic Equation
Radioactive Decay
Newton’s Law of Cooling/Warming
Mixing
Series Circuits

7. Population Dynamics
One of the earliest attempts to model human population growth by means of mathematics was by the English economist Thomas Malthus in 1798.
Basically, the idea behind the Malthusian model is the assumption that the rate at which the population of a country grows at a certain time is proportional to the total population of the country at that time.

8. Population Dynamics
In other words, the more people there are at time t, the more there are going to be in the future.
In mathematical terms, if P(t) denotes the total population at time t, then this assumption can be expressed as or

9. Population Dynamics where k is a constant of proportionality.
This simple model does not take into account factors such as immigration and emigration. Nevertheless, it is still used to model growth of small populations over short intervals of time (bacteria growing in a petri dish, for example).

10. Population Dynamics
The relative growth rate (or specific growth rate) is defined by k in

11. Population Dynamics
True cases of exponential growth over long periods of time are hard to find because the limited resources of the environment will at some time exert restrictions on the growth of a population.
Thus, for other models, (dP/dt)/P can be expected to decrease at the population P increases.

12. Example
The population of a town grows at a rate proportional to the population present at time t. The initial population of 500 increases by 15% in 10 years. What will be the population in 30 years? How fast is the population growing at t = 30?

16. Example
A bacterial culture is known to grow at a rate proportional to the amount present. After 1 hour, 1,000 strands of the bacteria are observed in the culture, and after 4 hours, 3,000 strands. Find (a) an expression for the number of strands of the bacteria in the culture and (b) the number of strands of the bacteria originally in the culture.

17. Solution:

21. Population Dynamics
In general, the assumption that the rate at which a population grows (or decreases) is dependent only on the number P present and not on any time-dependent mechanisms such as seasonal phenomena can be stated as

22. Population Dynamics
This differential equation is widely assumed in models of animal populations and is called the density-dependent hypothesis.

23. Logistic Equation
Suppose that an environment has a maximum relative growth rate of r in the beginning.
Suppose further than the environment is capable of sustaining no more than a fixed number K of individuals in its population.
The quantity K is called the carrying capacity of the environment.

24. Logistic Equation
This means at when the population reaches K, the relative growth rate becomes zero.
The simplest assumption we can make on the relative growth rate is that it is a linear function of the population P.

27. Logistic Equation
Thus, the equation becomes

28. Logistic Equation
Or, with relabeled constants,
This equation is known as the logistic equation, its solution the logistic function, and its graph the logistic curve.

29. Logistic Equation
Logistic curves have proved to be quite accurate in predicting the growth patterns, in limited space, of certain types of bacteria, protozoa, water fleas (Daphnia), and fruit flies (Drosophila).

30. Example
A variation on the population problem is when the population size has an inhibitory effect on the growth of the population. Hence, if x is the population size, then
This kind of model can be extended to many situations, called self-limiting problems. One person in a small town with a population of 200 proceeds to spread a rumor.

31. If the rate of which the rumor spreads is proportional to the number of people who know the rumor (x) as well as the number of people who haven’t heard of the rumor, find an expression for the number of people who know about the rumor after t days.
Fifty people know about the rumor after one day. How many people will have heard of the rumor after 2 days? Will the entire town eventually hear the rumor?

38. Radioactive Decay
The nucleus of an atom consists of combinations of protons and neutrons.
Many of these combinations of protons and neutrons are unstable – that is, the atoms decay or transmute into atoms of another substance.
Such nuclei are said to be radioactive.

39. Radioactive Decay
For example, over time the highly radioactive radium, Ra-226, transmutes into the radioactive gas radon, Rn-222.
In physics and chemistry, the half-life is a measure of the stability of a radioactive substance.
The half-life is simply the time it takes for one-half of the atoms in an initial amount to disintegrate or transmute into the atoms of another element.

40. Radioactive Decay
The longer the half-life of a substance, the more stable it is.
For example, the half-life of a highly radioactive radium, Ra-226, is about 1,700 years. This means that in 1,700 years, one half of a given quantity of Ra-226 is transmuted into radon, Rn-222.
The most commonly occurring uranium isotope, U-238, has a half-life of approximately 4.5 billion years, after which one-half of a quantity of U-238 is transmuted into lead, Pb-206.

41. Radioactive Decay
To model the phenomenon of radioactive decay, it is assumed that the rate dA/dt at which the nuclei of a substance decays is proportional to the amount A(t) of the substance remaining at time t:
Here, k is the proportionality constant.

42. Example
Radium disintegrates at a rate proportional to the amount present. If 100 mg are set aside now, 96 mg will be left 100 years hence. Find how much will be left t centuries from the time radium was set aside. Determine also the amount left after 2.56 centuries and the half-life of radium.

45. Growth and Decay
In general, the initial-value problem
where k is a constant of proportionality, serves as a model for diverse phenomenon involving either growth or decay

46. Problems describing growth (whether of populations, bacteria, or even capital) are characterized by a positive value of k, whereas problems involving decay (as in radioactive disintegration) yield a negative k value.

47. Accordingly, we say that k is either a growth constant (k > 0) or a decay constant (k < 0).

48. Growth and Decay The model for growth, , can also
be seen as the equation ,
which describes the growth of capital S when an annual rate of interest r is compounded continuously.

49. Growth and Decay The model for decay, , also
occurs in biological applications such as determining the half-life of a drug – the time that it takes for 50% of a drug to be eliminated from a body by excretion or metabolism.

50. Example
DNA (deoxyribonucleic acid) is a nucleic acid that serves as the main constituent of the chromosomes of living cells that determines individual heredity characteristics. The novel as well as the movie “Jurassic Park” assumes that dinosaur blood cells contain this genetic material and that dinosaurs can be cloned from fragments of preserved DNA. Scientific studies show that DNA deteriorates over time.

51. Example
If excavations and diggings of fossils indicate that only about 1.2% of DNA survive after 25 million years, what percent of an original sample would have been lost after 10 million years? In addition, estimate the half-life of the dinosaur DNA. Assume that dinosaur DNA starts deteriorating after a dinosaur’s death.

56. Newton’s Law of Cooling/Warming
According to Newton’s empirical law of cooling/warming, the rate at which the temperature of a body changes is proportional to the difference between the temperature of the body and the temperature of the surrounding medium, the so-called ambient temperature.

57. Newton’s Law of Cooling/Warming
If T(t) represents the temperature of a body at time t, Tm the temperature of the surrounding medium, and dT/dt the rate at which the temperature of the body changes, then Newton’s law of cooling/warming translates into
where k is a constant of proportionality.

58. Example
A thermometer reading 18oF is brought into a room, the temperature of which is 70oF. One minute later, the reading is 31oF. Determine the temperature reading as a function of time. Find the temperature reading 5 minutes after the thermometer is first brought into the room.

63. Exercise
A steel ball is placed at t = 0 in a medium which is maintained at a temperature of 30oC. Heat flows so rapidly within the ball that the temperature is essentially the same at all points of the ball. At the end of 3 minutes, the temperature of the ball is reduced to 70oC. After 15 minutes, the temperature of the ball becomes 42oC. Find the initial temperature of the steel ball.

64. Exercise
The body of a murder victim was “discovered” at 7:30 pm. The crime scene investigator on call arrived at 11:30 pm and immediately took the temperature of the body, which was 94.6oF. He again took the temperature after one hour, when it was 93.4oF, and he noted that the room temperature was constant at 70oF. Use Newton’s Law of Cooling to estimate the time of death, assuming the victim’s normal body temperature was 98.6oF.

65. Exercise
At 1:00 pm, a thermometer with a reading of 10oF was removed from a freezer and placed in a room whose temperature is 65oF. At 1:05 pm, the thermometer reads 25oF. Later, the thermometer is placed back in the freezer. At 1:30 pm, the thermometer reads 32oF. When was the thermometer returned to the freezer and what was the thermometer reading at that time?

66. Mixing
The mixing of two salt solutions of differing concentrations gives rise to a first-order differential equation for the amount of salt contained in the mixture.
Let us suppose that a large mixing tank initially holds Q gallons of brine (that is, water in which a certain amount of salt has been dissolved).

67. Mixing
Another brine solution is pumped into the mixing tank at a volumetric flow rate F1 with a salt concentration M1.
Assume that the solution in the tank is well-stirred and also be pumped out at a volumetric flow rate F2 with a salt concentration M2.
Let the amount of salt in the tank be a function of time, A(t).

68. Mixing
The input rate Rin at which salt enters the mixing tank is the product of the inflow concentration of salt and the inflow rate of liquid. In other words, Rin = M1F1.
If the liquid in the tank is well-mixed, then the concentration of salt in the tank is identical to the concentration of salt in the outflow. In other words, M2 = A/Q.

69. Mixing
The output rate Rin at which salt leaves the mixing tank is the product of the outflow concentration of salt and the outflow rate of liquid. In other words, Rout = M2F2.

70. Q(t) = volume of liquid in the container
F1, M1
Q(t) = volume of liquid in the container
F2, M2

72. Mixing
If Rin = Rout, there is no accumulation in the mixing tank. The volume of liquid in the tank is constant.
If Rin > Rout, there will be an accumulation of liquid in the mixing tank. Eventually, the liquid will overflow.
If Rin < Rout, then all the liquid in the tank will eventually drain. The tank will be emptied.

73. Example
A tank initially contains 100 gallons of water. Brine enters the tank at the rate of 3 gallons per minute. The mixture, thoroughly stirred, leaves the tank at the rate of 2 gallons per minute. If the concentration of the brine at the end of 20 minutes is to be 2 lb/gal, what should be the concentration of the brine entering the tank?

74. Solution: Given: Liquid Volume in tank (initial) = 100 gal
Flow rate into the tank = 3 gal/min
Concentration of incoming liquid = M1 lb/gal
Flow rate out of the tank = 2 gal/min
Concentration of outgoing liquid (after 20 minutes) = 2 lb/gal

80. Example
A tank initially holds 100 gallons of brine solution. At t = 0, fresh water is poured into the tank at the rate of 5 gpm, while the well-stirred mixture leaves the tank at the same rate. After 20 minutes, the tank contains 20/e lbs of salt. Find the initial concentration of the brine solution inside the tank.

81. Solution: Given: Liquid Volume in tank (initial) = 100 gal
Flow rate into the tank = 5 gal/min
Concentration of incoming liquid = 0 lb/gal
Flow rate out of the tank = 5 gal/min
Amount of salt in the tank (after 20 minutes) = 20/e lb

86. Practice Problem
Blood carries a drug into an organ at the rate of 3 cm3/sec and leaves at the same rate. The organ has a liquid volume of 125 cm3. If the concentration of the drug in the blood entering the organ is 0.2 g/cm3, what is the concentration of the drug in the organ at time t? After how many seconds will the concentration of the drug in the organ reach 0.1 g/cm3?

87. Practice Problem
A 300-gal capacity tank contains a solution of 200 gals of water and 50 lbs of salt. A solution containing 3 lbs of salt per gallon is allowed to flow into the tank at the rate of 4 gal/min. The mixture flows from the tank at the rate of 2 gal/min. How many pounds of salt are in the tank at the end of 30 min? When does the tank start to overflow? How much salt is in the tank at the end of 60 min?

88. Practice Problem
A tank contains 100 gal of water. In error, 300 lbs of salt was poured into the tank instead of 200 lb. To correct this condition, a stopper is removed from the bottom of the tank, allowing 3 gal of brine to flow out per minute. At the same time, 3 gallons of fresh water per minute are pumped into the tank. If the mixture is kept uniform by constant stirring, how long will it take for the brine to contain the desired amount of salt?

89. Electric Circuits
The basic laws governing the flow of electric current in a circuit or a network is presented in the succeeding slides without derivation.
The notation commonly used in most engineering texts are as follows:

90. Electric Circuits Symbol Unit Quantity t second s Time q coulomb C
Quantity of electricity (e.g., charge on a capacitor) I
ampere A
Current, time rate of flow of electricity E
volt V
Electromotive force or voltage R
ohm 
Resistance L
henry H
Inductance
farad F
capacitance

91. Electric Circuits
The current at each point in a network may be determined by solving the equations that result from applying Kirchhoff’s laws:
The sum of the currents into (or away from) any point is zero.
Around any closed path the sum of the instantaneous voltage drops in a specified direction is zero.

92. Electric Circuits
A circuit is treated as a network containing only one closed path.
For a circuit, Kirchhoff’s current law indicates that the current is the same throughout the circuit.
To apply Kirchhoff’s voltage law, it is necessary to know the contributions of each of the idealized elements in the circuit.

93. An LRC-series circuit

94. Electric Circuits
For a series circuit containing only an inductor and a resistor (LR series circuit), Kirchhoff’s second law states that the sum of the voltage drop across the inductor (L(di/dt)) and the voltage drop across the resistor (iR) is the same as the impressed voltage (E(t)) on the circuit.

95. LR-series circuit

96. Electric Circuits
For a series circuit containing only a resistor and a capacitor (RC series circuit), Kirchhoff’s second law states that the sum of the voltage drop across the resistor (iR) and the voltage drop across the capacitor (q/C) is the same as the impressed voltage (E(t)) on the circuit.

97. RC-series circuit

98. But current i and charge q are related by i = dq/dt, so the equation can be rewritten as

99. Electric Circuits L, C, and R are constants:
L  inductance
C  capacitance
R  resistance
The current i(t) is also called the response of the system.

100. Example
A 10-ohm resistor and a 5-henry inductor are connected to a 50-volt source at t = 0. Express the current I as a function of time. What happens to the value of I as time goes on indefinitely?

101. 10 
5 H
50 V

105. Example
A decaying emf (E = 200e-5t) is connected in series with a 20-ohm resistor and a 0.01 farad capacitor. Assuming q = 0 at t = 0, find the charge and current at any time. Show that the charge reaches a maximum; calculate it; and find when it is reached.

106. 20 
0.01 F
200e-5t

112. Example
A resistor (of unknown constant resistance) and a 0.02 farad capacitor are connected in series with an emf of E = 100 volts. The initial charge of the capacitor is 5 coulombs. If at t = 1/10 second the current flowing is 30/e amperes on the reverse direction, find the value of the resistance.

113. Example
An electric circuit contains an 8-ohm resistor in series with an inductor of 0.5 henry and a battery of E volts. At t = 0, the current is zero. Find E if the maximum current flowing in the circuit is 8 amperes.

114. Example
A 20-ohm resistor and a capacitor are connected in series with a battery of 60 volts. At t = 0, there is no charge on the capacitor. Find the capacitance if the current at t = 5 seconds is 3/e5 amperes.

115. Example
A coil of inductance 1 henry and a resistance of 20 ohms is connected to an emf E = 200sin20t. If I = 0 when t = 0, find I at t = 0.06 seconds.