ASSIGNMENT
The purpose of this assignment is for you to explore the results of modeling situations using systems of differential equations.
Introduction to Problems 1 and 2:
The battle of the Spanish Armada in 1588 was one of modern history's most important conflicts. It was part of an undeclared war between Spain, Europe's "superpower" at that time, and England. Spain was the richest, most powerful country in Europe at the time, and it was staunchly Roman Catholic.
The Spanish were ruled by King Philip II, who ruled a quarter of western Europe's population, plus Mexico, Peru, the Philippines and numerous islands and trading bases. Spain ruled the Netherlands in Europe, but in 1588, it was battling a Dutch revolt in the northern provinces of the Spanish Netherlands. In 1588, the most powerful, best-organized, best-equipped army in Europe - the Spanish 'Army of Flanders', under command of the respected and feared Duke of Parma - had almost crushed the Dutch revolt.
By contrast, England was a nation that had seen much turmoil in the early 1500's. Relative to Spain, it was a poor country with a small army. England was ruled by Queen Elizabeth I, who was Protestant. Elizabeth sought to advance the cause of Protestantism where possible, including supporting the Protestant Dutch Revolt against Spain. Additionally, England's privateershad great success in looting Spanish ships transporting treasure from the Spanish American colonies to Spain.
Philip II was a devout Roman Catholic. As a zealous Catholic, it was his deep wish to return England to Catholicism. In retaliation for supporting the rebels in the Netherlands and encouraging privateers to loot Spanish treasure ships, Philip planned an expedition to invade England and overthrow Elizabeth, thereby reversing the Protestant Reformation in England, ending the English support for the Dutch resistance to Spanish rule, and cutting off English attacks on Spanish trade and settlements in its American colonies. The Spanish Armada was a fleet of 130 ships under the command of the Duke of Medina Sidonia that was built to sweep the seas around the British Isles and transport an army under the Duke of Parma, from the Netherlands, across the Channel to invade England and overthrow Elizabeth. On May 28, 1588 the Armada set sail from Lisbon (Portugal) and headed for the English Channel.
The naval fleet defending England against the Spanish Armada consisted of 197 ships. Although the English fleet outnumbered the Spanish in ships, 197 to 130, the Spanish fleet outgunned the English: the Spanish had 5 guns on its ships to every 3 English guns on its ships.
Problem 1: The Spanish Armada Campaign
You are going to model the Spanish Armada campaign using a system of differential equations.
1. For this problem, let x(t) represent the number of English ships and y(t) represent the number of Spanish ships still operational t hours after the fighting begins in earnest.
x(0) is the number of ships that the English have at the beginning of the campaign;
y(0), the number of ships the Spanish have at the beginning.
Based on the introduction above, what are the initial conditions for the two sides?
Type or write "The initial conditions are x(0) = _____ and y(0) = _____.", where you fill in the blanks with the appropriate numbers.
2. Although the English outnumbered the Spanish in the number of ships, the Spanish had more total guns than the English: 2500 to 1500. If we assume that each gun on either side, on average, causes the same amount of damage in battle and once a ship becomes non-operational, it cannot be returned to the fleet, we can model the rate of loss of ships during the campaign by the differential equations:
dx/dt = -0.05y
dy/dt = -0.03x
Write a short paragraph explaining how this model makes sense.
3. Use a computer program or computing algebra system (CAS) calculator to find the solution equations of the initial-value system of differential equations given in paragraphs 1 and 2 above.
Type or write "The solution equations for the problem are", followed by your answers.
NOTES:
a. To solve an initial-value system of differential equations, in the command line, enter the equations and conditions, separated by commas, and click the "=" button.
b. You do NOT need to use an asterisk to represent multiplication operations; however, you may do so if you wish. For example, to enter "2x", type "2x".
c. You must use a carat "^" to raise an expression to a power. For example, to enter "2x", type "2^x".
d. You can use "e" to represent the natural base e. For example, to enter "e2x",type "e^2x".
e. You can use either differential or prime notation to enter a derivative expression. For example, you can express the derivative of y with respect to x as either "dy/dx" or "y'".
4. Using the solution equations you obtained in paragraph 3, evaluate the number of English and Spanish ships remaining 10, 20, and 30 hours after the fighting begins in earnest.
Type or write
"After 10 hours, there are _____ English ships and _____ Spanish ships.
After 20 hours, there are _____ English ships and _____ Spanish ships.
After 30 hours, there are _____ English ships and _____ Spanish ships.",
where you fill in the blanks with the appropriate numbers, rounded off to the nearest whole number.
To evaluate the solution equations for different values of t, enter only the right-hand sides of the equations and t = "value", separated by commas.
5. According to the solution equations, who wins? Assume that a country wins the campaign when the number of the opponent's operational ships is 20 or less.
Type or write "___________ wins.", where you fill in the blank with either "England" or "Spain".
6. How long, to the nearest hour, does the campaign last after the fighting begins in earnest?
Type or write "The campaign lasts ____ hours.", where you fill in the blank with the appropriate whole number.
7. Graphing each solution equation with respect to time, x(t) and y(t), would give you a picture of how the battle unfolds for each side. However, another useful way to see what happened, and what could have happened, in this situation is to graph the rates of change of the two dependent variables, x and y, on an xy-coordinate system. Such a plot is a two-dimensional form of a direction field, called a phase portrait. To create a phase portrait, short line segments are drawn at various points on the xy-coordinate system that display the instantaneous changes of x and ywith respect to tat the points. In other words, the short line segments show the rates of change of both x and y as functions of t, or (dy/dt)/(dx/dt) = dy/dx.
Produce a graph of the phase portrait (direction field) of the system of differential equations
dx/dt = -0.05y
dy/dt = -0.03x
that displays at least 400 slope segments on the intervals 0 ≤ t ≤ 80, 0 ≤ x ≤ 200, and 0 ≤ y ≤ 200.
Attach the phase portrait (direction field) graph to your assignment submission.
This site provides an "applet" that graphs phase portraits (direction fields) and performs numerical calculations for systems of two differential equations. It is the same applet that you used in Assignment #1, modified to allow you to enter a system of two differential equations. First, make sure that "x" is selected from the drop-down menu in the first line below the graphing box. Next, enter the formulas for the two derivatives, dx/dt and dy/dt, in the second and third lines below the graphing box.
NOTES:
a. You must use an asterisk to represent all multiplication operations. For example, to enter the expression "2x", you type 2*x".
b. You must use a carat "^" to raise most expressions to a power. For example, to enter the expression "2x", you type "2^x".
c. You must use "exp( )" to represent e raised to a power. For example, to enter the expression "e2x", you type "exp(2*x)".
Enter the maximum and minimum t-, x-, and y-values for the phase portrait on the fourth through sixth lines. The seventh line allows you to determine how many points in your display will display slope segments. For this problem, you should use "20" t, x, and y segments: this will plot slope segments at 400 points. Click the "Submit All" button at the end of the seventh line to obtain the phase portrait plot.
To print your phase portrait, click on the "Frame" button at the lower right of the window to open the phase portrait in a separate window. Click on "Edit" in the top menu line and select "Change Title" to give your graph a title. For this problem, you can use the title "Problem #1, paragraph 7". Next, click on "File" in the top menu line and select "Print Panel". Click the "OK" button in the "Print" window. I was unable to figure out how to copy these direction fields so that you could paste them into another document. Unless you can figure an easy way to do this, just print the frames that you want and attach them to your assignment submission.
8. Show the solution for the initial values you determined in paragraph 1.
Attach a second phase portrait graph showing the solution to your assignment submission.
Once you have graphed the phase portrait for a system of differential equations, you can graph individual numerical (approximate) solutions for initial conditions by filling in the x- and y-values of the conditions, one at a time, in the ninth line below the phase portrait box. Before you do this, change the entry in the drop-down box to the right of "Init. Conds" on the eighth line from "Mod. Euler" to "RK4" and change the "Step:" in the box on the far right of the row from "0.3" to "1". This has the program compute the solution values using the Runge-Kutta method of order 4 for systems of equations (we do NOT cover this in the course) with step sizes of 1. Using a step size of 1 shows the approximate values of x and y at the end of every hour of the campaign. After doing this, type in the values of the initial conditions on the ninth line and click on the "Submit" box on that line. You will see the solution on the graph. Title the graph "Problem #1, paragraph 8".
9. Now approximate the values of x and y when t = 10, 20, and 30 using the fourth-order Runge-Kutta Method with step size 1.
Type or write
"After 10 hours, there are _____ English ships and _____ Spanish ships.
After 20 hours, there are _____ English ships and _____ Spanish ships.
After 30 hours, there are _____ English ships and _____ Spanish ships.",
where you fill in the blanks with the appropriate numbers, rounded off to the nearest whole number.
Type a sentence comparing these approximate answers with the exact answers you obtained in paragraph 4 above.
If you have followed the steps above for this site, you have already set it up to provide fourth-order Runge-Kutta approximations with step sizes of 1. If necessary, re-enter the initial conditions on the ninth line below the phase portrait box and then click the "Show Table" box on that line. A "Solution Table" window will open that gives the results of the Runge-Kutta Method for all of the solution points plotted on the graph.
10. Who would have won, England or Spain, if England began with the same number of ships as it did historically and Spain had entered the campaign with 140 ships? 150 ships? 160 ships?
Assume that a country wins the campaign when the number of the opponent's operational ships is 20 or less.
Type or write
"If Spain had 140 ships, ________ would have won.
If Spain had 150 ships, ________ would have won.
If Spain had 160 ships, ________ would have won.",
where you fill in the blanks with either "England" or "Spain".
Change the appropriate initial condition to find out the answers.
11. According to this model, if England began with the same number of ships as it did historically, how many ships would Spain have needed at the start of the campaign for the campaign to have been a tie - the number of both countries' ships dropped to 20 or below in the same number of hours of fighting.
The number will need to be a number accurate to one decimal place!
Type or write "The campaign would have been a tie if Spain started with ____ ships.", where you fill in the blank with the appropriate number, accurate to one decimal place.
Problem 2: The Battle of Gravelines
After the English fleet unsuccessfully attempted to intercept the Armada shortly after it left Spain in May 1588, and after several skirmishes between the two navies as the Armada entered the English channel in July and August, the Spanish Armada reached the small port of Gravelines, part of Flanders in the Spanish Netherlands, where it was to pick up the Spanish invasion army to transport to England. On August 7, the Duke of Medina Sidonia, commander of the Armada, received word that the Duke of Parma's army would require six more days to be ready to board the ships to England. On August 8, 1588, the English navy attacked the Spanish Armada at Gravelines. The Battle of Gravelines was the crucial battle in the campaign of the Spanish Armada.
1. As in Problem #1, let x(t) represent the number of English ships and y(t) represent the number of Spanish ships still operational t hours after the battle begins. For this problem, x(0) is the number of ships that the English have at the beginning of the battle; y(0), the number of ships the Spanish have at the beginning. At the beginning of the Battle of Gravelines, the English had only 174 operational ships available, while the Spanish still had 130 operational ships. As the battle progressed, however, both England and Spain were able to bring additional ships to the battle. For this problem, we assume that English ship reinforcements arrive at the rate of 3 ships per hour and Spanish reinforcements arrive at the rate of 1 ship per hour from ships available to the Spanish Duke of Parma in the Netherlands. The differential equations used to describe the rates at which the numbers of ships change for the Battle of Gravelines are:
dx/dt = 3 - 0.05y
dy/dt = 1 - 0.03x
Use a computer program or computing algebra system (CAS) calculator to find the solution equations of the initial-value system of differential equations given in this paragraph.
Type or write "The solution equations for the problem are", followed by your answers.
NOTES: a. To solve an initial-value system of differential equations, in the command line, enter the equations and conditions, separated by commas, and click the "=" button.
b. If you obtain terms in the solutions whose coefficients are essentially 0, assume that the entire term is negligible and do not include it in your answer. For example, if you got as part of the solution, x(t) = (-3.8x10-12)t + 2.6 sin t, then that part of the solution would actually be x(t) = 2.6 sin t.
c. You do NOT need to use an asterisk to represent multiplication operations; however, you may do so if you wish. For example, to enter "2x", type "2x".
d. You must use a carat "^" to raise an expression to a power. For example, to enter "2x",type "2^x".
e. You can use "e" to represent the natural base e. For example, to enter "e2x", type "e^2x".
f. You can use either differential or prime notation to enter a derivative expression. For example, you can express the derivative of y with respect to x as either "dy/dx" or "y'".
2. Using the solution equations you obtained in paragraph 1, evaluate the number of English and Spanish ships remaining 10, 20, and 30 hours after the battle begins.
Type or write
"After 10 hours, there are _____ English ships and _____ Spanish ships.
After 20 hours, there are _____ English ships and _____ Spanish ships.
After 30 hours, there are _____ English ships and _____ Spanish ships.",
where you fill in the blanks with the appropriate numbers, rounded off to the nearest whole number.
To evaluate the solution equations for different values of t, enter only the right-hand sides of the equations and t = "value", separated by commas.
3. Produce a graph of the phase portrait (direction field) of the system of differential equations
dx/dt = 3 - 0.05y
dy/dt = 1 - 0.03x
that displays at least 400 slope segments on the intervals 0 ≤ t ≤ 200, 0 ≤ x ≤ 200, and 0 ≤ y ≤ 200.
Attach the phase portrait (direction field) graph to your assignment submission.
4. Show the solutions for three cases:
a. for the initial values you obtain from what you read in paragraph 1 of this problem, Problem #2.
b. where the initial number of English ships is the same, 174, but the initial number of Spanish ships is 155.
c. where the initial number of English ships is the same, 174, but the initial number of Spanish ships is 170.
Attach a second phase portrait graph showing the three solutions to your assignment submission.
See the notes at the end of paragraph 8 in Problem #1. Title the graph "Problem #2, paragraph 4".
5. Based on the phase portrait solution graph for the case where the Spanish began the battle with the number of ships given in paragraph 1 of this problem, estimate (to the nearest hour) the amount of time that passed before the battle turned to England's favor and estimate the number of ships on both sides that were engaged in the battle at the time that the battle turned.
Type or write "The battle turned in favor of the English after about ___ hours of fighting. At that time, the English had ___ ships and the Spanish had ____ ships.", where you fill in the blanks with the appropriate whole numbers.
HINT: Use the phase portrait to estimate the time of the "turning point". If the resolution of the graph is too small to determine the numbers of ships accurately, use the solutions you obtained in paragraph 1 of this problem to do this, if necessary.
6. What is the fewest number of ships Spain should have started with in order for it to have won the battle?
Type or write "The Spanish needed to start the battle with at least ____ ships in order to win.", where you fill in the blank with the appropriate whole number.
Epilogue to Problems 1 and 2
At the Battle of Gravelines, eleven Spanish ships were sunk or badly damaged, while the English escaped largely unscathed. The English attack at Gravelinesrattled the Spanish Armada's leadership. On August 9, with his fleet damaged and the wind backing to the south, Medina Sedonia abandoned the invasion plan. The Armada regrouped and withdraw north, with the English fleet harrying it for some distance up the east coast of England. It was then decided that the fleet should return to Spain and the fleet sailed around Scotland and Ireland. As the Armada sailed by Ireland, it encountered severe storms.Hammered by the wind and sea, at least 24 ships were driven ashore on the Irish coast, where many of the survivors were killed by Elizabeth's troops. More than 24 vessels were wrecked on the coasts of Ireland. Of the fleet's initial 130 ships, only 67 ships survived.
Problem 3 -
Three land-locked lakes are connected by channels flowing between them as shown in the diagram below. Lake 1 has a volume of 340 cubic miles, the volume of Lake 2 is 680 cubic miles, and Lake 3's volume is 204 cubic miles. Because of the geography of the lakes, water flows from Lake 1 into Lake 2, but not back. Likewise, water flows from Lake 2 to Lake 3, but not back. Water flows from Lake 1 into Lake 3 along one channel, and water flows back from Lake 3 to Lake 1 through another channel. The arrows show the directions of flow from one lake to another. The flows of water from Lake 1 into Lake 2 denoted as F21) and Lake 2 to Lake 3 (F32) are each 68 cubic miles per year; from is Lake 1 to Lake 3 (F31), 34 cubic miles per year; and Lake 3 to Lake 1 (F13) is 102 cubic miles per year. A pollutant flows into Lake 1 at a rate of 5e(7/100)t tons per year. Over time, the pollutant spreads into the other lakes by the flow of water between the lakes. You will be exploring a system of differential equations that can be used to determine the amount of pollutant in each of the three lakes at any time after the pollutant begins flowing into Lake 1.
Letx(t) be the number of tons of pollutant in Lake 1,y(t) be the number of tons of pollutant in Lake 2, and z(t) be the number of tons of pollutant in Lake 3, at any time t, in years, after the pollutant begins flowing into Lake 1. Assuming that the volumes of the lakes remain the same, the rates of change of the amount of pollutant in each lake can be described using the system of differential equations below.
1. Type or write the non-homogeneous system of equations on the right above in matrix form X' = AX + F(t), where you type or write out the matrices.
Type or write "The matrix form of the system is", followed by your answer.
2. Find the eigen-values and corresponding eigenvectors of the homogeneous system that corresponds to the system you expressed in paragraph 1.
Type or write "The eigenvalues and corresponding eigenvectors of the system are", followed by your answers. List each eigenvalue and its corresponding eigenvector on a separate line.
Suggested Internet site: https://www.wolframalpha.com/examples/Calculus.html
To find eigenvalues and eigenvectors of a matrix, in the command line type "eigenvectors {{row 1},{row 2},{row 3}, . . .}",where for each matrix row, you enter the values of the row, separated by commas, and click the "=" button.
3. Type or write the complementary function of the general solution to the problem. Type or write "The complementary function is yc=", followed by your answer.
4. Find the key pieces you would need to calculate the particular solution by following the steps below.
a. Determine a fundamental matrix, Φ(t), of the system. Type or write "A fundamental matrix is", followed by your answer.
b. Find the inverse of the fundamental matrix, Φ-1(t). Type or write "The inverse of the fundamental matrix is", followed by your answer.
To find the inverse of a 3 x 3 matrix, in the command line type "inv {{row 1},{row 2},{row 3}}", where for each matrix row, you enter the values of the row, separated by commas, and click the "=" button.
For WolframAlpha, in order to avoid a result expressed using long decimal numbers, type the entries in the rows using fractions only: NO decimal numbers!!
5. Using variation of parameters, the next step would be to multiply the inverse of the fundamental matrix, Φ-1(t), by F(t) from paragraph 1 above and take the integral of the resulting 3 x 1 matrix. You would then multiply this result by the fundamental matrix, Φ(t), to find the particular solution. At this point, the problem is getting so complex that your software or calculator may not be able to do these computations, and doing them by hand would take a very long time. Instead, skip these steps and use a computer program or computing algebra system (CAS) calculator to find the general solution equations of the non-homogeneous system of differential equations given at the beginning of this problem.
Type or write "The general solution equations for the problem are", followed by your answers.
NOTES: See the NOTES for Problem #1, paragraph 3 for solving systems of differential equations. To keep your results as simple as possible, type the system of equations using fractions only: NO decimal numbers!!
6. Assume that there is none of the pollutant in any of the lakes at the beginning of the problem, t = 0. Use a computer program or computing algebra system (CAS) calculator to find the solution equations of this initial-value non-homogeneous system of differential equations problem.
Type or write "The particular solution equations for the initial-value problem are", followed by your answers.
7. How much pollutant will be
a. in Lake 1 after 2 years?
b. in Lake 2 after 5 years?
c. in Lake 3 after 8 years?
Type or write
"a. After 2 years, there will be ______ tons of pollutant in Lake 1.
b. After 5 years, there will be ______ tons of pollutant in Lake 2.
c. After 8 years, there will be ______ tons of pollutant in Lake 3.",
where you fill in the blanks with the answers.
Attachment:- References.rar