Numerical methods provide a way to compute (approximate) values of solutions to differential equations, even when we cannot solve the equations exactly. The drawback is the large number of numerical calculations required to obtain a desired value and level of precision. In this project, you will use technology to implement the various numerical methods and use your technological solution to solve differential equations problems.
Contents
MAT 2680 Project
(Parts I-IV due Thursday, November 11, 2021) Use technology to compute approximate solutions to the initial value problem below using Euler’s Method, the Improved Euler’s Method, and the Runge-Kutta Method.
Initial Value Problem. Your project will be based on the initial value problem provided in the WeBWorK NumericalMethodsProject.
Part I: Use technology (see “Technology Options” below) to approximate the solution to your Initial Value Problem using Euler’s Method, the Improved Euler’s Method, and the Runge-Kutta Method. See “Project Requirements” below for more details.
Part II. Find the exact solution $y(x)$ to the Initial Value Problem and use it to determine the value requested in the problem. Round your answer to 8 decimal places.
Type or write your answer to this part and submit it with your project.
Part III. Compare the exact value obtained in Part II to the three numerical approximations obtained in Part I. How many correct decimal digits did each method produce? Which method was the most accurate?
Type or write your answer to this part and submit it with your project.
Part IV. Reflection. Write one or two paragraphs (minimum 300 words) responding to the following. Leave your response to this part as a comment on this post.
- Describe your project and how it works.
- Describe the process of building your numerical methods calculator. What kind of technology did you decide to use, and why? Did you encounter any unexpected challenges in completing this project?
- Why do we need numerical methods in addition to the other methods studied in the class?
- Why is this assignment included in the class (instead of just computing the various methods using a calculator)?
- Please include a link to your project (if it is online), or clearly state that you will be sending me the files by email (and don’t forget to do it!).
Technology Options
You can choose your technology tool for this job – use any one of the following:
- a spreadsheet (Excel, Google Sheets, or other spreadsheet)
- if you choose to create a spreadsheet, you should have columns for$x, y, f(x,y)$ , and so on, and each stage should appear in its own row (See Project Requirements below for more details)
- HOW TO SUBMIT: If your project is a spreadsheet, either share it with me (if it is in Google Sheets or a similar cloud-based platform), or email the file to me as an attachment.
- a programming language (Java, Perl, or other programming language)
- if you choose to write code, your program should output the values of $x, y, f(x,y)$ and so on at each stage (See Project Requirements below for more details)
- HOW TO SUBMIT: If your project is code, please submit it using an online coding site like ideone.com – once your code is working on the site, you can simply submit a link. If you are using a programming language not supported by ideone.com, you can email the source code to me (for Python, try this site).
- mathematical software (MatLab, Maple, Mathematica, or other mathematical software)
- if you choose to use mathematical software, your program should output the values of $x, y, f(x,y)$ and so on at each stage (See Project Requirements below for more details)
- HOW TO SUBMIT: If your project uses mathematical software, either share it with me (if it is in MatLab Online or a similar cloud-based platform), or email the file to me as an attachment.
Project Requirements
- Your solution must be able to carry out Euler’s Method, Improved Euler’s Method, and Runge-Kutta (you may implement these as three separate spreadsheets or programs if you wish).
- Your solution should display all the points $(x,y)$ found along the way, not just the final point.
- Your solution should also display other values found while carrying out each method:
- Euler’s Method: display the slope $f(x,y)$ at each stage
- Improved Euler’s: display the values of $k1, k2$ at each stage
- Runge-Kutta: display the values of $k1, k2, k3, k4$ at each stage
- You can display other values as well, if you wish (for example, the intermediate y-value in the Improved Euler method that we refer to as $z$).
- Your solution may NOT use any built-in version of these methods (for example, most mathematical software contains a built-in command for Euler’s Method – you can use this to check your work, but you need to create your own solution).
Test Your Project
Test your project. Data for the first few stages of calculation using each method can be found in WeBWorK.
The way my project works is that given an initial condition (x, y), can find an estimated y-value given another x-value. By using Excel, I was able to execute this numerical approximation in 3 different ways: Euler’s Method, Improved Euler’s Method and the Runge-Kutta Method. Throughout the execution of the project, the first objective for each numerical method was to make sure that the cells on each of the approximations shown on WebWork were the exact same as the ones I had on Excel. Once the first 2 lines were the same value on Excel as the three on WebWork, I would then simply drag down until I reached the desired x position. This process was the same for all numerical method approximations (the only difference is how many values are to be calculated in each row (slopes, z values, and the final new y value approximation).
I decided to use Excel for my project because to me it made the most sense. When you open the application, it greets you with rows and columns, and since there are multiple calculations to be executed in each row, I figured it was the best fit for this project. At first, the problem I was having was ensuring the first few inputs of the chart on Excel matched the values shown on WebWork. The reason why was mostly because the values were the same it was just that the value on Webwork was rounded sooner than that of Excel. For example, on Webwork I’d see the number 18.7961 whereas on Excel I’d see 18.79608, and that would be a little confusing when comparing the two numbers, and sometimes would make sure I had the right equation for that slope or z-value. Using numerical methods are an easy algebraic way to approximate the value at a point. Not only does the method physically show us a trend in a curve, but it also shows the process of how the curve became to be – since we analyze the slopes and their to-be point along the way. Whereas solving a differential equation taking the steps we would for a – in this case – a 1st order nonhomogeneous linear equation would simply give us an answer to the line that fits the scenario. Which at times on paper is just super confusing and just a lot of non-algebraic math used. Physical data portrayed by graphs especially on excel can be transported and modified to fit any scenario as well. I think this project was assigned to show that are multiple ways of solving a problem; to also see what happens when approximating 3 different ways to find an answer. On my excel file, I have the x and y values plotted on a graph, even though I cannot see the huge difference between the three, I can see that the Runge-Kutta for example is more acute than Euler’s Method.
I have imported the excel file to Google Spreadsheets you can find the “Technology” part of the assignment in the link:
https://docs.google.com/spreadsheets/d/1V3HsPzzNa6nar7TWEYG7MFRLJl90sdOvRPKcb1fOAuU/edit?usp=sharing
The work behind the exact solution as well as the comparison of each method to the exact value will be emailed.
In order to approximate solutions to the initial value problem using Euler’s Method, the Improved Euler’s Method, and the Runge-Kutta Method, I used the Matlab programming language. I created three separate scripts, one was for Euler’s, one for Improved Euler’s, and the last one for the Runge-Kutta Method. The scripts can’t all run at the same time, and it is good to clear the workspace after each code is run. The first thing I did was create the function using “ f ” as the variable to represent the function. I created some space then started putting in the initial x value which was 2.4, and after that I entered in the final value for “x” which I made 2.82. I entered in the “y” value, step size using “h” as the variable, i and then k1. I wanted the values to line up the way they were shown in the webwork, so I used the fprintf code to show the information stored in the variables and put them in that form. I had to use fprintf three times. The first one was for creating the data table header, the second was for defining the first row of the table, and the last one printed all of the next steps. Matlab has two different types of loops which are for loop and while loop, I used the for loop. The for loop is the best one to choose when you know the amount of times you want to repeat a loop. I started the for loop at the first point in order to compute the next one, then I stated the step size and also I imputed the equation xn-h, to stop it from going past the final “x” value, while constantly going to the next value. Inside the loop I wrote the equation for y,x,i, and k1, but before ending the loop I wrote the third fprintf. I only described how I set up the script for Euler’s instead of all of them because the scripts were set up the same way with the amount of variables and equations being the only differences.
I encountered challenges while creating two of the scripts. In the first script, I made some mistakes in the second fprintf line because I didn’t type the floating point and tab to represent the k1. When I couldn’t get the k1 to work I thought I wrote the equation wrong which made me write it over a few times, but I was able to do it correctly. The challenge that I faced creating the improved Euler’s script was entering the y and k2 formulas correctly inside the for loop. The k2 formula kept causing the y formula to be wrong, so I had to spend some time trying to type the correct k2 formula. The project made me see how valuable the numerical methods are because the program is able to list all of the values that you want even if it’s a long list of numbers quickly. It would be time consuming to use the methods studied in class when dealing with a long list of numbers and it will be easy to make mistakes. The assignment is included in class because we have to learn how to use the three methods and learning how those three methods work in a program completes the three topics.
The project will be sent by email because Matlab was installed on my laptop, I’m not using the online version.
The project was to design and create a calculator so to speak that replicates the numerical approximation methods used for differential equations. The methods being Euler’s Method, Improved Euler’s Method, and the Runge-Kutta Method. By using initial values, the slope, and intermediary steps between numbers or points an approximation of the given value can be found for an equation.
The very process of creating the calculators can be compared to using building blocks. To move from one point to another you first need to build up the table with data to reach another value. Hence, why the intermediate steps are important, depending on the method those steps can be crucial to how accurate the calculator will be. The design was as simple as putting together a table like we did in class to help keep things organized. Each part of the table has its own set of rules to calculate the values, so it was as simple as programming equations for specific data sets. Add the programmed equations with the given data we already have, and things all start to fall into place.
I would say my only real difficulty with the project would be solving the differential equation by hand. Seeing as how it’s been quite some time since we have worked on just basic first order linear equations getting used to the rules again took some time, but naturally it all started coming back to me and worked out.
It’s always important to have multiple methods to solve or determine something, either as an evaluation of sorts to double check or it can be a more viable option based on the data or information given. Some methods may give more accurate values, which in fact this project reinforces. I believe in addition this project was assigned to give us an analog method of verifying the values to be found in a differential equation, not to mention it will allow us to get a better understanding of the methods by designing our own calculator.
I personally chose to do this project using Microsoft Excel out of general ease and accessibility. I have a lot of experience with the program, and the way these methods are designed work perfectly for a spreadsheet software. I will be emailing an attachment of my technology side of the project.
The project works in base of the given equation, initial values, and the step size, computing these into a spreadsheet and using the Euler’s Method, Improved Euler’s Method and Runge-Kutta Method to get to the approximate value which was 2.38.
For each method I had to follow my class notes in order to have a organize work and same number as the webwork. For Euler’s Method, first I put the initial values as i,h,x, and y, in order to compute these given numbers into the differential equation to get the k value, after getting the k value I was able to compute this number into the Yi+1 formula which is the next point for the y coordinate, then I just drag down all the values, until the approximate value and was able to see the y value for that point. For Improved Euler’s Method, I did similar process but in this case it had some more extra variables, k1,z,and k2 where for k1 I had to use the differential equation with the initial values, after computing the k1 I used that value, and I plug it into the z formula which is the initial value for k2, then I found my next point Yi+1 using k1,k2 and the step size(h).For Runge-Kutta method, its similar to the Improved Method but it has some extra steps, and its different when computing initial values, I did all these steps carefully in order to get to the approximation value. I use google sheet to execute all these methods because I found it easier since I have some experienced on the program. The only challenged that I encounter while doing this project was solving the differential equation, since I had to go back on my notes and refresh.
The purpose of numerical methods is to find an approximated solution accurately. It’s necessary since we don’t have to compute all these numbers by hand and also is less time consuming.
This assignment is included in the class, to compare the differences of each method and to see which is the most precise one when comparing to the solved differential equation , also with this assignment we learned how to compute each method on spreadsheets or any programing, mathematical software.
The project will be send by email, this include the Part I, Part II and Part III.
The Numerical Method project was used to find the y value at an initial condition that was given. The purpose of this project is to show the importance of the different ways we can calculate numerical approximations. Given the information provided I used excel to apply Euler’s Method, Improved Euler’s Method, and Runge-Kutta method to display the slope (x,y) at each stage. Also displaying the values of z and K1 as well as the intermediate Y value in the Improved Euler’s method, and z3, k3, z4, k4 for the runge kutta method. To calculate these values, I set the values of X_0 and Y_0. To get k, I set the equation given to us and use excel to put the appropriate values. I did this for all Z, k, and Y_(n+1) values using the original formula. After I set the values that were given on webwork I would simply drag down that value to find the next number in sequence. I would continue to do this process until I found my desired x value. To create this numerical methods calculator was finding one value that led to another. For each method I would follow the equation that was created from the first line. Since the basic information was provided I used the formula Xn+1=Xn+h, Yn+1=Yn+h*f(Xn,Yn) to find the value of x at 5.4 for Euler’s method. To calculate Improved Euler’s method, I followed a similar formula, but calculated for z and K2. Lastly for the runge-kutta method it is like improved Euler in the sense that it required extra steps. I decided to use excel to calculate each method as I found it the easiest for me.
The only problem I encountered through this project was solving the first order differential equation. Since it required knowledge from prior calculus classes I needed to remember how to integrate by parts and using u substitution. I also believe that the most time-consuming part of this project was solving the first order non homogenous equation.
We need these methods to help us calculate an approximation answer, so we do not spend to much time solving them by hand. Throughout the process of this project, I realized if these calculations were done by hand, it would take hours to complete and finding the smallest error can mess up the ongoing calculations. By using numerical method, it can also show us graphs that gives a visual image of the calculations. In the end I noticed that the runge-kutta method must most similar answer which was 3.463792 and when calculated by hand I got 3.463833 which shows it was off only by three decimal places. Given this assignment showed me how we can use different software’s to help us when calculating. I will be emailing the technology part of the project as well as my calculations.
For the Numerical Methods project I had chosen the option to use either google sheets or excel. I decided to go with using google sheets because in my opinion it would be easier to share by providing a link that can be accessed by those who have it. Also I saw that it would be a great opportunity to practice with google sheets. In this project google sheets were used to demonstrate Euler’s method, Improved Euler’s Method , and the Runge-Kutta Method to find the y value when x is equal to 5.68. The way my project works is by inputting the formulas for each cell for each process in the numerical methods stated above.
For Euler’s method we start out with the initial point (5.2,0.3) with a step size of 0.08. To get the value of k1 we input (5.2,0.3) into the equation dy/dx =9x-6y and get the value of 45 then with this value we find the new Y value. The formula for yi+1 is h+yi*k1. Since we inputted all formulas we can copy our y value in the cell below 0.3 and drag down on the previous values because google sheets has the formulas for each step we did for the first row. We drag down until we find the y value for 5.68 which is 8.12666308. Next we explain how our Improved Euler’s method sheet works. Similar to the Euler method we start with the initial point given. To find k1 we take our initial point and plug in our x and y value to the original equation and we see k1 is also equal to 45. Next we need to calculate Z which is equal to h*(k1+y) which would equal to 3.2. Now we need to find k2 in sheets we used this equation 9*(D4+C4)-6*G4. D4 is the initial x value and g4 is the z value we found in the previous step. For the final step to find the new Y we take the average of the k values and multiply it by h our step size then we add our original y value to get 3.0648. After the first row is filled and the formulas are inputted we can copy our new value and drag down on each column until we get y(5.68) y which is 7.793787333. The final sheet demonstrates the Runge-Kutta Method. Like the other 2 methods we start with the initial values given. We input the formulas for each cell to find k1,z1,k2,z2,k3,z3 and k4. Using the values we take the average of k values and divide by 6 and multiply by the step size and then add it to the y value. After the first row is complete we copy our Y value below the initial Y value and we drag down due to the formula in each cell being inputted.
As I was working on my project I did not have many problems. Implementing Euler’s method and Improved Euler’s method were not that bad. I ran into some problems when inputting my formulas to the runge-kutta method. To fix my error I realized the importance of parenthesis especially when inputting the formula for the new y value. In order to make sure I was correct I used the tables provided in webwork to confirm my values.
In this class it is important we learn Numerical Methods for many reasons. Sometimes there may not be much given on a problem. We may not be able to identify a solution to a problem, especially a differential equation so we turn to Numerical Methods. It can help us reach an answer that is more accurate in a way we go through processes. I believe this project is included in this class for students to demonstrate how technology can be used to accurately perform Numerical Methods. Also it can help us show the differences between the three methods used in this project.
The Google Sheet will be provided in order with Links below. Parts 2 and 3 will be emailed
Euler’s Method:
https://docs.google.com/spreadsheets/d/1rg8OtzN2TSRa76LzVhTUj0xVTMTw8obboDYDse6Psow/edit?usp=sharing
Improved Euler’s Method:
https://docs.google.com/spreadsheets/d/1GzvqYqqwbrThqCs91SczaHRND8CRJtGSfV4yeIu5jXA/edit?usp=sharing
Runge-Kutta Method:
https://docs.google.com/spreadsheets/d/1cLemETYMSvur6yOVUpDQeDrxk6uAVGGzQxyCL3wX9Is/edit?usp=sharing
1. My project is comprised of four different parts. The first part use technology to calculate the solution to the given differential equation at a given point. The second part uses a traditional pen and paper method of using mathematics to solve the differential equation to find the exact value at the given point. The first part uses excel to create three spreadsheets that calculate the values of the differential equation at different values of x using Euler’s method, improved Euler’s method and the Runge Kutta method. The spreadsheets will automatically calculate the value of the differential equation at each point of X, as well as the intermediate values. A user can simply drag down the cells until they reach the desired X-Point to find the value of the differential equation at that point.
2. For this project, I decided to go with Microsoft Office Excel because I do not have knowledge of any programming language and apart from MatLab and SPSS, have no knowledge of mathematical software. However, my MatLab knowledge is at a beginner level since I have only started learning it this semester, and SPSS is used for statistical analysis, which Is not appropriate for solving these problems. To do this project in Excel, I looked at the tables that we did for each of the three methods when we were learning how they function in class and in the WebWork Homework. These tables were a prime fit for Excel, so I transferred the structure of the three tables into Excel, but instead of filling in every cell, column, and row by hand, I let Excel do the work by inputting the formulas for the different columns (of the intermediate values) into excel. I build these formulas by entering them into a way that Excel can understand. I started off by listing my given values, for example, the differential equation given, so that I can reference it, and the height in a manner that Excel can reference it (by including the number in a separate cell). In the first row, I entered each formula for each column that I made. Most of the columns however also use numbers from other columns, such as the initial X and Y values for K for instance. Thus, I simply wrote the given values in and had the formulas reference them. I made a separate column for Yi+1, which is the next value of Y and then cross-referenced points from that column into data for my Y column for every X value. After the first row was completed and the second set of X and Y values were available, I could simply drag the columns down and Excel would pick up on the data to calculate the next set of values. I was able to drag the columns down until I reached my “goal” value of Y at the point where X is equal to 2.7.
Some challenges I faced are mostly remembering the proper way to enter formulas and the different codes available. For instance, when cross referencing a cell that I did not want Excel to change, I forgot that I could use the $ symbols to keep the value in place. When I did it initially, Excel would treat it as any other cell and continue to pull values into it that did not make sense and messed up my table badly. Overall, most of the difficulty came from remembering the proper structuring of formulas for Excel to understand and from finding and fixing small mistakes that led to incorrect values and let me know something was wrong. The sample first few rows given in the assignment was a tremendous help in troubleshooting for this part of the project.
3. Although the methods that we learned in class are extremely useful and completely accurate, one reason we use numerical approximation is because we are mostly taking this class for real world use. With real world use come real world problems, such as time and financial constraints. While one can do these values by hand, the real world is all about saving time and money (time is money). Using a computer to do the work for us allows us to save time and resources. Solving more difficult differential equations with a much further value from the starting point could take a lot of manpower in terms of time used and wages paid.
Second, for those of us taking math for pleasure, and to become a mathematician, there are differential equations that are extremely complicated to solve or may even be impossible to solve using traditional methods. Fortunately, no matter how they may appear to us on the test, all the differential equations we have in this class are on the simpler side and thus, we can apply the traditional methods that we are learning about to solve them. However, a mathematician or physicist doing advanced work may run into differential equations that would require so much time to solve by hand that it simply better to approximate the value than solve for the actual exact value.
In real world application also, numerical methods allow us the flexibility of choosing how close to the actual value we want to be to see if the trade off in time is worth it. For example, for something where we must be extremely precise, like manufacturing of aerospace parts or work in nuclear energy, we would like to be as close as possible and would thus use a smaller step value of the H to get a better and more accurate value. Again, the trade off depends on what the differential equation is used for. For example, in the manufacturing process, for a nuclear power plant, you will want to be as precise as possible because any small defect in any part of the reactor could cause vast destruction. But if we are making spoons for Dollar Tree, we could approximate with a very large step size because it’s just not as important for us to be that precise. Unless we’re manufacturing for some spoon aficionados, is that even a thing? I digress.
4. This method is included in the class because from my experience doing it, this method is a learning method. It gives us a deeper look into how these problems are solved, how we get the values and the workings of the process of differential equations. It also introduces us to the concept of real-world applications of differential equations and that we don’t necessarily always solve every equation for the exact value.
5. I will be sending the files to your email. It will be a total of 3 Excel Spreadsheets and one PDF of work solving the differential equation for the exact value on paper and analyzing it in comparison to the numerical approximation project value I got.
My project is an excel spread sheet and most of the calculations are done atomically as long as some of the values are put in. It can be further edited to me completely automatic if some equations are tweaked. The process was relatively simple for me since it isn’t the first time I use excel and now a good amount of things in order to have excel do calculations for me. The most “challenging” aspect for me during completing the project was putting subscript in excel because of how tedious and repetitive it was, making it more annoying than it was a challenge. Numerical methods can be used to guess where a the next point of something is going to be as long as an equation is given to us and the use of linear algebra isn’t there for someone to use. The assignment is included in the class to understand the method because if we are given the same project but with no background knowledge the project would not be done correctly because none of the information presented would make sense unless prior and background knowledge is provided. The same thing can be said to everything that is computerized.
I will sending the files to the project since everything is done through applications, not online.
1. The point of the project is to solve an initial value problem using three different methods in excel or in code and see how accurate the solutions are using those methods to the exact solution. The three methods used were Euler’s, Improved Euler’s, and the Runge-Kutta.
2. I used Google Sheets to create a table and in each row have the values of h, x, y, k_1 etc.
so that it is more clear and organized to the reader which values are “n” (or in my case “i”).
I chose Sheets because I am more accustomed to using excel, although I do have experience with pyhton and c++ but not very comfortable with them, Honestly looked like such a hassle to program it while the excel option seemed so much easier. There was no need to complicate my life any further than it already has haha, so I just chose to go with using Google Sheets. One difficulty I had was with the Runge-Kutta Method. For some reason my Y(1.3) value does not look like the other two methods, kind of close but the second decimal point is very different. This leads me to believe that I have miscalculated somewhere along the way, though I have checked more than once(alot more) trying to make changes by messing with parenthesis and such but this is the closest solution I got that best represents the Exact solution.
3. We need numerical methods to double check (or even triple check) solutions to problems, we do not know which solution is truly correct or which method is accurate enough to be even considered an actual method to solving a solution until you check it by using different numerical methods, and if every solution is the same or very close, that tells us that it is indeed the true solution. It’s kind of like researchers/scientists peer reviewing their work to see how legitimate their work is and if it is in fact, correct.
4. This assignment is to prepare us for real life. In all likelihood we will not be using a physical calculator and a piece of paper to do calculations, instead we would likely be using technology and/or programming such technology then doing the work ourselves to see if our program functions how it is supposed to. This project ensures us that there are many methods that lead to the same solution, still in truth there are some that are more accurate than others.
5. I will send the project by email.
In the project we were given a differential equation which was y’= 6x-5y and the initial condition was given y(3.7)= 0.8 and asked to solve the differential equation and find the exact value y(4.06). I also asked to use the three numerical methods which were the Euler method, improved Euler’s method and runge kutta method with step size of h= 0.06 to approximate the value of y(4.06). Also we were the first few rows of the values for the all three methods which was helpful for us to start and see if we are getting the right value. What I have done to do this project was google sheets because I’m mostly familiar with it and I thought it would be a good idea using google sheet rather than trying something new which is unusual. Doing it in google sheets was easier for me because I was already familiar with it. What I have done is used the formula for all three methods and filled out the first row of the table and found the yi+1 at the end and when I found the yi+1 then it was easier to find all the other values by dragging the table. However, we had to be careful that the first few tables matched with the values that we were given in the webwork but the most important was finding the next y value and getting the first row right for all three methods. However, some of the struggle I have faced was finding the k values for the runge kutta method because we used different formula for between k values, for example the formula for K1 and K4 are different from K2and K3. For K1 and K4 we just added the x and y value to the differential equation but for K2and K3 we use the following formula f(xi+1,z) and the z value is the previous z value we found. Let say if we want to find K2 value we use the following 3.70+0.06/2=3.73, now plug into the differential equation, which was for me y’=6x-5y and now replace x= 3.73 and y= 1.346 which is my z2 so, y’=6*3.73-5*1.346=15.65 which is my K2 value. However other than finding the value for K2and K3 I didn’t have any other struggle with the project.
I think we need the numerical method in addition to the other method because with using numerical we can find the values of differential equations more easily than solving the differential equation. I also think that the numerical method gives us a variety of options to find the value for the given y(x) value or the differential equation. Also we can compare which method we can use that will give us the exact value when we solve the differential equation. As I saw in my project, the runge kutta method gives a close number then the other two methods. I personally think that it gives a more accurate number because we do more steps in this method then two other methods. We used this in class to have a better understanding hand to hand then with the computers. Even though it’s easier to solve in the computer, it’s not very helpful for our learning skills, especially learning the different numerical methods. So doing it in class helps to solve all the three numerical methods step by step and compare value to the value we got solving the differential equation. Even though doing it by hand is time consuming, it’s really helpful to learn because let’s say if we did not have a computer we had to solve it with hand, so knowing how to solve it is pretty helpful. However when we know the steps then we can solve it using our computer because it can save us time in the exam but I don’t support doing it on the computer before we haven’t learned in class and learn how to do step by step using math formulas. Also the project will be send by email.
Sorry in the first line I should have said “I were given” a differential equation instead of we were.
This project focuses on using a type computing technology in order to find a certain y value of an ordinary differential equation with an initial condition given. The goal was to use the technology in order to solve the equation with Euler’s Method, Improved Euler’s Method, and the Runge-Kutta Method. In my case, I elected to use Microsoft Excel in order to solve the given equation: dy/dx=3x-6y, with an initial condition of y(1.2)=0.2. The goal is to find the value of y(1.56).
I created the three tables that calculate the values of the differential equation at different values of x using Euler’s method, improved Euler’s method and the Runge Kutta method. The tables will automatically calculate the value of the equation at each point of x. You can input any value you want and the tables will change accordingly and you will find the value of the differential equation at that point.
The first step was to make sure that the first few lines of all three tables matched the tables given in WebWork so that I knew I was on the right track. The beauty of Excel kicks in after that, because when I pull down the cursor in each cell, the equations that I used will transfer down into the cells underneath, pulling the required numbers from the corresponding cells around it. This makes completing the tables the easy part. Of course, everything is easier on a computer! Writing out the equations and solving them on paper was more difficult. The importance of numerical methods in our everyday lives is that they can be used to create programs that solve real problems and provide reliable data for professionals.
I believe this assignment is included in this class because it shows another example of how we can use everyday technology in order to observe data and solve problems.
I will be submitting this project via email.
The Objective for the Numerical Methods Project was to approximate the value of solutions to differential equations, even when we cannot solve the equations exactly. The way my project works is that I was given the original differential equation ( dy / dx ) = 4x – 8y with an initial condition of y(4.7) = 0.5, my objective was to approximate the value y when x is equal to 5.12. The Numerical Methods used in this project were Euler’s Method, Improved Euler’s Method and Runge-Kutta Method, all of these methods have unique equations for determining the y value therefore the final answer for each method will not be equivalent but very close. The technology which I approached this problem with was Excel, I believe Excel is the most appropriate because the table format allows you to compare your results with the values shown in WebWork and if your first couple values match the table on webwork it’s on the right track.
Given the step size of 0.07 ( Which remains constant through the project ) we can use it alongside the given information of differential equation & initial condition, the step size will be represented by the variable “h” in the Excel table. I began with the Euler’s method and since we are given the initial “x” and “y values I had to compute “k” and “yi+1”, in order to compute these in excel I manually type it’s respective formulas in the column and was automatically give an answer for k and yi+1, after I would just compare the values in the WebWork table to the Excel table. Since the formulas are already input in the Excel column all that’s left is to drag down the column and it will automatically compute the values for you ( First we must manually input the previous values in that row before dragging down. Next was the Improved Euler’s method and the set up to this was almost identical except the need to add two extra rows for z and k2 with their respective formulas, this changes the dynamic of the problem and thus the y value will not be equivalent to the one found in the Euler’s Method table. Last method which was used in the project was the Runge-Kutta Method and here we had an additional 4 rows to the table of Improved Euler’s method, these were labeled as z3,k3,z4,and k4. As it turned out after completing my part 2 I compared my calculated value to the tables in Excel and out of the three methods used Runge-Kutta was the most accurate after all it is the method with the most variables used to make calculation (This is also my response to PART 3).
The Excel tables and part 2 will be emailed.
For my numerical methods project I decided to use Google Sheets a spreadsheet software. I chose this as my option because formulas are very simple to create in spreadsheets, and a table is created as you type. Just as we did in class. Therefore, I felt it was appropriate to use a spreadsheet software.
The only challenge I came across was when calculating the exact answer, my solution to not come anywhere close to the three approximations in the spreadsheet. After a few tries, I realized I made an error with the placement of “C” . After fixing my mistake, the solution was quite close to the three approximations. I used a technique called Integrating Factors to solve my differential equation.
I believe this project was assigned to us because if a differential equation is too difficult to solve (or impossible solve), we want to find a way to find a solution at a desired point.
I will be submitting Part 1, Part 2, and Part 3 via email. All three numerical method calculators are linked and on the same spreadsheet just under different “sheets”
For my Numerical Methods Project I choose to do it on Microsoft Excel. By doing it on Microsoft Excel I was able to use the features to solve most of the charts. In Microsoft Excel there are in cell functions that will take different cells and does multiplication, addition, subtraction, division, and so much more.
The process for building this is get the numbers from webwork along with the tables, copy the function, the Euler’s Method table, the Improved Euler’s Method, and the Rudra-Kutta Method into Microsoft Excel and some of the numbers were missing, to find the missing numbers I used my notes from class to find y and k in the Euler’s Method to find k by plugging in x and y into the equation, to find the missing numbers in the Improved Euler’s Method I used my notes and formulas to find y, k, z, k2 in the table, to find the missing numbers in the Runge-Kutta Method I used my notes and formulas to find y, k1, z2, k2, z3, k3, z4, k4 in the table, and after all that I wrote the designated y(5.22) for each table and highlighted them for easy location.
We need numerical methods in addition to the other methods studied in class because the more methods the easier the process is and the more precise the answers are. By using three methods and having the answers close together is a great sign that the answer is around those numbers. By finding the exact answer and seeing that the method that requires the most work was the closest to being correct informs us that depending on what the question asks for which method would be applicable.
This assignment is included in the class because it teaches us the difference between all 4 methods and going forward which method, we should use to get the answers to our problems.
I used the drop-box link to upload 3 files 1 excel, 1 word, and 1 image
This math project was designed to reflect the computation and functionality of numerical method processes to find approximations of solutions with given numbers. These methods are called Euler’s Method, Improved Euler’s Method and Runge-Kutta Method.
Using google sheets, I created individual cells that contained formulas that computed the x, y, the various ks, the various zs and yi+1, while incorporating the initial point values. Once one row for an initial point was completed, by holding the corner of the cell and dragging downwards, the computer automatically generated solutions for each column. The only challenge I encountered when completing this project was for the Runge-Kutta sheet where the Z required two formulas to generate an answer, one using the midpoint formula and the second where you find the value to use to find ks.
Multiple techniques to solve or decide anything are always vital, either as a type of double check or as a more plausible choice depending on the facts or information provided. Some approaches may provide more accurate results, which this project supports. This project, I believe, was given in class to examine the differences between each approach and determine which is the most precise when compared to the solved differential equation. This was also assigned to provide us with a means of checking the values to be obtained in a differential equation, as well as to help us to have a better knowledge of the processes by developing our own calculator.
Will be emailing project
The way in which my project works is that I was given the equation of dx/dy=7x-9y and with that I had to find 3 different tables that had to include using Euler’s Method for the dx/dy equation which was the smallest table out of all of them, then I had to move onto the Improved Euler’s Method which was in much more detail then Euler’s Method, and lastly the final table was the Runge-Kutta Method. Which I would have to say was the table that was giving me the most trouble.
For the process of me building my numerical method calculator was that I went into the path of using Excel which I found a lot better then all the other methods that were available. Some of my struggles I would have to say was that when I found out, that I am not really good at using Excel at all. I really found out when I was trying to use Excel and make the table give me different numbers when I would slide down the row, but I would see that I would have to watch and learn some YouTube videos on how to use Excel. I would have to say that this would have been like my 3rd time using Excel and I had to relearn how to use Excel.
I think the reason why we need numerical methods; it would have to be that numerical methods helps us solve and get really close to a huge number some of the methods that we have give us a better result then others, they get us close to the value that we are looking for. I think the reason why this assignment is included in this class is for us to understand how big a complex number is but we can take a step back and really break down the complex number instead of just computing the number and just getting the answer right away. And for the last part I will be sending all the files that I have for this project.
My project uses excel because excel is good for making tables. The project consists of four spreadsheets. The first one is where the user inputs the data. It also shows the final answer for all 3 methods and the manual calculation. The other three sheets are where the calculations are performed and the tables are drawn, one sheet for each method.
I designed the sheets with the intention of adding VBA code to allow it to solve other simple polynomials and to use a custom number of steps. I didn’t have time to write the code for that so it isn’t functional but the layout of the first sheet is designed for that purpose. It has a table for inputting polynomials in a format which can be easily computed.
The other sheets grab the coefficients for x and y from the first sheet and insert them into the equations. I had some trouble with this because I wasn’t used to using multiple sheets in excel. I tried selecting a cell from the first sheet and then going back to the sheet I was working on to complete the equation. When I clicked on the Euler tab after selecting a cell on the Input sheet, Excel changed the sheet reference from the Input sheet to the Euler sheet. I had to press enter before going back to the tab I was working on and editing the equation.
Methods of approximation are useful because we don’t know the solutions to all differential equations, we can at least get an approximate value for a particular point which we’re trying to find. They also help when the method for solving the equation is complicated and we just need one value. We can save time by approximating and with computers we can get more precision than we need for the application.
Presumably the reason we have this assignment is because it takes a lot of time and effort to calculate these manually and it’s easy to make an error. In real applications we wouldn’t think of doing it manually so it’s more important to know how to solve them using software.
I sent the files for parts 1, 2, and 3 by email.
The technology behind my project Is rooted in excel. Excel was my pick for this project because most people do not know how to code (including myself) so I wanted to make a user-friendly and simple way of calculating numerical methods. Using lecture notes and pretty much any information on these three topics I first wrote a foundation for how I wanted my project to look. My project consists of one file with 3 tables and the bread and butter lie in the cells. I wrote my formulas to interact with the cells in my table and produce values based on the initial formulas to solve the respective aspects of each method (Euler, improved Euler, and Runge Kutta). My cells were written very simply but got the job done and can be picked up by pretty much anyone. Above each table lies a sort of index that allows my cells to trace locked values and use them to solve formulas that involve that respective cell. Challenges that come to mind when working on this project are minimal for the most part just figuring out the correct cell structure for the formulas was tedious, however, part 2 of the project induced a fair bit of stress making stupid mistakes along the way…forced to refresh my memory on the steps needed to solve a first-order linear differential equation eventually I hit a concrete solution. I believe this assignment was given to us to help get exposure to applications for numerical methods, solidify the concepts, and compare and contrast the difference between the three methods and their accuracy. Personally meticulously crafting the project using the tech to make sure no errors were made perhaps even unconsciously built character and work ethic. I digress, once a certain level of completion is reached in the tables (mostly just inputting initial values given) the user is able to drag down the corners of the table and the computer generates perfect values using the formulas written. I will be sending the project files through email, files included will consist of Part 1,2,3, and 4.
My project is a simple numerical calculator, where you insert the number of your x, h, and Y1. It could automatically calculate the data required to fill in the various other values needed for the completion of parts 1 and 2 by dragging the column back down. If I were to describe the process, I would describe it as akin to many others who choose excel as their tech template of choice, namely programming in the various asset required to assemble into a coherent formula. For example, to calculate the value of k1 for the Euler’s Method, the calculator must combine and place the value of x and y in place of their letter counterpart, and this must be achieved by inserting their respective column into the appropriate column of their own, this poses a challenge of unfamiliarity for me as I have little experience working with excel and sees this as an opportunity to familiarize myself with it, coupled with the throat infection I suffered during the time of the midterm exam which needless to say caused a significant delay in the completion of this project and thus it also resulted in my failure to complete the project on time. As the reason why we needed the numerical method in the first place, the only reason I could think of is it serves the purpose of broadening our options one would naturally assume with additional methods comes a firmer grasp of the subjects at hand, which I would also wager is also the reason for this project’s inclusion namely to further our understanding of its various numerical formula beyond that of simply telling us to use a calculator as we needed all the formula to compete for part 1 and 2 of the project. Lastly, I will submit my project shortly after this post is posted via email.