User manual for Iode - Department of Mathematics
Contents
1. T The remaining option is more important mathematically for it lets you change the top harmonic i e the number of terms used in computing the approximate series solution Increasing top harmonic will yield a more accurate solution though at some computational cost 28 CHAPTER 5 PARTIAL DIFFERENTIAL EQUATIONS Appendix A Installing Iode Iode runs under either Matlab or Octave If you wish to use the graphical user interface of Iode you will need Matlab Version 6 0 or later If your computer already has Matlab or Octave installed then you only need to install Iode Obtaining and installing Iode On your machine create a directory called my_iode or similar Then e Unix systems From www math uiuc edu iode download the file iode zip and unpack it into the directory my_iode with unzip or sim ilar e Windows systems From www math uiuc edu iode download the file iode_dos zip and unpack it into the directory my_iode with WinZip or similar lode is free software available under the GNU General Public License Obtaining and installing Matlab Matlab is commercial software available at www mathworks com for Unix Windows and Mac It is already installed on many Unix systems in universi ties Matlab is available to students at a discounted price see the downloads page at www math uiuc edu iode 29 30 APPENDIX A INSTALLING IODE After installing Matlab on a Windows system it is helpful to set it up to2. error x f x 5 J An cos T B sin T n 1 Notice that the vertical scale on this error plot is generally different from the scale on the top plot in which the function is plotted In fact the vertical scale on the error plot will change as you step through the partial sums increasing or decreasing the top harmonic Across the top of both graphs you will find the function written out along with the basic period interval x lt x lt x The function is extended periodically by Iode and is shown over two periods Controls e Plot partial sums and errors plots f and its partial sum in the top graph and the error difference in the bottom plot e Plot coefficients A_n and B n plots the A coefficients in the top plot and the B coefficients in the bottom one for 0 lt n lt top harmonic Notice Ag is never plotted because it is the least interesting Fourier coefficient affecting only how much the graphs are translated up or down in the y direction e Plot coefficients C_n A_n 2 B_n 2 1 2 plots the Fourier mag nitudes C A B2 in the top graph The bottom graph can be used to investigate the rate of decay of the Ch using an arbitrary com parison function see Options below e Current top harmonic increase or decrease this value by clicking on the arrow buttons in the middle of the window Alternatively you can type a number directly into the box between the arrows You can change the top
3. and auxiliaries providing the graphical user interface The second and main kind are the mathematical modules which actually perform the calcu lations needed to solve the differential equations numerically and calculate the Fourier coefficients numerically and so on The mathematical modules euler m and rk m are particularly instructive they implement the Euler and Runge Kutta methods for solving a first order ODE You can implement different solver methods yourself as outlined in Appendix C In order to develop a better understanding of what the various modules do you can look at the file doc txt Together with Figure D 1 it should give you a rather complete picture of the structure of Iode Another useful file to look at is sample m which contains the transcript of a short Matlab session that shows how to use the direction fields module and numerical solvers without resorting to any menus 37 38 APPENDIX D STRUCTURE OF IODE lode GUI Version sinef mixednd a7 cosef dirichlet periodicef lt EAN ie neumann Aine ei S heat d solplot trapsum df diguie ___ ay ee er mvgui S ei partialsum mp e b pP ppaux trajplot Figure D 1 Relationships between the most important modules of the GUI version of Iode Any manual can only take you so far If you have followed this manual up to this point you are ready to go exploring on your own Good luck and have fun
4. dx a T g x y z Y F u a The module shows the phase portrait associated with such a system and it computes and plots numerical solutions of the system in this phase plane It can also plot x and y versus t either individually or together After selecting Phase planes from the Iode main menu the Phase planes window will open up It shows a plot of the phase portrait for the system with the system itself written across the top of the plot When you first enter the module you will find Iode has already chosen a system and some option settings Next we explain how to change these settings and plot solutions Controls These are the almost same as in the Direction fields module See Chapter 1 The pop up menu that appears upon right clicking in the graph has one additional entry for continuing plots 15 16 CHAPTER 2 PHASE PLANES Equation menu This menu is very similar to the Equation menu in the Direction fields mod ule See Chapter 1 Below are noted a few differences e Enter differential equation prompts you to enter the right hand side functions f x y and g x y for the system using valid Matlab syntax see Appendix C You do not input the letters f or g you just input the expressions that you want on the right hand side of the system Remark 2 1 Note for advanced users Iode can also handle non autonomous systems in which the expressions for f and g depend also on t Iode will correctly compute num
5. the Matlab Sym bolic Toolbox is installed Also most equations do not have explicit solution formulas anyway e In graph controls By default left clicking in the graph will plot a solution through the click point To change this default see the Options menu below Dragging the mouse i e moving the mouse while holding down the left button over the graph creates a rectangle and when you release the mouse button Iode will zoom in on this rectangle Right clicking in the graph will display a pop up menu allowing you to erase solutions and undo zooms Remark 1 1 Plots of exact solutions can yield unexpected results For instance the exact symbolic solution of the default equation oe sin y x involves the inverse tangent atan The atan function is only defined up to an additive multiple of 7 and so the symbolic solution is only correct when the proper multiple of m is added Moreover different multiples of 7 might need to be added in different regions of the solution graph Now when Matlab evaluates atan numerically it always yields values between 5 and 5 which is usually the wrong choice and sometimes even results in discontinuous plots Even worse the plot of the numerical evalu ation of a symbolic solution with initial values xo yo does not always pass through the point xo yo This behavior is not strictly a bug in Matlab 13 but it is certainly a flaw that makes symbolic solutions less us
6. then click on the Plot solution button Second you can enter the desired initial slope x t into the relevant Initial conditions box and then click on the graph where you want the solution to start Iode will plot a solution through the click point whose initial slope is the value you entered in the box Third you can press down the mouse button at the desired initial point in the graph and then drag the mouse a short distance at the desired slope When you release the mouse button Iode will plot a solution starting at the point where you first pressed the button with the initial slope of that solution being given by the line going from where you first pressed the button to where you finally released it Play around and see how it works Equation menu When you select Enter differential equation you will be prompted for the coefficients m c and k to be used in equation 3 1 These coefficients are allowed to involve the independent variable but are often just constants You will also be prompted to enter the forcing function f Otherwise this menu is very similar to the corresponding part of the direction fields module described in Chapter 1 Options menu This menu is the same as the corresponding part of the direction fields module described in Chapter 1 except for one additional menu item Show forcing function used for either displaying or hiding the plot of the forcing function Chapter 4 Fourier series The Fourier
7. ENDIX C REFERENCE MATERIALS Some solver codes recognized by Iode euler Euler method for systems of first order equations rk Runge Kutta for systems of first order equations In addition to the pre installed solvers listed above you can also create your own For example if you devise a new method for solving differential equations then you could program it in a new file called my_algorithm n in your lode directory The structure of my_algorithm m should follow that of euler m You will simply need to change the update steps which in euler m are kl feval fs x tc i x x hxk1 Then after creating the file my_algorithm m you can input my_algorithm when Iode prompts you for a numerical method Appendix D Structure of Iode For users who like to look under the hood we now provide some information on the structure of Iode We encourage users to modify Iode and create new modules Figure D 1 breaks Iode down into its constituent modules Each module occurs as an m file with the same name For example dfgui in the figure means the file dfgui m that comes as part of the Iode package If you type doctool guidoc at the Matlab prompt you ll see Figure D 1 in a Matlab figure and if you click on the name of a module its help message will appear in Matlab s help window This will give you an idea of what Iode s modules do and how they work together There are two kinds of module The first kind are the GUI modules
8. User manual for lode graphical user interface Peter Brinkmann and Richard Laugesen Department of Mathematics University of Illinois Urbana Champaign U S A March 5 2010 Copyright c 2003 The Triode iode math uiuc edu Permission is granted to copy distribute and or modify this document under the terms of the GNU Free Documentation License Version 1 2 or any later version published by the Free Software Foundation with no Invariant Sec tions no Front Cover Texts and no Back Cover Texts A copy of the license is available at http www fsf org copyleft fdl html Contents 1 Direction fields 2 Phase planes 3 Second order linear ODEs 4 Fourier series On Partial differential equations Installing Iode Running Iode Reference materials I a WwW p Structure of Iode 11 15 19 21 25 29 31 33 37 CONTENTS Overview Iode rhymes with diode is a software package that enables you to explore direction fields phase planes second order linear ODEs Fourier series and heat and wave equations The name lode is supposed to be reminiscent of Illinois and ODE Iode runs under either Matlab or Octave their programming languages are mostly compatible For instructions on downloading installing and running all the needed software see Appendix A If your computer already has Matlab or Octave installed then all you need get is Iode Experienced users can proceed directly to www mat
9. ab or Octave running just type iode at the Matlab or Octave prompt The graphical user interface for Iode will launch if it can otherwise the text based user interface will launch Some develop ment versions of Octave sometimes fail to work with the part of Iode that determines whether the graphical user interface can be launched If you are using a version of Octave that reports an error when processing the command iode you can launch the text based interface right away by typing iodetxt Running Iode under Windows Log in to the machine and launch Matlab or Octave through the Start Programs menu Remark B 1 Important note Before launching Matlab or Octave for the very first time it is helpful to right click on the program name in the Start Programs menu then click on Properties and put the cor rect filepath into the Start in box Your filepath should look something 31 32 APPENDIX B RUNNING IODE like C My files MathClass my_iode The point is that from now on when you launch Matlab or Octave it will be able to find the iode files Once you have gotten Matlab or Octave running just type iode at the Matlab or Octave prompt If this does not work then you probably need to change into the directory my_iode where you have stored the Iode files You can do this from within Matlab or Octave using Unix commands such as cd my_iode at the Matlab or Octave prompt Appendix C Reference materials Mathematical e
10. annotations e g your name or the number of a homework assignment Miscellaneous features Error handling In any of Iode s dialog boxes if Iode cannot make sense of the information you provide then it will produce an error message and simply keep the previous values For example a common mistake is to forget CONTENTS 9 to type to indicate multiplication If you are asked to enter a function and type 2xy as 2xy then Iode will report an error and will continue using the previous function The correct input for 2xy is 2 x y Interrupting Matlab lacks proper mechanisms for interrupting lengthy computations so that there are only two solutions if a computation takes a long time to finish Either you wait until the computation is done or you quit from Matlab altogether The good news is that you won t encounter lengthy computations unless you explicitly request them e g by choosing an inordinately small step size for numerical computations 10 CONTENTS Chapter 1 Direction fields This module deals with general first order ODEs that is equations of the form dy The module plots the associated direction or slope field and it computes and plots numerical solutions of the equation Exact solutions can also be calculated and plotted provided Matlab s Symbolic Toolbox is installed After selecting Direction fields from the Iode main menu the Direc tion fields window will open up It shows a plot of
11. ble and y for the dependent variable In the second case input t for the independent variable and x for the dependent variable 14 CHAPTER 1 DIRECTION FIELDS Variable names should consist of only one letter to avoid the risk of conflicting with Iode s internal variables Options menu e Clicking on figure lets you choose what happens when you left click on the plot The four possibilities are plots solution through click point this is the default ac tion zooms in on click point zooms out on click point recenters figure on click point Incidentally the time taken by the zoom out operation grows ex ponentially with the number of clicks because zooming out requires a recomputation of existing solutions over the new larger domain If the domain doubles in size when you zoom out then recomputing the solutions will take twice as long as previously because the number of steps has to be doubled also If you zoom out again the time needed to recompute all solutions doubles again and so on e Change zoom factors this only matters if you have set the clicking option to zoom e Clear plot to erase all curves e Refresh plot try this if Matlab didn t update your plot properly This sometimes happens when other windows partially cover a Matlab window Chapter 2 Phase planes This module deals with autonomous systems of two equations that is sys tems of equations of the form
12. eating initial values for partial differential equations hat x a b equals 0 for x lt a and x gt b and equals 1 for a lt x lt b To make sense hat requires a lt b triangle x a b m a triangular shaped function equalling 0 for x lt a then rising linearly to height 1 at m and falling back linearly to zero at x b then equalling zero for x gt b To make sense triangle requires a lt m lt b The parameter m is optional and defaults to the midpoint oth 35 bump x a b m is a continuously differentiable function that equals 0 for x lt a and x gt b is positive for a lt x lt b has a maximum of 1 at x m and is strictly increasing between a and m and strictly decreasing between m and b To make sense bump requires a lt m lt b The parameter m is optional and defaults to ott Logical expressions in Matlab Octave and Iode Expressions like x gt 2 are treated as logical functions and return a value of either 1 true or 0 false So x gt 2 is the step function that equals 1 es ene f 0 otherwise Example C 2 Logical functions help us create functions defined in pieces ift lt 4 0 otherwise pl t 2 t lt 4 means l because t lt 4 equals 1 if t lt 4 and equals 0 otherwise Color codes recognized by Matlab Octave and Iode blue green red cyan magenta yellow black wK BakK eA oe Note that yellow plots are usually hard to see due to lack of contrast 36 APP
13. eful in practice than you might have expected Equation menu e Enter differential equation prompts you to enter the right hand side function f x y of the differential equation using valid Matlab syntax consult Appendix C for examples For example to study the equation dy dx yx you should input y x 2 Notice you do not input the letter f here You just input the expression that you want on the right hand side of the ODE Warning 1 2 Your expression for f should involve the independent or dependent variables or both but no other variables e Change display parameters prompts you to enter the domain and range of the plot and the number of line segments to be shown in the direction field e Plot arbitrary function allows you to plot any function for which you input the formula For example to plot y sinx you just input sin x Consult Appendix C for more examples of functions you can use equation other e Relabel variables this prompts you for letters for the independent variable and dependent variable When you input the names for the independent horizontal and de pendent vertical variables you are specifying the form in which you want the equation written for example e dy dx f x y in which case the solution will be a function y of a variable x or e dx dt f t x in which case the solution will be a function x of a variable t In the first case you would input x for the independent varia
14. erical solutions for such systems but will not display the full phase portrait which is 3 dimensional the phase portrait displayed is the slice at t t Also see next item e Change plot duration the plot of a solution curve can be thought of as tracing the path of a moving particle from time t tmin to time t tmarx The time tg for the initial condition should fall somewhere between tmin and tmar The default setting in Iode is to take tmin to 0 and tmar 27 meaning the particle flows forwards from the initial point starting at time 0 for a duration of 27 time units If you want to plot more of the solution curve just increase tmaz If you want to plot backwards in time from the initial point choose to tmaz To plot both backwards and forwards from the initial point choose a value of t strictly between tmin and tmar Options menu This is identical to the Options menu in the Direction fields module see Chapter 1 except for one new option e Show coordinate maps causes a new window to open up containing plots of x versus t y versus t and x and y versus t If you want to look at the 3 dimensional plot of x and y versus t from a different angle you can rotate it by dragging the mouse across the 3D plot 17 Finally as opposed to the behavior in the Direction fields module zooming out is not exponential in the number of clicks because the duration of solution plots is not controlled by the size o
15. f the display 18 CHAPTER 2 PHASE PLANES Chapter 3 Second order linear ODEs This module deals with second order linear nonhomogeneous ODEs ma t ex t ke t f t 3 1 Equations of this sort arise in models of simple mechanical vibrations and electrical circuits The application to mechanical vibrations explains why the module has filename mvgui m The module can plot the forcing function f and it computes and plots numerical solutions of 3 1 The user can input the equation and the initial conditions and choose the method used for computing solutions After selecting Second order linear ODEs from the Iode main menu the Second order linear ODEs window opens up The current ODE is dis played across the top along with the current options for plotting solutions The options in this module are either self explanatory or else are very similar to the corresponding parts of the direction fields module Controls These are almost the same as in the Direction fields module see Chapter 1 except for the default action of mouse clicks on the graph which we now explain The initial conditions for solving a second order linear ODE consist of the initial time to the initial position x to and the initial velocity x to You can give lode this initial data in three ways 19 20 CHAPTER 3 SECOND ORDER LINEAR ODES First you can enter the values to to and 2 to into the Initial conditions boxes and
16. h uiuc edu iode to download it The graphical user interface for Iode is supported only under Matlab version 6 0 or later and is described in this manual The text based user interface runs under earlier versions of Matlab and also under Octave and is described in another manual The graphical user interface offers certain features and plotting options that are unavailable in the text based interface A good way to learn Iode is simply to explore it Choose various options and see what happens Also the built in help features in Matlab and Octave are your friends For example typing help euler at the prompt will give information on the module euler m Please let us know if you encounter problems with Iode or if you want to suggest new features email us at iode math uiuc edu From a programming perspective the main point of Iode is that it is mod ular and has well defined interfaces This has several useful consequences For example it is easy for the user to customize Iode or to create and plug in new modules Simple modules of this sort can be created even by users with out much programming experience In fact lode was born out of frustration with other educational packages that conceal their inner workings from the user The mathematical modules such as df m or euler m can be used in 5 6 CONTENTS Matlab and Octave without the graphical user interface and much of the code is easily extensible Users are encouraged and in s
17. harmonic while in any one of the three plotting modes above 23 Function menu e Enter function You will be asked first for the left and right endpoints of the basic period interval x lt lt z for your function and then you enter the function using Matlab syntax as usual e g exp x for e Very often the interval of interest is just m lt x lt m in which case you enter pi and pi for the left and right endpoints Remark 4 1 The interval z lt lt z is part of the definition of the function For example it tells us that the period of the function is P v2 T e Comparison functions This item is only accessible when you are in the Plot coefficients C_n mode Its purpose is to compare the rate of decay of the Fourier coefficients with a function like L It has two subitems Enter comparison function prompts you to enter a function of n The idea is to enter a function that might be decaying at the same rate as the Fourier coefficients which in most cases means you should enter something like 1 n or 1 n or 1 n Show comparison function plots the ratio of C over your com parison function in the bottom graph You will typically need top harmonic to be 25 or so to see a convincing pattern in this ratio graph To get rid of the ratio graph just choose Show comparison function again If the ratio graph is decaying steeply overall then the coeffi cients C are decaying faster than yo
18. ions x The computed times are determined by the resolution see Options below Current coordinate You can step through the snapshots and sec tions using the arrow buttons or else you can enter a coordinate value inside the box Iode will round your input to the nearest com puted value of the coordinate Equation menu e Enter equation and boundary conditions First you choose the type of differential equation Wave Heat or Other The Other option lets you call on user created modules for handling other differential equations You can create such modules by copying and modifying the files wave m or heat m in your Iode directory Next you choose the boundary conditions Dirichlet u 0 t 0 and u L t 0 for all t Neumann u 0 t 0 and u L t 0 for all t Periodic u 0 t u L t and u 0 t uz L t for all t or Other e g you might create a module for mixed boundary con ditions To create such a boundary conditions module just copy and suitably modify the file dirichlet m Enter parameters and initial data this will prompt you for the wavespeed c if you are working with the wave equation or the thermal diffusivity k if you are working with the heat equation Then it prompts you for the length L of the interval 0 lt xz lt L on which you are solving the equation and for the duration T of the time interval 0 lt t lt T on which you want to examine the solution Finally you are as
19. ked to enter the intial data which consists of the ini tial displacement u x 0 f x and the initial velocity u x 0 g z 27 if working with the wave equation or the initial temperature con centration function u x 0 f x if working with the heat diffusion equation As always functions must be entered using valid Matlab syntax such as sin x for sin z Iode has some built in functions that make for interesting initial data hat triangle and bump See Appendix C for details Remark 5 1 Iode computes its approximate solutions by separation of vari ables For example for the heat equation with Dirichlet boundary conditions Iode will use N Et X perk sin n 1 where N is the value of top harmonic which can be changed using the Options below and where the b are the Fourier sine coefficients of the initial temperature f x For the wave equation with Dirichlet boundary conditions the approximate solution is N NTT nret L nret u x t X fon cost T 4 Barac sin T sin 7 gt n 1 where the bn are the Fourier sine coefficients of the initial displacement f z and the Bn are the Fourier sine coefficients of the initial velocity g x Options menu e Change resolution You can enter the number of points to be plotted in each coordinate direction This is a merely a question of display resolution Note that this number is the number of points in the interval 0 lt x lt L resp 0 lt t lt
20. launch in the correct directory See Appendix B Obtaining and installing Octave Octave is free software and you can obtain it from www octave org for Unix Windows and Mac OS X Installing Octave on Unix systems If your home machine runs Unix then chances are that you are already expert enough to download and install Octave yourself following the directions at www octave org If yourun into trouble your local Unix geeks will be happy to help Installing Octave on Windows systems You can find an easy to install Windows executable at http www site uottawa ca adler octave After downloading the exe file chances are that your browser will offer to run the installation program If not then you should double click on the file to activate the installation If something goes wrong with the installation of Octave you may be able to get some help at http octave sourceforge net Octave Windows htm After installing Octave under Windows it is helpful to set it up to launch in the correct directory See Appendix B Appendix B Running Iode Running Iode under Unix Log in to the machine and get to a Unix prompt Change directory to get into my_iode by using a Unix command such as cd my_iode Then launch Matlab or Octave by typing matlab or octave at the prompt Note that the order of operations matters here you need to change into the directory containing Iode before you launch Matlab or Octave Once you have gotten Matl
21. ome of our classes expected to look at the code and modify it take it apart put it back together and so on For example Appendix D discusses how to create your own solver module for numerically solving ODEs On the other hand Iode s graphical user interface gives you access to most of the mathematical power of the underlying modules without requiring you to program at all This manual concentrates mostly on explaining the graphical user interface General features Launching the Iode graphical user interface see instructions in Appendix B gets you to the main menu Direction fields Phase planes Second order linear ODEs Fourier series Partial differential equations Exit You can launch these modules independently and can even launch mul tiple copies of each module Before looking at these modules in detail we explain some features that are common to all or most of them Every window contains one or two graphs as well as a number of controls for affecting those graphs And across the top of each window is a menu bar whose entries include File Equation or Function and Options The File menu deals with the outside world such as files and printers The Equation menu deals with the mathematical content of your window i e it controls what you re looking at The Options menu contains the display options i e it controls how you re looking at the contents of your window Next we describe those menu items that are sha
22. red by all modules Then we ll come to more detailed chapters on each module in turn File menu The File menu is the same for all components of Iode e Open a file containing previously saved settings for the window 7 8 CONTENTS e Save a file containing the current settings of the window The file name should be entered in the Selection window of the dialog box and should have extension mat The point is that you can Save your work and then Open it again later e Print By default this will print the entire current window If you just want to print the graphs and not the rest of the window choose Print and then click on the Options button In the resulting dialog check the box labeled Suppress printing of user interface controls then click on OK If you choose to print to a file instead of to a printer then you can write your plot as a ps file which can then be opened and viewed with Ghostview For example on a Unix system you would get to a Unix prompt outside of Matlab and type ghostview my_plot eps e Quit will exit from the module Equation or Function menu The Equation or Function menus have just one entry in common e Relabel variables this item allows you to choose new names for the independent as well as dependent variables Options menu The Options menus have one entry in common e Enter caption this item lets you put a caption at the very bottom of the window and is useful for adding
23. s This module computes and plots solutions of the wave equation tie Clee and the heat or diffusion equation Ut Kure Solutions are found on an interval 0 lt x lt L for times 0 lt t lt T The method is separation of variables When you select Partial differential equations from the Iode main menu the Partial differential equations window opens up showing two graphs The top graph shows a 3D plot of a solution of either the wave or heat equation Try rotating this 3D plot by dragging the mouse across the graph The equation boundary conditions and initial conditions are written above the top graph The bottom graph is a cross section of the 3D solution graph either a t snapshot which shows the solution as a function of x at some time f or else a x section which shows the solution as a function of t at some position x Try stepping through these snapshots and sections with the arrow buttons or else enter a coordinate value into the box Controls You can rotate the 3D plot the top graph by dragging the mouse across it The following control buttons relate to the bottom graph only 25 26 CHAPTER 5 PARTIAL DIFFERENTIAL EQUATIONS e Plot t snapshots these are graphs of the solution as a function of x at various computed times t The computed times are determined by the resolution see Options below Plot x sections these are graphs of the solution as a function of t at various computed posit
24. series module can compute and graph the Fourier coefficients of a periodic function f and can plot partial sums of the Fourier series and the error difference between these partial sums and the function f For concreteness we will write f x for the function being considered even though you can change the name of the independent variable from x to something else like t if you want After selecting Fourier series from the Iode main menu the Fourier series window will open up showing two graphs The top graph plots a function f x in dark blue and a partial sum of its Fourier series in red These are plotted over two period lengths Recall that a partial sum of the Fourier series is an expression of the form N Ag NTT NTT D X An COS TES Bn s n F 4 1 n 1 where 1 T2 1 T2 n T f x cos de and Bn an f x sin de are the n Fourier coefficients and L is the half period L a2 x1 2 The top harmonic number N tells you the highest frequency that is in cluded in the partial sum Try increasing or decreasing the value of the top harmonic used in your plot by clicking on the arrow buttons in the mid dle of the window Alternatively you can type a number directly into the Current top harmonic box between the arrows 21 22 CHAPTER 4 FOURIER SERIES The bottom graph in the window plots the error between the function f x and the partial sum of its Fourier series N Ag NTT NTT E a Ay TR n j Sap
25. the direction field for the ODE with the equation itself written across the top of the plot When you first enter the module you will find Iode has already chosen a default ODE as well as reasonable option settings Next we explain how to change these settings and plot solutions Controls e Solution method specifies the method to be used when computing solutions Numerical methods include Euler Runge Kutta and Other for user defined methods as in Appendix C the final method is Exact available provided the Matlab Symbolic Toolbox is installed See also Remark 1 1 below e Step size enter your desired step size usually a small positive num ber for the numerical solution method 11 12 CHAPTER 1 DIRECTION FIELDS e Plot color pick a color any color for your next solution plot Note Yellow tends not to show up very well e Initial conditions and Plot solution You can tell Iode to plot a solution of the differential equation in two ways Either enter the coordinates of the initial point into the Initial conditions boxes and then click on the Plot solutions button or else simply click on the plot itself at the initial condition point i e where you want the plot to begin Note In either case Iode will compute and plot the solution curve both forwards and backwards from the initial condition e Exact solution click here to try to obtain a formula for the general solution of the equation This feature only works if
26. ur comparison function so you should enter a faster decaying comparison function such as a larger power of 1 n If the ratio graph is growing steeply overall then the coefficients C are decaying slower than your compari son function so you should enter a slower decaying comparison function such as a smaller power of 1 n If the ratio graph re mains bounded but does not approach zero as n gets bigger then congratulations You have probably hit upon the correct rate of decay of the Fourier coefficients 24 CHAPTER 4 FOURIER SERIES Remark 4 2 It is interesting to plot the values of C and determine their rate of decay because C equals the amplitude of the combined oscillations at the nth frequency level in the Fourier series of f To see this observe that A Cos B sin Gy cos an 4 2 where the angle a is defined by requiring cosa An Cn and sina B Cp Formula 4 2 is proved just by substituting the identity cos G a cos 3 cosa sin 8 sina on the right hand side Options menu e Change plot resolution this item is only accessible when you are in the Plot partial sums and errors mode You should probably increase the plot resolution if you are studying rapidly oscillating func tions or if you choose top harmonic bigger than say 100 The prompt Number of points to be plotted refers to the number of plot points in the interval 7 lt lt T Chapter 5 Partial differential equation
27. xpressions in Matlab Octave and lode For simple expressions we use the usual keyboard characters 2X means 22 x73 1 6 means x 1 6 Or instead of the usual division we can use left division pi 3 means 3 3 pi also means 3 Built in functions exp x exponential e log x natural logarithm In x log10 x base 10 logarithm log abs x absolute value sqrt x square root yx 1 ifz gt 0 sign x signum function which equals 0 ifzr 0 1 ifz lt 0 sin x sinh x 33 34 APPENDIX C REFERENCE MATERIALS cos x trigonometric cosh x hyperbolic tan x functions tanh x trigonometric cot x x in radians coth x functions sec x sech x csc x csch x asin x asinh x acos x inverse acosh x inverse atan x trigonometric atanh x hyperbolic acot x functions acoth x trigonometric asec x asech x functions acsc x acsch x besselj nu z Bessel function of the first kind bessely nu z Bessel function of the second kind besseli nu z Modified Bessel function of the first kind besselk nu z Modified Bessel function of the second kind Example C 1 sin exp y 74 means sin e acos exp 1 1 means arccos e No matter whether you re using Octave or Matlab you can always find more information on a function by typing help function Additional Iode functions The installation of Iode includes the following functions useful for studying Fourier series and for cr
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