MaplePrimes Posts

MaplePrimes Posts are for sharing your experiences, techniques and opinions about Maple, MapleSim and related products, as well as general interests in math and computing.

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  • These worksheets provide the volume calculations  of a small causal diamond near the tip of the past light cone, using dimensional analysis and particular test metrics.

    I recommend them for anyone working in causet theory on the problem of finding higher order corrections.

    2D.mw

    4D.mw

    4Dflat.mw

     

     

     

     

    With this app you will be able to interpret the curvatures generated by two position vectors, either in the plane or in space. Just enter the position vectors and drag the slider to calculate the curvature at different times and you will of course be able to observe its respective graph. At first I show you how it is developed using the natural syntax of Maple and then optimize our
     app with the use of buttons. App made in Maple for engineering students. In spanish.

    Plot_of_Curvature.mw

    Videotutorial:

    https://www.youtube.com/watch?v=SbXFgr_5JDE

    Lenin Araujo Castillo

    Ambassador of Maple

    There is a problem with mapleprimes favorites. 

    Looking at my list of favorites it cites I have 6 pages, however when I get to page 3 it shows a list of only six items and no more page listings after clicking the back button and going again to page 3 even less appear.  I was looking for specific questions and posts I have favorited which seem to be abscent from the listing. 

    Can mapleprimes administrators please address the problem?

    This is Maple:

    These are some primes:

    22424170499, 106507053661, 193139816479, 210936428939, 329844591829, 386408307611,
    395718860549, 396412723027, 412286285849, 427552056871, 454744396991, 694607189303,
    730616292977, 736602622363, 750072072203, 773012980121, 800187484471, 842622684461
    

    This is a Maple prime:


    In plain text (so you can check it in Maple!) that number is:

    111111111111111111111111111111111111111111111111111111111111111111111111111111
    111111111111111111111111111111116000808880608061111111111111111111111111111111
    111111111111111111111111111866880886008008088868888011111111111111111111111111
    111111111111111111111116838888888801111111188006080011111111111111111111111111
    111111111111111111110808080811111111111111111111111118860111111111111111111111
    111111111111111110086688511111111111111111111111116688888108881111111111111111
    111111111111111868338111111111111111111111111111880806086100808811111111111111
    111111111111183880811111111111111111100111111888580808086111008881111111111111
    111111111111888081111111111111111111885811188805860686088111118338011111111111
    111111111188008111111111111111111111888888538888800806506111111158500111111111
    111111111883061111111111111111111116580088863600880868583111111118588811111111
    111111118688111111111001111111111116880850888608086855358611111111100381111111
    111111160831111111110880111111111118080883885568063880505511111111118088111111
    111111588811111111110668811111111180806800386888336868380511108011111006811111
    111111111088600008888688861111111108888088058008068608083888386111111108301111
    111116088088368860808880860311111885308508868888580808088088681111111118008111
    111111388068066883685808808331111808088883060606800883665806811111111116800111
    111581108058668300008500368880158086883888883888033038660608111111111111088811
    111838110833680088080888568608808808555608388853680880658501111111111111108011
    118008111186885080806603868808888008000008838085003008868011111111111111186801
    110881111110686850800888888886883863508088688508088886800111111111111111118881
    183081111111665080050688886656806600886800600858086008831111111111111111118881
    186581111111868888655008680368006880363850808888880088811111111111111111110831
    168881111118880838688806888806880885088808085888808086111111111111111111118831
    188011111008888800380808588808068083868005888800368806111111111111111111118081
    185311111111380883883650808658388860008086088088000868866808811111111111118881
    168511111111111180088888686580088855665668308888880588888508880800888111118001
    188081111111111111508888083688033588663803303686860808866088856886811111115061
    180801111111111111006880868608688080668888380580080880880668850088611111110801
    188301111111111110000608808088360888888308685380808868388008006088111111116851
    118001111111111188080580686868000800008680805008830088080808868008011111105001
    116800111111118888803380800830868365880080868666808680088685660038801111180881
    111808111111100888880808808660883885083083688883808008888888386880005011168511
    111688811111111188858888088808008608880856000805800838080080886088388801188811
    111138031111111111111110006500656686688085088088088850860088888530008888811111
    111106001111111111111111110606880688086888880306088008088806568000808508611111
    111118000111111111111111111133888000508586680858883868000008801111111111111111
    111111860311111111111111111108088888588688088036081111860803011111111863311111
    111111188881111111111111111100881111160386085000611111111888811111108833111111
    111111118888811111111111111608811111111188680866311111111111811111888861111111
    111111111688031111111111118808111111111111188860111111111111111118868811111111
    111111111118850811111111115861111111111111111888111111111111111080861111111111
    111111111111880881111111108051111111111111111136111111111111188608811111111111
    111111111111116830581111008011111111111111111118111111111116880601111111111111
    111111111111111183508811088111111111111111111111111111111088880111111111111111
    111111111111111111600010301111111111111111111111111111688685811111111111111111
    111111111111111111111110811801111111111111111111158808806881111111111111111111
    111111111111111111111181110888886886338888850880683580011111111111111111111111
    111111111111111111111111111008000856888888600886680111111111111111111111111111
    111111111111111111111111111111111111111111111111111111111111111111111111111111
    

    This is a 3900 digit prime number. It took me about 400 seconds of computation to find using Maple.

    It turns out be be really easy to do because prime numbers are realy quite common.  If you have a piece of ascii art where all the characters are numerals, you could just call on it and get a prime number that is still ascii art with a couple digits in the corner messed up (for a number this size, I expect fewer than 10 of the least significant digits would be altered).  You may notice, however, that my Maple Prime has beautiful corners!  This is possible because I found the prime in a slightly different way.

    To get the ascii art in Maple, I started out by using to import ( )  and process the original image.  First then and to get a nice 78 pixel wide image.  Then to make it a pure 1-bit black or white image.

    Then, from the image, I create a new Array of the decimal digits of the ascii art and my prime number.  For each of the black pixels I randomly use one of the digits or and for the white pixels (the background) I use 's.  Now I convert the Array to a large integer and test if it is prime using (it probably isn't) so, I just randomly change one of the black pixels to a different digit (there are 4 other choices) and call again. For the Maple Prime I had to do this about 1000 times before I landed on a prime number. That was surprisingly fast to me! It is a great object lesson in how dense the prime numbers really are.

    So that you can join the fun without having to replicate my work, here is a small interactive Maple document that you can use to find prime numbers that draw ascii art of your source images. It has a tool that lets you preview both the pixelated image and the initial ascii art before you launch the search for the prime version.

    Prime_from_Picture.mw

    With this application you will learn the beginning of the study of the vectors. Graphing it in a vector space from the plane to the space. You can calculate its fundamental characteristics as triangle laws, projections and strength. App made entirely in Maple for engineering students so they can develop their exercises and save time. It is recommended to first use the native syntax then the embedded components. In Spanish.

    Vector_space_with_projections_and_forces_UPDATED.mw

    Movie #01

    https://www.youtube.com/watch?v=VAukLwx_FwY

    Movie #02

    https://www.youtube.com/watch?v=sIxBm_GN_h0

    Movie #03

    https://www.youtube.com/watch?v=LOZNaPN5TG8

    Lenin Araujo Castillo

    Ambassador of Maple

    I'd like to present the following bugs in the IntTutor command.

    1. Initialize

    Student[Calculus1]:-IntTutor((1+cos(3*x))^(3/2), x);

    then press the All Steps button. The command produces the answer (see Bug1_in_IntTutor.mw)

    (4/9)*sqrt(2)*sin((3/2)*x)^3-(4/3)*sqrt(2)*sin((3/2)*x)

    which is not correct in view of

    plot(diff((4/9)*sqrt(2)*sin((3/2)*x)^3-(4/3)*sqrt(2)*sin((3/2)*x), x)-(1+cos(3*x))^(3/2), x = 0 .. .2);

    One may compare it with the Mathematica result Step-by-step2.pdf.

    2. Initialize

    Student[Calculus1]:-IntTutor(cos(x)^2/(1+tan(x)), x);

    In the window press the Next Step button. This crashes (The kernel connection has been lost) my comp in approximately an half of hour (see screen2.docx). One may compare it with the Mathematica result Step-by-step.pdf .

    Indeed,  "We wanted the best, but it turned out like always" .

    This is a toy example illustrating three possible ways to define a group.

    An icosahedron:

    with(GroupTheory): with(Student[LinearAlgebra]): with(geom3d):
    
    icosahedron(ii): vv := vertices(ii):
    
    PLOT3D(POLYGONS(op(evalf(faces(ii))), TRANSPARENCY(.75)),
      op(zip(TEXT, 1.1*evalf(vv), [`$`(1..12)])), AXESSTYLE(NONE));
    

    The group of rotations is generated by the rotation around the diagonal (1,4) and the rotation around the line joining the midpoints of the edges (1,2) and (3,4), by the angles 2*Pi/5 and Pi respectively.

    Define the group by how the two rotations permute the 6 main diagonals of the icosahedron:

    gr := PermutationGroup({[[2, 5, 3, 4, 6]], [[1, 2], [5, 6]]});
    
    IdentifySmallGroup(gr);
                                 60, 5
    

    Or define the group by the relations between the two rotations:

    gr2 := FPGroup([a, b], [[a$5], [b$2], [a, b, a, b, a, b]]);
    
    IdentifySmallGroup(gr2);
                                 60, 5
    

    Finally, define a group with elements that are rotation matrices:

    m1 := RotationMatrix(2*Pi/5, Vector(op(4, vv) - op(1, vv))):
    m2 := RotationMatrix(Pi, Vector(add(op(3..4, vv) - op(1..2, vv)))):
    m1, m2 := op(evala(convert([m1, m2], radical))):
    
    gr3 := CustomGroup([m1, m2], `.` = evala@`.`, `/` = evala@rcurry(`^`, -1), `=` = Equal);
    
    IdentifySmallGroup(gr3);
                                 60, 5
    
    AreIsomorphic(SmallGroup(60, 5), AlternatingGroup(5));
                                  true
    

    One question is how to find out that the group is actually A5 without looking up the group (60, 5) elsewhere.

    Also, it doesn't matter for this example, but adding `1`=IdentityMatrix(3) to the CustomGroup definition gives an error.

     

    With the launch of Maple 2017, we really wanted to showcase some of the amazing people that work so hard to make Maple. We wanted to introduce our developers to our awesome user community, put names to faces, and have some fun in the process.

    We’ll be doing this Q&A session from time to time with team members from the Maple, MapleSim, Maple T.A. and Möbius development groups.

    My first Q&A is with Math Architect, Paulina Chin. If you’re a regular MaplePrimes user, you’ll know her as @pchin. Let’s get right into the questions.

    1. What do you do at Maplesoft?

    I’m a member of the Math Software group. Much of my time goes toward developing and maintaining parts of the Maple library, but I occasionally develop Maple content related to math education as well.

    1. What did you study in school?

    I started in Applied Mathematics and then continued with graduate work in Computer Science and Electrical Engineering. My graduate and post-doc research  were in the area of numeric computation.

    1. What area(s) of Maple are you currently focusing on in your development?

    For many years, I’ve been working on the plotting and typesetting features in Maple. I also work on the Grading package and related applications.

    1. What’s the coolest feature of Maple that you’ve had a hand in developing?

    The Typesetting (2-D math) system in Maple is undoubtedly the most challenging and complex project I’ve worked on, and it involves careful coordination among a team of developers. I’m not sure others would see it as cool, because the features are not flashy like some of the visualization features I’ve worked on. However, whenever we implement a new feature and it works well, it’s really satisfying because it makes mathematics that much more accessible to users.

    1. What do you like most about working at Maplesoft? How long have you worked here?

    I’ve been at Maplesoft 17 years and my work has never been boring. I especially enjoy being surrounded by a very diverse and dedicated group of co-workers, and it’s terrific when we get new students, interns and visitors who come from all parts of the world. All of these people contribute to the great atmosphere here.

    1. Favorite hobby?

    I like discussing books as much as reading them. I run several book clubs, including the one here at Maplesoft. I also enjoy working with young people and volunteer at my daughter’s high school, helping students train for programming contests.

    1. What do you like on your pizza?

    Pineapple and hot peppers.

    1. What’s your favourite movie?

    I have so many favourites that it’s hard to answer this question. At the moment, I might say Notorious, The Empire Strikes Back, and Annie Hall, but ask me again next week and I’ll probably give you a different list.

    1. What skill would you love to learn? (That you haven’t already) Why?

    I wish I could play a musical instrument. I know a number of highly skilled amateur and professional musicians, and I’ve always admired their abilities.

    1. Who’s your favourite mathematician?

    I’d have to say it’s Euclid. When I was in Grade 6, my teacher saw I was bored with the math exercises we were doing and gave me a book on geometric constructions. That was the start of a life-long fascination with math. I even named my cat Euclid but she didn’t live up to the name, as she turned out to be lovable but not very smart.

    It can be interesting to consider a directional derivative of f(z) in the direction w:

    %ddiff(f(z), z, w) = %limit((f(z + w*h) - f(z))/(w*h), h = 0);
    ddiff := proc (fz0, z, dir) local rule, fz, dfz, ans;
      dfz := %ddiff(fz, z, dir)*dir;
      rule :=
       [abs(1, fz::anything) = (conjugate(fz)*dfz + fz*conjugate(dfz))/(2*abs(fz)*dfz),
        signum(1, fz::anything) = (conjugate(fz)*dfz - fz*conjugate(dfz))/(2*abs(fz)*conjugate(fz)*dfz)];
      ans := applyrule(rule, diff(fz0, z));
      ans := value(ans);
      ans := [ans, op(convert~(ans, [abs, argument, Re, Im, signum, conjugate]))];
      op(1, sort(simplify(ans, size), length)) end proc;
    

    For analytic functions the derivative is the same in any direction:

    ddiff(sqrt(Re(f(z))^2+Im(f(z))^2)*exp(I*argument(f(z))), z, w); # f(z) in disguise
                                 d      
                                --- f(z)
                                 dz     
    

    For non-analytic functions that's no longer the case:

    ddiff(conjugate(z), z, w);
                                   1     
                               ----------
                                        2
                               signum(w) 
    
    ddiff(conjugate(f(z)), z, 1+I);
                                 ________
                                  d      
                              -I --- f(z)
                                  dz     
    
    ddiff(abs(z), z, z);
                                   1    
                               ---------
                               signum(z)
    
    ddiff(ln(abs(z)), z, z);
                                   1
                                   -
                                   z
    

    Some of those derivates have simple geometric interpretations: the derivative of argument(z) in the direction z is zero, since argument(z) doesn't change when moving in the direction z from the point z; the derivative of abs(z) in the direction I*z is zero, since the direction is tangent to the circle abs(z)=constant; since signum(z) is a function of argument(z) only, its derivative in the direction z is zero as well.

    Interestingly, the derivative taken twice in the direction z is zero for each of the six basic functions:

    map(fz -> ddiff(ddiff(fz, z, z), z, z), [abs, argument, Re, Im, signum, conjugate](z));
                           [0, 0, 0, 0, 0, 0]
    

    Does that have some simple geometric interpretation as well?

     

    I was asked if I would put together a list of top resources to help students who are using Maple for the first time.  An awful lot of students will be cracking Maple open in the next few weeks (the ones who are keeping up with their assignments, at least – for others, it sometimes takes little longer :-), so it seemed like a good idea.

    So then I had to decide what to do. I know Top N lists are very popular (Ten Things that Will Shock You about Your Math Software!), and there are tons of Maple training resources available to fill such a list without any difficulties.  But personally, I don’t always like Top N lists. What are the chances that there are exactly N things you need to know, for nice values of N? And how often you are really interested in all N items? I just want to get straight to the points I care about.

    I decided I’d try a matrix. So here you go: a mini “choose your own adventure” guide for getting to know Maple.  Pick the row that corresponds to what you want to do, and the column for how you want to do it.  All on a single, page, and ad-free!

    And best of luck for the new school year.

     

     

    I like words

    I like videos

    Just let me try it

    Product Overview

    Inside Maple, from the Help menu, select Take a Tour of Maple then click on the Ten Minute Tour button.

     

    (Okay, even though I like words, too, you might also want to watch the video in the next column. The whole “picture is worth a thousand words” does have some truth to it, much as I don’t always like to admit it. J)

    Watch Clickable Math

     

    Keep in mind that if you prefer to use commands instead of these Clickable Math tools, you can do that too.  Personally, I mix and match.

    You’ll figure it out.

    Getting Started Info

    Read the Maple Quick Start Tutorial Guide, as a PDF, or from the Help system. To access this guide from within Maple, start Maple, click on the Getting Started icon the left, then select the Quick Start Guide (first icon in the second row).

    Watch the Maple Quick Start Tutorial Video.

    The most important things to remember are

    1. Right click on your math expression to bring up a menu of things you can do, like plotting or integrating or solving your expression
    2. If you have just entered an exponent or the denominator of a fraction, use the right arrow key to get out of it.

    How do I? Essentials

    Look at the “How do I” section of the Maple Portal (Start Maple, click on the Getting Started icon, click on the Maple Portal icon; or search for “MaplePortal” in the help system).  Also look at the Maple Portal for Students, using the button from the Maple Portal.

    Check out the dozens of videos in the Maple Training Video collection.

    You can do a lot with the context menus and the various tools you’ll find on the Tools menu. But when in doubt, look at the list of “How do I” tasks from the Maple Portal described in the “words” column and pull out what you need from there.

    What now?

    The help system is your friend. Not only does it have help pages for every feature and every command, but it includes both the Maple User Manual and the Maple Programming Guide (also available as PDFs).

    Check out the collection of videos on the Maplesoft YouTube channel.  (And the help system is your friend, too. We can’t make videos to cover every last thing, and if we did, you wouldn’t have time to watch them all!)

    Maple comes with many examples and applications you can look at and modify.  You can browse through the Start page resources, or search for “examples,index” in the help system to see the full list.

     

    And yes, the help system is your friend, too.  But don’t worry, no one is going to make you read the manual.

     

     

     

    Download New_ReportGeneration_with_ExcelData.mw

    Dear Users,

    I have received a congratulations from a Mapleprime user for my post (on Finite Element Analysis - Basics) posted two years earlier. I  did not touch that subject for two years for obvious reasons. Now that a motivation has come, I have decided to post my second application using embedded components. This I was working for the past two years and with the support from Maplesoft technical support team and Dr.RobertLopez. I thank them here for this workbook has come out well to my satisfaction and has given me confidence to post it public.

    About the workbook

    I have tried to improve the performance of a 2-Stroke gasoline engine to match that of a four stroke engine by using exhaust gas recirculation. Orifice concept is new and by changing the orifice diameter and varying the % of EGR, performance was monitored and data stored in Excel workbook. These data can be imported to Maple workbook by you as you want for each performance characteristic. The data are only my experimental and not authentic for any commercial use.

    This Maple workbook generates curves from data for various experiments conducted by modifying the field variables namely Orifice diameter, % Exhaust gas Recirculation and Heat Exchanger Cooling. Hence optimum design selection is possible for best performance.

    Thanks for commenting, congratulating or critisising!! All for my learning and improving my Maple understanding!! 

    In this app you can use from the creation of curve, birth of the position vector and finally applied to the displacement and the distance traveled. All this application revolves around the creation of a path and the path of a particle over this generated by vectors. You will only have to insert the vector components and the times to evaluate. Designed for engineering students guided through Maple. In Spanish.

    Displacement_and_distance_traveled_with_vectors.mw Updated

    Video

    https://www.youtube.com/watch?v=jOcKYZ5EEM0

    Lenin Araujo C

    Ambassador of Maple

    Here in this video you can observe the correct insertion of vectors; Making use of the keyboard, ascii code and tool palette of our Maple program. As our worksheet is very large, I made the explanation in two parts; I recommend that you observe this first part of performing any execution on your Maple worksheet. You can contrast your results with the apps also made in this software. In Spanish.

    Shortcut_in_Vectors_for_Engineering.mw

    Movie # 01

    https://www.youtube.com/watch?v=EJtAli54q_A

    Movie # 02

    https://www.youtube.com/watch?v=m-JUmhkbWI8

    Lenin Araujo Castillo

    Ambassador of Maple

    On 5/July/2017, Kitonum responded to the 3/July/2017 MaplePrimes question "How to perform double integration over subdomain" by providing code for a procedure IntOverDomain that implements Green's theorem applied to a planar region whose boundary is a simple, closed, rectifiable, oriented curve (SCROC by some authors).

    I was intrigued. First, this is a significant extension of existing Maple functionalities. Second, the implementation admits boundaries defined piecewise with sections defined parametrically; or sections that are polygonal lines defined by a list of nodes.

    But how was the line integral around such boundaries coded? In the worksheet "IntOverDomain_Deconstructed," I summarize the existing Maple functionality for implementing iterated double integrals over specified domains, then analyze how Kitonum coded Green's theorem as an extension of Maple's capabilities. After recognizing the great coding skills of Kitonum, I conclude with a short wishlist of related extensions that I would like to see added to Maple in the future.

     

    Download the worksheet: IntOverDomain_Deconstructed.mw

    A new code based on higher derivative method has been implemented in Maple. A sample code is given below and explained. Because of the symbolic nature of Maple, this method works very well for a wide range of BVP problems.

    The code solves BVPs written in the first order form dy/dx = f (Maple’s dsolve numeric converts general BVPs to this form and solves).

    The code can handle unknown parameters in the model if sufficient boundary conditions are provided.

    This code has been tested from Maple 8 to Maple 2017. For Digits:=15 or less, this code works in all of the Maple versions tested.

    Most problems can be solved with Digits:=15 with atol = 1e-10 or so. This code can be used to get a tolerance value of 1e-20 or any high precision as needed by changing the number of Digits accordingly. This may be needed if the original variables are not properly sacled. With arbitrarily high Digits, the code fails in Maple 18 or later version, etc because Maple does not support SparseDirect Solver at high precision in some of the versions (hopefuly this bug can be removed in the future versions).

    For simple problems, Maple’s dsolve/numeric is superior to the code developed as it is implemented in hardware floats. For large scale problems and stiff problems, the method developed is much more superior to Maple and comparable to (and often times better than) state of the art codes for BVPs - bvp4c (MATLAB), COLSYS,TWPBVP, etc.

    The code, as written, cannot be used for problems with a singularity at end points (doable in the future). In addition, mixed boundary conditions are not supported in this version of the code (for example, y1(1)=y2(0)). Future updates will include the application of this approach for DAE-BVPs, currently not supported by Maple’s dsolve/numeric command.

    A paper has been submitted to JCAM. I welcome feedback on the code and solicit input from Mapleprimes members if they are able to test (and break this code) for any BVP.

     

    PDF of the paper submitted, example maple code and the solver as a text file needed are uploaded here. Additional examples are hosted on my website at http://depts.washington.edu/maple/HDM.html


     

     

    ##################################################################################

    Troesch's problem
    This is an inherently unstable, difficult, nonlinear, two-point BVP formulated by Weibel and Troesch that describes the confinement of a place column by radiation pressure. Increasing epsilon increases the stiffness of the ODE.
    1. E.S. Weibel, On the confinement of a plasma by magnetostatic fields, Phys. Fluids. 2 (1959) 52-56.
    2. B. Troesch, A simple approach to a sensitive two-point boundary value problem, J. Comput. Phys. 21 (1976) 279-290.

    Introduction
    The package HDM solves boundary value problems (BVPs) using higher derivative methods (HDM) in Maple®. We explain how to solve BVPs using this package. HDM can numerically solve BVPs of ordinary differential equations (ODEs) of the form shown is the fowllowing example.

    ###################################################################################

     

     

    Reset the program to clear the memory from previous execution command.

    restart:

     

    Read the txt file which contains the HDM solver for BVPs.

    read("HDM.txt");

     

    Declare the precision for the entire Maple® sheet.

    Digits:=15;

    Digits := 15

    (1)

     

    Enter the first-order ODEs into EqODEs list.

    EqODEs:=[diff(y1(x),x)=y2(x),diff(y2(x),x)=epsilon*sinh(epsilon*y1(x))];

    EqODEs := [diff(y1(x), x) = y2(x), diff(y2(x), x) = epsilon*sinh(epsilon*y1(x))]

    (2)

     

    Define the left boundary condition (bc1), and the right boundary condition (bc2). One should collect all the terms in one side.

    bc1:=evalf([y1(x)]);

    bc1 := [y1(x)]

    (3)

    bc2:=evalf([y1(x)-1]);

    bc2 := [y1(x)-1.]

    (4)

     

    Define the range (bc1 to bc2) of this BVP.

    Range:=[0.,1.];

    Range := [0., 1.]

    (5)

     

    List any known parameters in the list.

    pars:=[epsilon=2];

    pars := [epsilon = 2]

    (6)

     

    List any unknown parameters in the list. When there is no unknown parameter, use [ ].

    unknownpars:=[];

    unknownpars := []

    (7)

     

    Define the initial derivative in nder (default is 5 for 10th order) and the number of the nodes in nele (default is 10 and distributed evenly across the range provided by the user). The code adapts to increase the order. For many problems, 10th order method with 10 elements are sufficient.

    nder:=5;nele:=10;

    nder := 5

    nele := 10

    (8)

     

    Define the absolute and relative tolerance for the local error. The error calculation is done based on the norm of both the 9th and 10th order simulation results.

    atol:=1e-6;rtol:=atol/100;

    atol := 0.1e-5

    rtol := 0.100000000000000e-7

    (9)

     

    Call HDMadapt procedure, input all the information entered above and save the solution in sol. HDMadapt procedure does not need the initial guess for the mesh.

    sol:= HDMadapt(EqODEs,bc1,bc2,pars,unknownpars,nder,nele,Range,atol,rtol):

     

    Present some details of the solution.

    sol[4]; # final derivative

    5

    (10)

    sol[5]; # Maximum local RMSE

    0.604570329905172e-8

    (11)

     

    Store the dimension of the solution (after adjusting the mesh) to NN.

    NN:=nops(sol[3])+1;

    NN := 11

    (12)

     

    Plot the interested variable (the ath ODE variable will be sol[1][i+NN*(a-1)] )

    node:=nops(EqODEs);
    odevars:=select(type,map(op,map(lhs,EqODEs)),'function');

    node := 2

    odevars := [y1(x), y2(x)]

    (13)

    xx:=Vector(NN):

    xx[1]:=Range[1]:

    for i from 1 to nops(sol[3]) do xx[i+1]:=xx[i]+sol[3][i]: od:

    for j from 1 to node do
      plot([seq([xx[i],rhs(sol[1][i+NN*(j-1)])],i=1..NN)],axes=boxed,labels=[x,odevars[j]],style=point);
    end do;

     

     


     

    Download Example_3_Troesch.mws

     

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