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I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra - 2017" . It was a very interesting event. This second presentation, about "Differential algebra with mathematical functions, symbolic powers and anticommutative variables", describes a project I started working in 1997 and that is at the root of Maple's dsolve and pdsolve performance with systems of equations. It is a unique approach. Not yet emulated in any other computer algebra system.

At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.
 

Differential algebra with mathematical functions,

symbolic powers and anticommutative variables

 

Edgardo S. Cheb-Terrab

Physics, Differential Equations and Mathematical Functions, Maplesoft

 

Abstract:
Computer algebra implementations of Differential Algebra typically require that the systems of equations to be tackled be rational in the independent and dependent variables and their partial derivatives, and of course that A*B = A*B, everything is commutative.

 

It is possible, however, to extend this computational domain and apply Differential Algebra techniques to systems of equations that involve arbitrary compositions of mathematical functions (elementary or special), fractional and symbolic powers, as well as anticommutative variables and functions. This is the subject of this presentation, with examples of the implementation of these ideas in the Maple computer algebra system and its ODE and PDE solvers.

 

 

restartwith(PDEtools); interface(imaginaryunit = i)

sys := [diff(xi(x, y), y, y) = 0, -6*(diff(xi(x, y), y))*y+diff(eta(x, y), y, y)-2*(diff(xi(x, y), x, y)) = 0, -12*(diff(xi(x, y), y))*a^2*y-9*(diff(xi(x, y), y))*a*y^2-3*(diff(xi(x, y), y))*b-3*(diff(xi(x, y), x))*y-3*eta(x, y)+2*(diff(eta(x, y), x, y))-(diff(xi(x, y), x, x)) = 0, -8*(diff(xi(x, y), x))*a^2*y-6*(diff(xi(x, y), x))*a*y^2+4*(diff(eta(x, y), y))*a^2*y+3*(diff(eta(x, y), y))*a*y^2-4*eta(x, y)*a^2-6*eta(x, y)*a*y-2*(diff(xi(x, y), x))*b+(diff(eta(x, y), y))*b-3*(diff(eta(x, y), x))*y+diff(eta(x, y), x, x) = 0]

 

declare((xi, eta)(x, y))

xi(x, y)*`will now be displayed as`*xi

 

eta(x, y)*`will now be displayed as`*eta

(1)

for eq in sys do eq end do

diff(diff(xi(x, y), y), y) = 0

 

-6*(diff(xi(x, y), y))*y+diff(diff(eta(x, y), y), y)-2*(diff(diff(xi(x, y), x), y)) = 0

 

-12*(diff(xi(x, y), y))*a^2*y-9*(diff(xi(x, y), y))*a*y^2-3*(diff(xi(x, y), y))*b-3*(diff(xi(x, y), x))*y-3*eta(x, y)+2*(diff(diff(eta(x, y), x), y))-(diff(diff(xi(x, y), x), x)) = 0

 

-8*(diff(xi(x, y), x))*a^2*y-6*(diff(xi(x, y), x))*a*y^2+4*(diff(eta(x, y), y))*a^2*y+3*(diff(eta(x, y), y))*a*y^2-4*eta(x, y)*a^2-6*eta(x, y)*a*y-2*(diff(xi(x, y), x))*b+(diff(eta(x, y), y))*b-3*(diff(eta(x, y), x))*y+diff(diff(eta(x, y), x), x) = 0

(2)

casesplit(sys)

`casesplit/ans`([eta(x, y) = 0, diff(xi(x, y), x) = 0, diff(xi(x, y), y) = 0], [])

(3)

NULL

Differential polynomial forms for mathematical functions (basic)

   

Differential polynomial forms for compositions of mathematical functions

   

Generalization to many variables

   

Arbitrary functions of algebraic expressions

   

Examples of the use of this extension to include mathematical functions

   

Differential Algebra with anticommutative variables

   


 

Download DifferentialAlgebra.mw

Download DifferentialAlgebra.pdf

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

 

  1. I have a set of equations described on different lines and I want them to be executed together and their answers to be displayed in seperate lines after that. How do I do that?
  2. This is particulalry important because, at times I will have to document multiple equations sequentially and later on have their answers reffere to. This way its more structured and I can group equations falling into certain category.

Please find the attached Minimum working example. Any advise is appreciated. File link: [MWE.mw]

 

Thanks

 

 

Dear all

I have a linear system and I would like to generate the matrix from the linear system of equations

MatrixAgenerating.mw

Many thanks for your help

 

Hi all,

I am working on a Maple file to find the right force excerted in a specifik angle (theta). This is the script Maple than has to work out:

 

eq4 := Fh1 = (1/2)*(solFh2*sqrt(2)-40)/sin(theta);
eq5 := Fh1 = (1/2)*(solFh2*sqrt(2)-100)/cos(theta);
sol := solve({eq4, eq5}, {Fh1, theta});

Next it gives me the answers as following:

sol := {Fh1 = 121.6477702, theta = .9606764638}, {Fh1 = -121.6477702, theta = -2.180916190}

Which is correct: I get a force (Fh1 = ± 121.6477...) with 2 angles (theta = .9696... or theta=-2.1809...)

 

If i want to continue working with Fh1 it gives an error saying it has 2 values for it (obviously a positive and a negative value). Is there a way to continue working with the positive values of Fh1 and theta?

 

I was thinking of solving the intersect equation on the positive 'theta'-axis in a form like:

 

sol := solve({eq4, eq5}, {Fh1, theta>0}); as theta is my horizontal axis and a positve theta gives me a positive Fh1 but Maple doesn't work that straightforward. 

 

Thanks a lot!

I am trying to solve a constrained maximization problem. 

The starting function is the one at the top in bold. Whenever I use (1/2) or (1) as an exponent for either of the variables I get unwanted results. For x2 (below) I am getting that result. I should be getting x2= (3m)/(5P2)

However, whenever I input an exponent that does not equal (1/2), I get the results I want.

 

What am I doing wrong?

Guys,

I am not very familiar with Maple and have to solve quite a complex equation.

I have an equation which is complex ,containing I . I split this equation up in Re=0 an Im=0 . I have to get an answer in function of other parameters, in order to plot these... Maybe it s easier if you look at the work sheet


 

restart

with(LinearAlgebra):

Student(NumericalAnalysis)

(module Student () description "a package to assist with the teaching and learning of standard undergraduate mathematics"; local ModuleLoad, localColors, GetColor, SelectColor, UpdateColor, GetCaption, colorNum, colorDefaults, Defaults, PlotOptionsWindow, InitAnimation, EndAnimation, DoPlayPause, IncrSpd, DecrSpd, Colours, CheckPoint, CheckRange, CheckTextField, CleanFloat, CombineRanges, EvaluateFunction, FindHRange, FindHRange3d, FindVRange, FindVRange3d, GetSpecPoints, EvaluateFunctionNumeric, EvaluateFunctionNumeric3d, VRangeCmp, MaximizePointList, MinimizePointList, FindHRange3dCrossSections, FindVRangeSymbolic, SymEvalFunc, SymLimits, FindAllSpecialPoints, FindHRangeRatPoly, GetRealDomain, GetTextField, GetVariable, IsColour, MapletGenericError, MapletNoInputError, MapletTypeError, ProcessCharacter, ProcessVisual, RequiredError, RemovePlotOptions, mapletColor, mapletDarkColor, mapletLightColor, mapletHelpColor, IsMac, ProcessColorNames; export _pexports, SetColors, SetDefault, SetDefaults, Precalculus, MultivariateCalculus, VectorCalculus, LinearAlgebra, Statistics, Calculus1, NumericalAnalysis, Basics; global x, y, z, r, t, p; option package, `Copyright (c) Maplesoft, a division of Waterloo Maple Inc. 2005`; end module)(NumericalAnalysis)

(1)

wn := 50

50

(2)

Np := 2

2

(3)

``

``

w0 := v0*wn

50*v0

(4)

V0 := 230*sqrt(2)

230*2^(1/2)

(5)

Ep0 := 1.5*V0

345.0*2^(1/2)

(6)

NULL

delta := 0

0

(7)

phi[q0] := 0

0

(8)

Iq0 := 0

0

(9)

L := proc (p) options operator, arrow; Ls*(p*tau[r]+1)/(p*tau[r]/sigma+1) end proc

proc (p) options operator, arrow; Ls*(p*tau[r]+1)/(p*tau[r]/sigma+1) end proc

(10)

Rs := 2.43

2.43

(11)

Rr := 2.43

2.43

(12)

Lr := 0.12e-1+.237

.249

(13)

Ls := Lr

.249

(14)

M := sqrt(.92*Ls^2)

.2388324099

(15)

sigma := 1-M^2/(Ls*Lr)

0.799999996e-1

(16)

``

tau[r] := sigma*Lr*wn/Rr

.4098765412

(17)

tau[s] := sigma*Ls*wn/Rs

.4098765412

(18)

alpha := tau[r]/tau[s]

1.000000000

(19)

``

``

``

``

assume(v0, 'real', nu, 'real')

``

345.0*2^(1/2)

(20)

phi[d0] := Ls*Id0-Ep0/w0

.249*Id0-6.900000000*2^(1/2)/v0

(21)

Vd0 := V0*sin(delta)

0

(22)

Vq0 := V0*cos(delta)

230*2^(1/2)

(23)

Id0 := (Rs*Vd0-wn*v0*Ls*(Vq0-Ep0))/(Ls^2*w0^2+Rs^2)

1431.7500*v0*2^(1/2)/(155.002500*v0^2+5.9049)

(24)

Dp := (Rs+p*wn*L(p))^2+v0*wn^2*L(p)^2

(2.43+12.450*p*(.4098765412*p+1)/(5.123456791*p+1))^2+155.002500*v0*(.4098765412*p+1)^2/(5.123456791*p+1)^2

(25)

simplify(Dp)

(26.04023075*p^4+254.1275544*p^3+(644.8103998+26.04023074*v0)*p^2+(121.014+127.0637772*v0)*p+5.9049+155.0025*v0)/(5.123456791*p+1.)^2

(26)

N := -(3/2)*Np*[[L(p)^2*(Id0*phi[d0]+Iq0*phi[q0])-L(p)*(phi[d0]^2-phi[q0]^2)]*(p^2+v0^2)*wn^2+Rs*[p*L(p)^2*(Id0^2+Iq0^2)-p*(phi[d0]^2+phi[q0]^2)]*wn+Rs^2*[L(p)*Iq0^2+L(p)*Id0^2-Id0*phi[d0]-Iq0*phi[q0]]]

[-7500*[88.76993175*(.4098765412*p+1)^2*v0*2^(1/2)*(356.5057500*v0*2^(1/2)/(155.002500*v0^2+5.9049)-6.900000000*2^(1/2)/v0)/((5.123456791*p+1)^2*(155.002500*v0^2+5.9049))-.249*(.4098765412*p+1)*(356.5057500*v0*2^(1/2)/(155.002500*v0^2+5.9049)-6.900000000*2^(1/2)/v0)^2/(5.123456791*p+1)]*(p^2+v0^2)+[-18084126.16*(.4098765412*p+1)*v0^2/((5.123456791*p+1)*(155.002500*v0^2+5.9049)^2)+25363.02172*v0*2^(1/2)*(356.5057500*v0*2^(1/2)/(155.002500*v0^2+5.9049)-6.900000000*2^(1/2)/v0)/(155.002500*v0^2+5.9049)-92653238.97*p*(.4098765412*p+1)^2*v0^2/((5.123456791*p+1)^2*(155.002500*v0^2+5.9049)^2)+364.5000000*p*(356.5057500*v0*2^(1/2)/(155.002500*v0^2+5.9049)-6.900000000*2^(1/2)/v0)^2]]

(27)

char := (p*J*wn^2/Np*p)*Dp+N

1250*p^2*J*((2.43+12.450*p*(.4098765412*p+1)/(5.123456791*p+1))^2+155.002500*v0*(.4098765412*p+1)^2/(5.123456791*p+1)^2)+[-7500*[88.76993175*(.4098765412*p+1)^2*v0*2^(1/2)*(356.5057500*v0*2^(1/2)/(155.002500*v0^2+5.9049)-6.900000000*2^(1/2)/v0)/((5.123456791*p+1)^2*(155.002500*v0^2+5.9049))-.249*(.4098765412*p+1)*(356.5057500*v0*2^(1/2)/(155.002500*v0^2+5.9049)-6.900000000*2^(1/2)/v0)^2/(5.123456791*p+1)]*(p^2+v0^2)+[-18084126.16*(.4098765412*p+1)*v0^2/((5.123456791*p+1)*(155.002500*v0^2+5.9049)^2)+25363.02172*v0*2^(1/2)*(356.5057500*v0*2^(1/2)/(155.002500*v0^2+5.9049)-6.900000000*2^(1/2)/v0)/(155.002500*v0^2+5.9049)-92653238.97*p*(.4098765412*p+1)^2*v0^2/((5.123456791*p+1)^2*(155.002500*v0^2+5.9049)^2)+364.5000000*p*(356.5057500*v0*2^(1/2)/(155.002500*v0^2+5.9049)-6.900000000*2^(1/2)/v0)^2]]

(28)

eval(char)

1250*p^2*J*((2.43+12.450*p*(.4098765412*p+1)/(5.123456791*p+1))^2+155.002500*v0*(.4098765412*p+1)^2/(5.123456791*p+1)^2)+[-7500*[88.76993175*(.4098765412*p+1)^2*v0*2^(1/2)*(356.5057500*v0*2^(1/2)/(155.002500*v0^2+5.9049)-6.900000000*2^(1/2)/v0)/((5.123456791*p+1)^2*(155.002500*v0^2+5.9049))-.249*(.4098765412*p+1)*(356.5057500*v0*2^(1/2)/(155.002500*v0^2+5.9049)-6.900000000*2^(1/2)/v0)^2/(5.123456791*p+1)]*(p^2+v0^2)+[-18084126.16*(.4098765412*p+1)*v0^2/((5.123456791*p+1)*(155.002500*v0^2+5.9049)^2)+25363.02172*v0*2^(1/2)*(356.5057500*v0*2^(1/2)/(155.002500*v0^2+5.9049)-6.900000000*2^(1/2)/v0)/(155.002500*v0^2+5.9049)-92653238.97*p*(.4098765412*p+1)^2*v0^2/((5.123456791*p+1)^2*(155.002500*v0^2+5.9049)^2)+364.5000000*p*(356.5057500*v0*2^(1/2)/(155.002500*v0^2+5.9049)-6.900000000*2^(1/2)/v0)^2]]

(29)

p := I*nu

I*nu

(30)

R := Re(char)

Re(-1250*nu^2*J*((2.43+(12.450*I)*nu*((.4098765412*I)*nu+1)/((5.123456791*I)*nu+1))^2+155.002500*v0*((.4098765412*I)*nu+1)^2/((5.123456791*I)*nu+1)^2)+[-7500*[88.76993175*((.4098765412*I)*nu+1)^2*v0*2^(1/2)*(356.5057500*v0*2^(1/2)/(155.002500*v0^2+5.9049)-6.900000000*2^(1/2)/v0)/(((5.123456791*I)*nu+1)^2*(155.002500*v0^2+5.9049))-.249*((.4098765412*I)*nu+1)*(356.5057500*v0*2^(1/2)/(155.002500*v0^2+5.9049)-6.900000000*2^(1/2)/v0)^2/((5.123456791*I)*nu+1)]*(-nu^2+v0^2)+[-18084126.16*((.4098765412*I)*nu+1)*v0^2/(((5.123456791*I)*nu+1)*(155.002500*v0^2+5.9049)^2)+25363.02172*v0*2^(1/2)*(356.5057500*v0*2^(1/2)/(155.002500*v0^2+5.9049)-6.900000000*2^(1/2)/v0)/(155.002500*v0^2+5.9049)-(92653238.97*I)*nu*((.4098765412*I)*nu+1)^2*v0^2/(((5.123456791*I)*nu+1)^2*(155.002500*v0^2+5.9049)^2)+(364.5000000*I)*nu*(356.5057500*v0*2^(1/2)/(155.002500*v0^2+5.9049)-6.900000000*2^(1/2)/v0)^2]])

(31)

im := Im(char)

Im(-1250*nu^2*J*((2.43+(12.450*I)*nu*((.4098765412*I)*nu+1)/((5.123456791*I)*nu+1))^2+155.002500*v0*((.4098765412*I)*nu+1)^2/((5.123456791*I)*nu+1)^2)+[-7500*[88.76993175*((.4098765412*I)*nu+1)^2*v0*2^(1/2)*(356.5057500*v0*2^(1/2)/(155.002500*v0^2+5.9049)-6.900000000*2^(1/2)/v0)/(((5.123456791*I)*nu+1)^2*(155.002500*v0^2+5.9049))-.249*((.4098765412*I)*nu+1)*(356.5057500*v0*2^(1/2)/(155.002500*v0^2+5.9049)-6.900000000*2^(1/2)/v0)^2/((5.123456791*I)*nu+1)]*(-nu^2+v0^2)+[-18084126.16*((.4098765412*I)*nu+1)*v0^2/(((5.123456791*I)*nu+1)*(155.002500*v0^2+5.9049)^2)+25363.02172*v0*2^(1/2)*(356.5057500*v0*2^(1/2)/(155.002500*v0^2+5.9049)-6.900000000*2^(1/2)/v0)/(155.002500*v0^2+5.9049)-(92653238.97*I)*nu*((.4098765412*I)*nu+1)^2*v0^2/(((5.123456791*I)*nu+1)^2*(155.002500*v0^2+5.9049)^2)+(364.5000000*I)*nu*(356.5057500*v0*2^(1/2)/(155.002500*v0^2+5.9049)-6.900000000*2^(1/2)/v0)^2]])

(32)

``

simplify(im)

Im(((32550.28842*nu^4*J-(158829.7215*I)*nu^3*J-193753.125*nu^2*J)*v0-32550.28844*nu^6*J+(317659.4430*I)*nu^5*J+806012.9998*nu^4*J-(151267.5000*I)*nu^3*J+(-7381.125*J+[196873.5712*[-177.5398635*((.4098765412*I)*nu+1.)^2*(713.0115000*v0^2+40.74381000)/(((5.123456791*I)*nu+1.)^2*(155.002500*v0^2+5.9049)*(155.0025000*v0^2+5.904900000))-.4979999998*((.4098765412*I)*nu+1)*(713.0115*v0^2+40.74381)^2/(((5.123456791*I)*nu+1.)*v0^2*(155.0025*v0^2+5.9049)^2)]*(-nu^2+v0^2)+[474704866.5*((.4098765412*I)*nu+1)*v0^2/(((5.123456791*I)*nu+1.)*(155.002500*v0^2+5.9049)^2)-26.24980949*(-36168252.32*v0^2-2066772.276)/(155.0025*v0^2+5.9049)^2+(2432129872.*I)*nu*((.4098765412*I)*nu+1.)^2*v0^2/(((5.123456791*I)*nu+1.)^2*(155.002500*v0^2+5.9049)^2)-(19136.11111*I)*nu*(713.0115*v0^2+40.74381)^2/(v0^2*(155.0025*v0^2+5.9049)^2)]])*nu^2+(10.24691358*I)*[-7500*[-177.5398635*((.4098765412*I)*nu+1.)^2*(713.0115000*v0^2+40.74381000)/(((5.123456791*I)*nu+1.)^2*(155.002500*v0^2+5.9049)*(155.0025000*v0^2+5.904900000))-.4979999998*((.4098765412*I)*nu+1)*(713.0115*v0^2+40.74381)^2/(((5.123456791*I)*nu+1.)*v0^2*(155.0025*v0^2+5.9049)^2)]*(-nu^2+v0^2)+[-18084126.16*((.4098765412*I)*nu+1)*v0^2/(((5.123456791*I)*nu+1.)*(155.002500*v0^2+5.9049)^2)+(-36168252.32*v0^2-2066772.276)/(155.0025*v0^2+5.9049)^2-(92653238.97*I)*nu*((.4098765412*I)*nu+1.)^2*v0^2/(((5.123456791*I)*nu+1.)^2*(155.002500*v0^2+5.9049)^2)+(728.9999996*I)*nu*(713.0115*v0^2+40.74381)^2/(v0^2*(155.0025*v0^2+5.9049)^2)]]*nu+[-7500*[-177.5398635*((.4098765412*I)*nu+1.)^2*(713.0115000*v0^2+40.74381000)/(((5.123456791*I)*nu+1.)^2*(155.002500*v0^2+5.9049)*(155.0025000*v0^2+5.904900000))-.4979999998*((.4098765412*I)*nu+1)*(713.0115*v0^2+40.74381)^2/(((5.123456791*I)*nu+1.)*v0^2*(155.0025*v0^2+5.9049)^2)]*(-nu^2+v0^2)+[-18084126.16*((.4098765412*I)*nu+1)*v0^2/(((5.123456791*I)*nu+1.)*(155.002500*v0^2+5.9049)^2)+(-36168252.32*v0^2-2066772.276)/(155.0025*v0^2+5.9049)^2-(92653238.97*I)*nu*((.4098765412*I)*nu+1.)^2*v0^2/(((5.123456791*I)*nu+1.)^2*(155.002500*v0^2+5.9049)^2)+(728.9999996*I)*nu*(713.0115*v0^2+40.74381)^2/(v0^2*(155.0025*v0^2+5.9049)^2)]])/((5.123456791*I)*nu+1.)^2)

(33)

solve(im = 0)

Warning, solve may be ignoring assumptions on the input variables.

 

Error, (in Engine:-Dispatch) badly formed input to solve: not fully algebraic

 

``

``

Error, (in fsolve) b is in the equation, and is not solved for

 

-I*b/l

(34)

``


 

Download operationele_inductanties.mwoperationele_inductanties.mw
 

Hi, I am new in Maple. If I have an electric network as in the figure, I want to get the Transfer Function V2(s)/Vi(s) from this equation system:

Vi=R1.I1+V1

Vi=R2.I2+V2

I1-I2=C1.dV1/dt

I2=C2.dV2/dt

Which are the commands that I may write to get this?? Before hand, Thanks by your answers!

I have four matrix equations

P1, P2, P3 are known 4x4 matrix.

A1 A2 A3 A4 are known 1x4 matrix.

x1 x2 x3 x4 are 1x1 known matrix.

U is 4x4 unknown matrix.

These equations are 

(A1T*U*P1*A1) +( (P2*A1)T*U*P1*A1) + ( (P3*A1)T*U*A1) + ( ( P3*A 2)T*U*P1*A1) + x1 =0;

(A2T*U*P1*A2) +( (P2*A2)T*U*P1*A2) + ( (P3*A2)T*U*A2) + ( ( P3*A2 )T*U*P1*A2) + x2 =0;

(A3T*U*P1*A3) +( (P2*A3)T*U*P1*A3) + ( (P3*A3)T*U*A3) + ( ( P3*A3 )T*U*P1*A3) + x3 =0;

(A4T*U*P1*A4) +( (P2*A4)T*U*P1*A4) + ( (P3*A4)T*U*A4) + ( ( P3*A4 )T*U*P1*A4) + x4 =0;

How can i find 4x4 matrix U by using these above four equations??

Thank you

E_T := (2/mu-2/r)*exp(-r/mu)*Pi^2;

How do I extract the numbers out of the equation, so it becomes

2*Pi^2*(1/mu-1/r)*exp(-r/mu) instead? 


 

restart; with(plots); beta := 0.1e-1; Bi := 1; Pr := 3.0; L0 := 1; w = 0.2e-1

Eq1 := diff(f(eta), eta, eta, eta)+f(eta)*(diff(f(eta), eta, eta))-(diff(f(eta), eta))^2+beta*H(eta)*(F(eta)-(diff(f(eta), eta))) = 0

diff(diff(diff(f(eta), eta), eta), eta)+f(eta)*(diff(diff(f(eta), eta), eta))-(diff(f(eta), eta))^2+0.1e-1*H(eta)*(F(eta)-(diff(f(eta), eta))) = 0

(1)

Eq2 := G(eta)*(diff(F(eta), eta))+F(eta)^2+beta*(F(eta)-(diff(f(eta), eta))) = 0

G(eta)*(diff(F(eta), eta))+F(eta)^2+0.1e-1*F(eta)-0.1e-1*(diff(f(eta), eta)) = 0

(2)

Eq3 := G(eta)*(diff(G(eta), eta))+beta*(f(eta)+G(eta)) = 0

G(eta)*(diff(G(eta), eta))+0.1e-1*f(eta)+0.1e-1*G(eta) = 0

(3)

Eq4 := H(eta)*F(eta)+H(eta)*(diff(G(eta), eta))+G(eta)*(diff(H(eta), eta)) = 0

H(eta)*F(eta)+H(eta)*(diff(G(eta), eta))+G(eta)*(diff(H(eta), eta)) = 0

(4)

Eq5 := (diff(theta(eta), eta, eta))/Pr+f(eta)*(diff(theta(eta), eta))+(2*beta*H(eta)*(1/3))*(theta[p](eta)-theta(eta)) = 0

.3333333333*(diff(diff(theta(eta), eta), eta))+f(eta)*(diff(theta(eta), eta))+0.6666666667e-2*H(eta)*(theta[p](eta)-theta(eta)) = 0

(5)

Eq6 := G(eta)*(diff(theta[p](eta), eta))+L0*beta*(theta[p](eta)-theta(eta)) = 0

G(eta)*(diff(theta[p](eta), eta))+0.1e-1*theta[p](eta)-0.1e-1*theta(eta) = 0

(6)

bcs1 := f(0) = 0, (D(f))(0) = 1, (D(theta))(0) = -Bi*(1-theta(0)), (D(f))(5) = 0, F(5) = 0, G(5) = -f(5), H(5) = w, theta(5) = 0, theta[p](5) = 0

f(0) = 0, (D(f))(0) = 1, (D(theta))(0) = -1+theta(0), (D(f))(5) = 0, F(5) = 0, G(5) = -f(5), H(5) = w, theta(5) = 0, theta[p](5) = 0

(7)

p := dsolve({Eq1, Eq2, Eq3, Eq4, Eq5, Eq6, bcs1}, numeric)

Error, (in dsolve/numeric/process_input) system must be entered as a set/list of expressions/equations

 

odeplot(p, [eta, f(eta)], 0 .. 10);

odeplot(p, [eta, f(eta)], 0 .. 10)

(8)

``

 

 


 

Download from_net.mw

Hi guys,

 

I am trying to solve a Fredholm equation of the second kind using Maple. An analytical expression cannot be in principle found. I was wondering whether Maple does numerical evaluation of such integral equations. Please see the equation in attach. Any help is highly appreciated.

Thanks

F

 

Question.mw

The material below was presented in the "Semantic Representation of Mathematical Knowledge Workshop", February 3-5, 2016 at the Fields Institute, University of Toronto. It shows the approach I used for “digitizing mathematical knowledge" regarding Differential Equations, Special Functions and Solutions to Einstein's equations. While for these areas using databases of information helps (for example textbooks frequently contain these sort of databases), these are areas that, at the same time, are very suitable for using algorithmic mathematical approaches, that result in much richer mathematics than what can be hard-coded into a database. The material also focuses on an interesting cherry-picked collection of Maple functionality, that I think is beautiful, not well know, and seldom focused inter-related as here.

 

 

Digitizing of special functions,

differential equations,

and solutions to Einstein’s equations

within a computer algebra system

 

Edgardo S. Cheb-Terrab

Physics, Differential Equations and Mathematical Functions, Maplesoft

Editor, Computer Physics Communications

 

 

Digitizing (old paradigm)

 

• 

Big amounts of knowledge available to everybody in local machines or through the internet

• 

Take advantage of basic computer functionality, like searching and editing

 

 

Digitizing (new paradigm)

• 

By digitizing mathematical knowledge inside appropriate computational contexts that understand about the topics, one can use the digitized knowledge to automatically generate more and higher level knowledge

 

 

Challenges


1) how to identify, test and organize the key blocks of information,

 

2) how to access it: the interface,

 

3) how to mathematically process it to automatically obtain more information on demand

 

 

 

 

                                           Three examples


Mathematical Functions

 

"Mathematical functions, are defined by algebraic expressions. So consider algebraic expressions in general ..."

The FunctionAdvisor (basic)

 

"Supporting information on definitions, identities, possible simplifications, integral forms, different types of series expansions, and mathematical properties in general"

Examples

   

General description

   

References

   

 

Differential equation representation for generic nonlinear algebraic expressions - their use

 

"Compute differential polynomial forms for arbitrary systems of non-polynomial equations ..."

The Differential Equations representing arbitrary algebraic expresssions

   

Deriving knowledge: ODE solving methods

   

Extending the mathematical language to include the inverse functions

   

Solving non-polynomial algebraic equations by solving polynomial differential equations

   

References

   

 

Branch Cuts of algebraic expressions

 

"Algebraically compute, and visualize, the branch cuts of arbitrary mathematical expressions"

Examples

   

References

   

 

Algebraic expresssions in terms of specified functions

 

"A conversion network for arbitrary mathematical expressions, to rewrite them in terms of different functions in flexible ways"

Examples

   

General description

   

References

   

 

Symbolic differentiation of algebraic expressions

 

"Perform symbolic differentiation by combining different algebraic techniques, including functions of symbolic sequences and Faà di Bruno's formula"

Examples

   

References

   

 

Ordinary Differential Equations

 

"Beyond the concept of a database, classify an arbitrary ODE and suggest solution methods for it"

General description

   

Examples

   

References

   

 

Exact Solutions to Einstein's equations

 

 

Lambda*g[mu, nu]+G[mu, nu] = 8*Pi*T[mu, nu]

 

"The authors of "Exact solutions toEinstein's equations" reviewed more than 4,000 papers containing solutions to Einstein’s equations in the general relativity literature, organized the whole material into chapters according to the physical properties of these solutions. These solutions are key in the area of general relativity, are now all digitized and become alive in a worksheet"


The ability to search the database according to the physical properties of the solutions, their classification, or just by parts of keywords (old paradigm) changes the game.

More important, within a computer algebra system this knowledge becomes alive (new paradigm).

• 

The solutions are turned active by a simple call to one commend, called the g_  spacetime metric.

• 

Everything else gets automatically derived and set on the fly ( Christoffel symbols  , Ricci  and Riemann  tensors orthonormal and null tetrads , etc.)

• 

Almost all of the mathematical operations one can perform on these solutions are implemented as commands in the Physics  and DifferentialGeometry  packages.

• 

All the mathematics within the Maple library are instantly ready to work with these solutions and derived mathematical objects.

 

Finally, in the Maple PDEtools package , we have all the mathematical tools to tackle the equivalence problem around these solutions.

Examples

   

References

   

 

Download:  Digitizing_Mathematical_Information.mw,    Digitizing_Mathematical_Information.pdf

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

hi....how i can extract Coefficients  (i.e. {f1[2],f2[2],f2[3],f3[2],.....f3[6]}) from every algebric equations and create matrix A ,in form AX=0, (X are f1[2],f2[2],f2[3],f3[2],.....f3[6] ) then the determinant of the matrix of coefficients (A) set to zero for obtaining unknown parameter omega.?

Note that  if m=3 then 6 equations is appeare and if m=4 then 9 equations is appeare.thus i need a procedure that works for every arbitary value of ''m''.

in attached file below m=4 thus we have 9 equations, i.e. 3 for eq1[k_] and 3 for eq2[k_] and so on...

also we should use boundary conditions for some amount of fi[j] (i=1,2,3 and j=2,3,...,7)

be extacting above Coefficients for example from first equation ,

''**:= (1/128)*f1[2]*omega^2-(1/4)*f2[2]-(1/2)*f2[3]+(1/4)*f2[4]+(1/4)*f3[2]-(1/2)*f3[3]+(1/4)*f3[4]+140*f1[2]-80*f1[3]+20*f1[4]'''

must compute

coeff(**, f1[2]); coeff(**, f2[2]) and so on...

 

 

 

 

 

fdm-maple.mw

 

 ############################Define some parameters

 

 
restart; Digits := 15; A1 := 10; A2 := 10; A3 := 10; A4 := 1; A5 := 1; A6 := 1; A7 := 1; A8 := 1; A9 := 1; A10 := 1; A11 := 1; B1 := 10; B2 := 10; B3 := 10; B4 := 1; B5 := 1; B6 := 1; B7 := 1; B8 := 1; B9 := 1; B10 := 1; B11 := 1; C1 := 10; C2 := 10; C3 := 10; C4 := 1; C5 := 1; C6 := 1; C7 := 1; C8 := 1; C9 := 1; C10 := 1; C11 := 1; C12 := 1; C13 := 1; C14 := 1; C15 := 1; C16 := 1; A12 := 1; B12 := 1; C18 := 1; C17 := 1; C19 := 1; n := 1; U := proc (x, theta) options operator, arrow; f1(x)*cos(n*theta) end proc; V := proc (x, theta) options operator, arrow; f2(x)*sin(n*theta) end proc; W := proc (x, theta) options operator, arrow; f3(x)*cos(n*theta) end proc; n := 1; m := 4; len := 1; h := len/m; nn := m+1
 ############################Define some equation

eq1[k_] := -2*f1[k]*(-A11*n^4+A10*n^2+A12*omega^2)*h^4+(A6*(f2[k-1]-f2[k+1])*n^3+A9*(f3[k-1]-f3[k+1])*n^2-A5*(f2[k-1]-f2[k+1])*n-A8*(f3[k-1]-f3[k+1]))*h^3+(4*(f1[k]-(1/2)*f1[k-1]-(1/2)*f1[k+1]))*(A3*n^2-A2)*h^2+(-A4*(f2[k-2]-2*f2[k-1]+2*f2[k+1]-f2[k+2])*n-A7*(f3[k-2]-2*f3[k-1]+2*f3[k+1]-f3[k+2]))*h+12*A1*(f1[k]+(1/6)*f1[k-2]-(2/3)*f1[k-1]-(2/3)*f1[k+1]+(1/6)*f1[k+2]):
  ``

 

 

 

 

                                     ######################################  APPLY BOUNDARY CONDITIONS

f1[nn+1] := f1[m]:
 

for k from 2 to m do eq1[k_]; eq2[k_]; eq3[k_] end do

-(1/64)*f2[4]+(1/128)*f2[3]+(1/64)*(f3[4]-(1/2)*f3[3])*(omega^2-1)-(1/64)*f1[2]+(1/32)*f1[3]+(1/64)*f1[4]-280*f3[4]-120*f3[2]+300*f3[3]+20*f3[7]

(1)

``



Download fdm-maple.mw

 

hi...how i can convert 3 couple equations to 1 equation with Placement each other?

thanks...

 

3-1.mw

pd1 := A1*(diff(U(x, theta), x, x, x, x))+A2*(diff(U(x, theta), x, x))+A3*(diff(U(x, theta), x, x, theta, theta))+A4*(diff(V(x, theta), x, x, x, theta))+A5*(diff(V(x, theta), x, theta))+A6*(diff(V(x, theta), x, theta, theta, theta))+A7*(diff(W(x, theta), x, x, x))+A8*(diff(W(x, theta), x))+A9*(diff(W(x, theta), x, theta, theta))+A10*(diff(U(x, theta), theta, theta))+A11*(diff(U(x, theta), theta, theta, theta, theta))-A12*omega^2*U(x, theta)

A1*(diff(diff(diff(diff(U(x, theta), x), x), x), x))+A2*(diff(diff(U(x, theta), x), x))+A3*(diff(diff(diff(diff(U(x, theta), theta), theta), x), x))+A4*(diff(diff(diff(diff(V(x, theta), theta), x), x), x))+A5*(diff(diff(V(x, theta), theta), x))+A6*(diff(diff(diff(diff(V(x, theta), theta), theta), theta), x))+A7*(diff(diff(diff(W(x, theta), x), x), x))+A8*(diff(W(x, theta), x))+A9*(diff(diff(diff(W(x, theta), theta), theta), x))+A10*(diff(diff(U(x, theta), theta), theta))+A11*(diff(diff(diff(diff(U(x, theta), theta), theta), theta), theta))-A12*omega^2*U(x, theta)

(1)

pd2 := B1*(diff(V(x, theta), x, x, x, x))+B2*(diff(V(x, theta), x, x))+B3*(diff(V(x, theta), theta, theta, theta, theta))+B4*(diff(V(x, theta), theta, theta))+B5*(diff(V(x, theta), x, x, theta, theta))+B6*(diff(U(x, theta), x, x, x, theta))+B7*(diff(U(x, theta), x, theta, theta, theta))+B8*(diff(U(x, theta), x, theta))+B9*(diff(W(x, theta), x, x, theta))+B10*(diff(W(x, theta), theta, theta, theta))+B11*(diff(W(x, theta), theta))-B12*omega^2*V(x, theta)

B1*(diff(diff(diff(diff(V(x, theta), x), x), x), x))+B2*(diff(diff(V(x, theta), x), x))+B3*(diff(diff(diff(diff(V(x, theta), theta), theta), theta), theta))+B4*(diff(diff(V(x, theta), theta), theta))+B5*(diff(diff(diff(diff(V(x, theta), theta), theta), x), x))+B6*(diff(diff(diff(diff(U(x, theta), theta), x), x), x))+B7*(diff(diff(diff(diff(U(x, theta), theta), theta), theta), x))+B8*(diff(diff(U(x, theta), theta), x))+B9*(diff(diff(diff(W(x, theta), theta), x), x))+B10*(diff(diff(diff(W(x, theta), theta), theta), theta))+B11*(diff(W(x, theta), theta))-B12*omega^2*V(x, theta)

(2)

pd3 := C1*(diff(W(x, theta), x, x, x, x, x, x))+C2*(diff(W(x, theta), x, x, x, x))+C3*(diff(W(x, theta), x, x, x, x, theta, theta))+C4*(diff(W(x, theta), x, x))+C5*(diff(W(x, theta), x, x, theta, theta))+C6*(diff(W(x, theta), x, x, theta, theta, theta, theta))+C7*(diff(U(x, theta), x, x, x))+C8*(diff(U(x, theta), x))+C9*(diff(U(x, theta), x, theta, theta))+C10*(diff(V(x, theta), x, x, theta))+C11*(diff(V(x, theta), theta))+C12*(diff(V(x, theta), theta, theta, theta))+C13*W(x, theta)+C14*(diff(W(x, theta), theta, theta))+C15*(diff(W(x, theta), theta, theta, theta, theta))+C16*(diff(W(x, theta), theta, theta, theta, theta, theta, theta))-C19*omega^2*W(x, theta)-C18*omega^2*(diff(W(x, theta), theta, theta))-C17*omega^2*(diff(W(x, theta), x, x))

C1*(diff(diff(diff(diff(diff(diff(W(x, theta), x), x), x), x), x), x))+C2*(diff(diff(diff(diff(W(x, theta), x), x), x), x))+C3*(diff(diff(diff(diff(diff(diff(W(x, theta), theta), theta), x), x), x), x))+C4*(diff(diff(W(x, theta), x), x))+C5*(diff(diff(diff(diff(W(x, theta), theta), theta), x), x))+C6*(diff(diff(diff(diff(diff(diff(W(x, theta), theta), theta), theta), theta), x), x))+C7*(diff(diff(diff(U(x, theta), x), x), x))+C8*(diff(U(x, theta), x))+C9*(diff(diff(diff(U(x, theta), theta), theta), x))+C10*(diff(diff(diff(V(x, theta), theta), x), x))+C11*(diff(V(x, theta), theta))+C12*(diff(diff(diff(V(x, theta), theta), theta), theta))+C13*W(x, theta)+C14*(diff(diff(W(x, theta), theta), theta))+C15*(diff(diff(diff(diff(W(x, theta), theta), theta), theta), theta))+C16*(diff(diff(diff(diff(diff(diff(W(x, theta), theta), theta), theta), theta), theta), theta))-C19*omega^2*W(x, theta)-C18*omega^2*(diff(diff(W(x, theta), theta), theta))-C17*omega^2*(diff(diff(W(x, theta), x), x))

(3)

``


Download 3-1.mw

Dear All,

I have a problem solving the attached nonlinear system of equations using shooting method.
I will be grateful if you could help me finding the solutions out.

 

restart; Shootlib := "C:/Shoot9"; libname := Shootlib, libname; with(Shoot);
with(plots);
N1 := 1.0; N2 := 2.0; N3 := .5; Bt := 6; Re_m := N1*Bt; gamma1 := 1;
FNS := {f(eta), fp(eta), fpp(eta), g(eta), gp(eta), m(eta), mp(eta), n(eta), np(eta), fppp(eta)};
ODE := {diff(f(eta), eta) = fp(eta), diff(fp(eta), eta) = fpp(eta), diff(fpp(eta), eta) = fppp(eta), diff(g(eta), eta) = gp(eta), diff(gp(eta), eta) = N1*(2.*g(eta)+(eta-2.*f(eta)).gp(eta)+2.*g(eta)*fp(eta)+2.*N2.N3.(m(eta).np(eta)-n(eta).mp(eta))), diff(m(eta), eta) = mp(eta), diff(mp(eta), eta) = Re_m.(m(eta)+(eta-2.*f(eta)).mp(eta)+2.*m(eta)*fp(eta)), diff(n(eta), eta) = np(eta), diff(np(eta), eta) = Re_m.(2.*n(eta)+(eta-2.*f(eta)).np(eta)+2.*N2/N3.m(eta).gp(eta)), diff(fppp(eta), eta) = N1*(3.*fpp(eta)+(eta-2.*f(eta)).fppp(eta)-2.*N2.N2.m(eta).(diff(mp(eta), eta)))};
blt := 1.0; IC := {f(0) = 0, fp(0) = 0, fpp(0) = alpha1, g(0) = 1, gp(0) = beta1, m(0) = 0, mp(0) = beta2, n(0) = 0, np(0) = beta3, fppp(0) = alpha2};
BC := {f(blt) = .5, fp(blt) = 0, g(blt) = 0, m(blt) = 1, n(blt) = 1};
infolevel[shoot] := 1;
S := shoot(ODE, IC, BC, FNS, [alpha1 = 1.425, alpha2 = .425, beta1 = -1.31, beta2 = 1.00, beta3 = 1.29]);
Error, (in isolate) cannot isolate for a function when it appears with different arguments
p := odeplot(S, [eta, fp(eta)], 0 .. 15);
Error, (in plots/odeplot) input is not a valid dsolve/numeric solution
display(p);
Error, (in plots:-display) expecting plot structure but received: p
p2 := odeplot(S, [eta, theta(eta)], 0 .. 10);
Error, (in plots/odeplot) input is not a valid dsolve/numeric solution
display(p2);
Error, (in plots:-display) expecting plot structure but received: p2

 

 

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