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mskalsi

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@nm The assume command does not wor...

mskalsi 290
March 04 2022
0 0

@nm 

The assume command does not work in the desired way when used in partial derivatives, please see simplification below:

> 

with(PDEtools):
> 

DepVars := [u(x, y, t), U(xi, eta)]; 1; alias(u = u(x, y, t))

[u(x, y, t), U(xi, eta)]

  

u

 (1)
> 

xi[1] := 1/2*(x^2+y^2); 1; xi[2] := t; 1; u := (h(t)+(x^2+y^2)*(1/2))*arccos(x/sqrt(x^2+y^2))/t+U(xi[1], xi[2])

(1/2)*x^2+(1/2)*y^2

  

t

  

(h(t)+(1/2)*x^2+(1/2)*y^2)*arccos(x/(x^2+y^2)^(1/2))/t+U((1/2)*x^2+(1/2)*y^2, t)

 (2)
> 

(diff(u, x))*(diff(u, y))

(x*arccos(x/(x^2+y^2)^(1/2))/t-(h(t)+(1/2)*x^2+(1/2)*y^2)*(1/(x^2+y^2)^(1/2)-x^2/(x^2+y^2)^(3/2))/((1-x^2/(x^2+y^2))^(1/2)*t)+(D[1](U))((1/2)*x^2+(1/2)*y^2, t)*x)*(y*arccos(x/(x^2+y^2)^(1/2))/t+(h(t)+(1/2)*x^2+(1/2)*y^2)*x*y/((x^2+y^2)^(3/2)*(1-x^2/(x^2+y^2))^(1/2)*t)+(D[1](U))((1/2)*x^2+(1/2)*y^2, t)*y)

 (3)
> 

`assuming`([simplify(subs(1/2*(x^2+y^2) = xi, t = eta, (x*arccos(x/(x^2+y^2)^(1/2))/t-(h(t)+(1/2)*x^2+(1/2)*y^2)*(1/(x^2+y^2)^(1/2)-x^2/(x^2+y^2)^(3/2))/((1-x^2/(x^2+y^2))^(1/2)*t)+(D[1](U))((1/2)*x^2+(1/2)*y^2, t)*x)*(y*arccos(x/(x^2+y^2)^(1/2))/t+(h(t)+(1/2)*x^2+(1/2)*y^2)*x*y/((x^2+y^2)^(3/2)*(1-x^2/(x^2+y^2))^(1/2)*t)+(D[1](U))((1/2)*x^2+(1/2)*y^2, t)*y)))], [x::real, y::real, y > 0])

(1/4)*(2*(D[1](U))(table( [( 1 ) = (1/2)*x^2+(1/2)*y^2, ( 2 ) = eta ] ), eta)*eta*x^3+2*(D[1](U))(table( [( 1 ) = (1/2)*x^2+(1/2)*y^2, ( 2 ) = eta ] ), eta)*eta*x*y^2+2*arccos(x/(x^2+y^2)^(1/2))*x^3+2*arccos(x/(x^2+y^2)^(1/2))*x*y^2-x^2*y-y^3-2*h(eta)*y)*(2*(D[1](U))(table( [( 1 ) = (1/2)*x^2+(1/2)*y^2, ( 2 ) = eta ] ), eta)*eta*x^2*y+2*(D[1](U))(table( [( 1 ) = (1/2)*x^2+(1/2)*y^2, ( 2 ) = eta ] ), eta)*eta*y^3+2*arccos(x/(x^2+y^2)^(1/2))*x^2*y+2*arccos(x/(x^2+y^2)^(1/2))*y^3+x^3+x*y^2+2*h(eta)*x)/((x^2+y^2)^2*eta^2)

 (4)
> 

NULL

Download Ques_Mapleprime.mw

I am sorry for replying late...

mskalsi 290
December 03 2017
0 0

@Tatowills 

Thanks for your comment, by taking a11=0 and a21=0 worked for me. Sorry, as I saw your comment so late.

Singular case...

mskalsi 290
May 27 2017
0 0

@Torre 

I to deal with the singular case? I have tried but an error occurred.


 

> 

with(DifferentialGeometry)with(JetCalculus)NULL
> 

DGsetup([x, y, t], [u], E, 1)
E > 

M := u[1, 2, 3, 3]/u[]-2*u[1, 2, 3]*u[3]/u[]^2+2*u[1, 2]*u[3]^2/u[]^3-u[1, 2]*u[3, 3]/u[]^2-u[1, 1, 1, 2]/u[]+2*u[1, 1, 2]*u[1]/u[]^2-2*u[1, 2]*u[1]^2/u[]^3+u[1, 2]*u[1, 1]/u[]^2+4*u[3]*u[1]+4*u[]*u[1, 3]

u[1, 2, 3, 3]/u[]-2*u[1, 2, 3]*u[3]/u[]^2+2*u[1, 2]*u[3]^2/u[]^3-u[1, 2]*u[3, 3]/u[]^2-u[1, 1, 1, 2]/u[]+2*u[1, 1, 2]*u[1]/u[]^2-2*u[1, 2]*u[1]^2/u[]^3+u[1, 2]*u[1, 1]/u[]^2+4*u[3]*u[1]+4*u[]*u[1, 3]

 (1)
E > 

EulerLagrange(M)

[((24*u[1]*u[2]/u[]^5-6*u[1, 2]/u[]^4)*u[3]-6*u[2]*u[1, 3]/u[]^4-6*u[1]*u[2, 3]/u[]^4+2*u[1, 2, 3]/u[]^3)*u[3]+(-6*u[3]*u[2]/u[]^4+2*u[2, 3]/u[]^3)*u[1, 3]+(-6*u[3]*u[1]/u[]^4+2*u[1, 3]/u[]^3)*u[2, 3]+(-6*u[1]*u[2]/u[]^4+2*u[1, 2]/u[]^3)*u[3, 3]+((-24*u[1]^2/u[]^5+6*u[1, 1]/u[]^4)*u[1]+12*u[1]*u[1, 1]/u[]^4-2*u[1, 1, 1]/u[]^3)*u[2]+(18*u[1]^2/u[]^4-6*u[1, 1]/u[]^3)*u[1, 2]-((48*u[3]*u[1]/u[]^5-12*u[1, 3]/u[]^4)*u[2]-12*u[1, 2]*u[3]/u[]^4-12*u[1]*u[2, 3]/u[]^4+4*u[1, 2, 3]/u[]^3)*u[3]-(-12*u[3]*u[2]/u[]^4+4*u[2, 3]/u[]^3)*u[1, 3]-(-12*u[1]*u[2]/u[]^4+4*u[1, 2]/u[]^3)*u[3, 3]-((-48*u[1]^2/u[]^5+12*u[1, 1]/u[]^4)*u[1]+24*u[1]*u[1, 1]/u[]^4-4*u[1, 1, 1]/u[]^3)*u[2]-(36*u[1]^2/u[]^4-12*u[1, 1]/u[]^3)*u[1, 2]+(-6*u[1, 2]*u[3]/u[]^4+2*u[1, 2, 3]/u[]^3)*u[3]+((24*u[3]^2/u[]^5-6*u[3, 3]/u[]^4-24*u[1]^2/u[]^5+6*u[1, 1]/u[]^4)*u[1]+12*u[1]*u[1, 1]/u[]^4-12*u[3]*u[1, 3]/u[]^4-2*u[1, 1, 1]/u[]^3+2*u[1, 3, 3]/u[]^3)*u[2]+(18*u[1]^2/u[]^4-6*u[3]^2/u[]^4+2*u[3, 3]/u[]^3-6*u[1, 1]/u[]^3)*u[1, 2]-2*u[2]*u[1, 3, 3]/u[]^3-2*u[1, 1, 2]*u[1]/u[]^3+4*u[1, 2, 3]*u[3]/u[]^3-6*u[1, 2]*u[3]^2/u[]^4+6*u[1, 2]*u[1]^2/u[]^4+(6*u[1, 2]*u[1]/u[]^4-2*u[1, 1, 2]/u[]^3)*u[1]-(4*u[1, 2, 3]/u[]^3-12*u[1, 2]*u[3]/u[]^4)*u[3]-(-4*u[1, 1, 2]/u[]^3+12*u[1, 2]*u[1]/u[]^4)*u[1]]

 (2)
E > 

simplify(op([((24*u[1]*u[2]/u[]^5-6*u[1, 2]/u[]^4)*u[3]-6*u[2]*u[1, 3]/u[]^4-6*u[1]*u[2, 3]/u[]^4+2*u[1, 2, 3]/u[]^3)*u[3]+(-6*u[3]*u[2]/u[]^4+2*u[2, 3]/u[]^3)*u[1, 3]+(-6*u[3]*u[1]/u[]^4+2*u[1, 3]/u[]^3)*u[2, 3]+(-6*u[1]*u[2]/u[]^4+2*u[1, 2]/u[]^3)*u[3, 3]+((-24*u[1]^2/u[]^5+6*u[1, 1]/u[]^4)*u[1]+12*u[1]*u[1, 1]/u[]^4-2*u[1, 1, 1]/u[]^3)*u[2]+(18*u[1]^2/u[]^4-6*u[1, 1]/u[]^3)*u[1, 2]-((48*u[3]*u[1]/u[]^5-12*u[1, 3]/u[]^4)*u[2]-12*u[1, 2]*u[3]/u[]^4-12*u[1]*u[2, 3]/u[]^4+4*u[1, 2, 3]/u[]^3)*u[3]-(-12*u[3]*u[2]/u[]^4+4*u[2, 3]/u[]^3)*u[1, 3]-(-12*u[1]*u[2]/u[]^4+4*u[1, 2]/u[]^3)*u[3, 3]-((-48*u[1]^2/u[]^5+12*u[1, 1]/u[]^4)*u[1]+24*u[1]*u[1, 1]/u[]^4-4*u[1, 1, 1]/u[]^3)*u[2]-(36*u[1]^2/u[]^4-12*u[1, 1]/u[]^3)*u[1, 2]+(-6*u[1, 2]*u[3]/u[]^4+2*u[1, 2, 3]/u[]^3)*u[3]+((24*u[3]^2/u[]^5-6*u[3, 3]/u[]^4-24*u[1]^2/u[]^5+6*u[1, 1]/u[]^4)*u[1]+12*u[1]*u[1, 1]/u[]^4-12*u[3]*u[1, 3]/u[]^4-2*u[1, 1, 1]/u[]^3+2*u[1, 3, 3]/u[]^3)*u[2]+(18*u[1]^2/u[]^4-6*u[3]^2/u[]^4+2*u[3, 3]/u[]^3-6*u[1, 1]/u[]^3)*u[1, 2]-2*u[2]*u[1, 3, 3]/u[]^3-2*u[1, 1, 2]*u[1]/u[]^3+4*u[1, 2, 3]*u[3]/u[]^3-6*u[1, 2]*u[3]^2/u[]^4+6*u[1, 2]*u[1]^2/u[]^4+(6*u[1, 2]*u[1]/u[]^4-2*u[1, 1, 2]/u[]^3)*u[1]-(4*u[1, 2, 3]/u[]^3-12*u[1, 2]*u[3]/u[]^4)*u[3]-(-4*u[1, 1, 2]/u[]^3+12*u[1, 2]*u[1]/u[]^4)*u[1]]))

0

 (3)
E > 

omega5 := evalDG(M*`&w`(`&w`(Dx, Dy), Dt))

_DG([["biform", E, [3, 0]], [[[1, 2, 3], (4*u[]^4*u[1, 3]+4*u[]^3*u[1]*u[3]-u[]^2*u[1, 1, 1, 2]+u[]^2*u[1, 2, 3, 3]+2*u[]*u[1]*u[1, 1, 2]-2*u[]*u[3]*u[1, 2, 3]+u[]*u[1, 1]*u[1, 2]-u[]*u[1, 2]*u[3, 3]-2*u[1]^2*u[1, 2]+2*u[3]^2*u[1, 2])/u[]^3]]])

 (4)

Since u[]is in denumerator need to change Strategy

E > 

opt := intmethod = "ExteriorDerivativeHomotopy", path = "zigzag", variableorder = [u[], u[1], u[3], u[1, 1], u[1, 2], u[1, 3], u[3, 3], u[1, 1, 2], u[1, 2, 3], u[1, 1, 1, 2], u[1, 2, 3, 3]], initialpoint = [u[] = 1]

intmethod = "ExteriorDerivativeHomotopy", path = "zigzag", variableorder = [u[], u[1], u[3], u[1, 1], u[1, 2], u[1, 3], u[3, 3], u[1, 1, 2], u[1, 2, 3], u[1, 1, 1, 2], u[1, 2, 3, 3]], initialpoint = [u[] = 1]

 (5)
E > 

HorizontalHomotopy(omega5, opt)

Error, (in DifferentialGeometry:-JetCalculus:-HorizontalHomotopy) the list of variables specified by the keyword "variableorder"  must agree (apart from order) with the  jet variables [x, y, t, u[], u[1], u[2], u[3], u[1, 1], u[1, 2], u[1, 3], u[2, 2], u[2, 3], u[3, 3], u[1, 1, 1], u[1, 1, 2], u[1, 1, 3], u[1, 2, 2], u[1, 2, 3], u[1, 3, 3], u[2, 2, 2], u[2, 2, 3], u[2, 3, 3], u[3, 3, 3], u[1, 1, 1, 1], u[1, 1, 1, 2], u[1, 1, 1, 3], u[1, 1, 2, 2], u[1, 1, 2, 3], u[1, 1, 3, 3], u[1, 2, 2, 2], u[1, 2, 2, 3], u[1, 2, 3, 3], u[1, 3, 3, 3], u[2, 2, 2, 2], u[2, 2, 2, 3], u[2, 2, 3, 3], u[2, 3, 3, 3], u[3, 3, 3, 3]] for the  current frame E. Receive...

  
E > 

NULL


 

Download [1267]_Horizontal_Homotopy.mw

Negative sign and three independent vari...

mskalsi 290
May 27 2017
0 0

@Torre 

Please see my queries below:

  1. Why there is extra negative sign in one flux?
  2. How to deal with case of three independent variables.


 

> 

with(DifferentialGeometry)with(JetCalculus)NULL
> 

DGsetup([x, t], [u], E, 1)
> 

L := alpha*u[]*u[1]+beta*u[1]-u[]*u[1, 1, 1]-3*u[1]*u[1, 1]+u[2]-u[1, 1, 2]

alpha*u[]*u[1]+beta*u[1]-u[]*u[1, 1, 1]-3*u[1]*u[1, 1]+u[2]-u[1, 1, 2]

 (1)
> 

EulerLagrange(L)

[0]

 (2)
> 

omega5 := evalDG((alpha*u[]*u[1]+beta*u[1]-u[]*u[1, 1, 1]-3*u[1]*u[1, 1]+u[2]-u[1, 1, 2])*`&w`(Dx, Dt))

_DG([["biform", E, [2, 0]], [[[1, 2], alpha*u[]*u[1]+beta*u[1]-u[]*u[1, 1, 1]-3*u[1]*u[1, 1]+u[2]-u[1, 1, 2]]]])

 (3)
E > 

HorizontalHomotopy(omega5)

_DG([["biform", E, [1, 0]], [[[1], (1/3)*u[1, 1]-u[]], [[2], (1/2)*alpha*u[]^2-u[]*u[1, 1]-u[1]^2-(2/3)*u[1, 2]+beta*u[]]]])

 (4)

How  can I identify respective fluxes? As I can see from "TotalDiff" command, the component multipled with Dx is t-flux(ignoring negative sign) and component multiplied with Dt is x-flux. Please see below

E > 

TotalDiff((1/3)*u[1, 1]-u[], t)+TotalDiff((1/2)*alpha*u[]^2-u[]*u[1, 1]-u[1]^2-(2/3)*u[1, 2]+beta*u[], x)

-u[2]-(1/3)*u[1, 1, 2]+(alpha*u[]+beta-u[1, 1])*u[1]-2*u[1]*u[1, 1]-u[]*u[1, 1, 1]

 (5)

Which is not equal to original expression (1). But if I ignore negative sign in Dx component of (4), then I have

E > 

TotalDiff(-(1/3)*u[1, 1]+u[], t)+TotalDiff((1/2)*alpha*u[]^2-u[]*u[1, 1]-u[1]^2-(2/3)*u[1, 2]+beta*u[], x)

u[2]-u[1, 1, 2]+(alpha*u[]+beta-u[1, 1])*u[1]-2*u[1]*u[1, 1]-u[]*u[1, 1, 1]

 (6)
E > 

simplify(u[2]-u[1, 1, 2]+(alpha*u[]+beta-u[1, 1])*u[1]-2*u[1]*u[1, 1]-u[]*u[1, 1, 1])

alpha*u[]*u[1]+beta*u[1]-u[]*u[1, 1, 1]-3*u[1]*u[1, 1]+u[2]-u[1, 1, 2]

 (7)

Which is actually an original expression (1). Is ignoring negative sign is correct?

I guess the negative has some relation to limits of integrals in definition of homotopy operator, am I right?
What is correct way to extract fluxes? So that the correct divergence expression may be achived like (7).

Please also explain, how to extract fluxes when there are three independent variables. Please see Khokhlov–Zabolotskaya equation.

E > 

restart
E > 

with(DifferentialGeometry):with(JetCalculus):``NULL
E > 

DGsetup([x, y, t], [u], E, 1):
E > 

M := -u[]*u[1, 1]-u[1]^2+u[1, 2]-u[2, 2]

-u[]*u[1, 1]-u[1]^2+u[1, 2]-u[2, 2]

 (8)
E > 

EulerLagrange(M)

[0]

 (9)
E > 

omega5 := evalDG((-u[]*u[1, 1]-u[1]^2+u[1, 2]-u[2, 2])*`&w`(`&w`(Dx, Dy), Dt))

_DG([["biform", E, [3, 0]], [[[1, 2, 3], -u[]*u[1, 1]-u[1]^2+u[1, 2]-u[2, 2]]]])

 (10)
E > 

HorizontalHomotopy(omega5)

_DG([["biform", E, [2, 0]], [[[1, 3], u[2]-(1/2)*u[1]], [[2, 3], (1/2)*u[2]-u[1]*u[]]]])

 (11)

Now how can I identify fluxes corresponding x, y, and t components?

E > 

``


 

Download [1289]_Homotopy.mw

Degraded image quality in exported JPEG ...

mskalsi 290
April 15 2017
0 0

@Kitonum 

When I exported plot in JPEG format, the image quality is degraded, why it is so?

 

Another site !!...

mskalsi 290
March 04 2017

Why another site for Maple when MaplePrime is performing good ?

Maxima minima from plot....

mskalsi 290
February 23 2017
0 0

@Markiyan Hirnyk 

My question has been answered by calculating critical points of given function by using standard definitions of differential calculus. The Max and Min are visible from plot of function. What if only plot is given i.e. when function is not given, then how I can find critical points ?

I am sorry as this query is not reflected in my qriginal question.

Derivative...

mskalsi 290
February 23 2017
0 0

@Markiyan Hirnyk 

I am interested in derivative. In fact my aim is to extract Maxima and Minima from plot.

Hmmm......!...

mskalsi 290
December 08 2016
0 0

@John Fredsted 

That is true but how should I proceed in my case ?

Solved !!...

mskalsi 290
November 27 2016
0 0

In meantime my problem is solved using JetCalculus[Euler-Lagrange].

Yeah it is nonsense !!...

mskalsi 290
November 19 2016
0 4

@Carl Love 

I am sorry I could not express my problem properly. In my second reply I have explained what actually I am wanting to do.

Thank you ver much. Please respond to second reply too.

Actually I trying for power series solut...

mskalsi 290
November 19 2016
0 0

@Carl Love 

I asked this query becuase I am trying to perform Maple calculations for power series solution of an ODE and I need to verifiy coefficients with my manual caluclations. Here I give you one example.


 

> 

F := sum(P[n]*xi^n, n = 0 .. infinity);

sum(P[n]*xi^n, n = 0 .. infinity)

 (1)
> 

ode := F^2+2*b*(diff(F, xi))+F*(diff(F, xi, xi))

(sum(P[n]*xi^n, n = 0 .. infinity))^2+2*b*(sum(P[n]*xi^n*n/xi, n = 0 .. infinity))+(sum(P[n]*xi^n, n = 0 .. infinity))*(sum(P[n]*xi^n*n^2/xi^2-P[n]*xi^n*n/xi^2, n = 0 .. infinity))

 (2)
> 

``

Actually I want to construct power series solution for above ODE by considering expansion (1). For this I need to find coefficient of xi^0 .. xi^nin the expansion (2)". "There is power series solution option in dsolve which I have not tried yet. Rather I would like to make manual calculations.


 

Download Power_Series_solution.mw

But this isn't the coefficient !!...

mskalsi 290
November 19 2016
0 0

@Carl Love 
Suppose I go with coefficient of xi^2, then it must be like:


 

Suppose I have a expression whose terms are in summation:

> 

sum(P[n]*xi^n, n = 0 .. infinity)+(sum(P[n]*xi^n, n = 0 .. infinity))^2+sum(P[2*n+1]*P[2*n]*xi^n, n = 0 .. infinity)

sum(P[n]*xi^n, n = 0 .. infinity)+(sum(P[n]*xi^n, n = 0 .. infinity))^2+sum(P[2*n+1]*P[2*n]*xi^n, n = 0 .. infinity)

 (1)
> 

diff(sum(P[n]*xi^n, n = 0 .. infinity)+(sum(P[n]*xi^n, n = 0 .. infinity))^2+sum(P[2*n+1]*P[2*n]*xi^n, n = 0 .. infinity), xi)

sum(P[n]*xi^n*n/xi, n = 0 .. infinity)+2*(sum(P[n]*xi^n, n = 0 .. infinity))*(sum(P[n]*xi^n*n/xi, n = 0 .. infinity))+sum(P[2*n+1]*P[2*n]*xi^n*n/xi, n = 0 .. infinity)

 (2)

In above summation, how I can write specific term ? For example, how I can display term for n = 2etc.

> 

coeff(eval(%, infinity = 2), xi, 2)*xi^2;

6*P[1]*P[2]*xi^2

 (3)

But is expression (2) the coefficient of xi^2must be like:

> 

3*P[3]+2*(3*P[0]*P[3]+2*P[1]*P[2]+P[1]*P[2])+3*P[7]*P[6]

6*P[0]*P[3]+6*P[1]*P[2]+3*P[6]*P[7]+3*P[3]

 (4)
> 

``


 

Download Summation.mw

Seems too sophisticated !!...

mskalsi 290
October 27 2016
0 1

@acer 

I think you are addressing my stackoverlow query which I had written in simple manner as I was expecting some clue from there.

Although you did great by printing the results as required but how I can modify above code to run for composite function actually given in worksheet. For example,

U(xi(x,y,t),eta(x,y,t),tau(x,y,t)) ?

Further, after displaying results in desired manner I will also collect coefficient with respect various derivatives like U[xi], U[eta], U[tau], U[xi,xi].......etc.

A failed try...

mskalsi 290
October 24 2016
0 0

I am trying to do like this:


 

> 

xi := proc (x, y, t) options operator, arrow; xi(x, y, t) end proc; 1; tau := proc (x, y, t) options operator, arrow; tau(x, y, t) end proc; 1; tau := proc (x, y, t) options operator, arrow; tau(x, y, t) end proc

proc (x, y, t) options operator, arrow; xi(x, y, t) end proc

  

proc (x, y, t) options operator, arrow; tau(x, y, t) end proc

  

proc (x, y, t) options operator, arrow; tau(x, y, t) end proc

 (1)
> 

U := proc (xi, tau, eta) options operator, arrow; U(xi, tau, eta) end proc

proc (xi, tau, eta) options operator, arrow; U(xi, tau, eta) end proc

 (2)
> 

diff(U(xi, tau, eta), x)

Error, (in U) too many levels of recursion

  
> 

``


 

Download A_trial.mw

Regards

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