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salim-barzani

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These are questions asked by salim-barzani

How remove error when changing delta par...

salim-barzani 1640 Maple 2024
finance spam numeric
December 17 2024
0 14

This is my first time working with plotting data from a matrix. However, with the help of a friends on MaplePrimes, I learned how to plot the data in both Maple and MATLAB. Despite this, I am having trouble with visualization. When I change the delta value, my function experiences vibrations or noise, which is clearly visible in the plot. But when I change delta, I encounter errors with my matrix data. How can I fix this problem? and there is any way for get better visualization by Explore ? also How show this vibration or noise in 2D?

> 

restart;
> 

randomize():
> 

local gamma;

gamma

 (1)
> 

currentdir(kernelopts(':-homedir'))
> 

NULL
> 

T3 := (B[1]*(tanh(2*n^2*(delta^2-w)*k*t/((k*n-1)*(k*n+1))+x)-1))^(1/(2*n))*exp(I*(-k*x+w*t+delta*W(t)-delta^2*t))

(B[1]*(tanh(2*n^2*(delta^2-w)*k*t/((k*n-1)*(k*n+1))+x)-1))^((1/2)/n)*exp(I*(-k*x+w*t+delta*W(t)-delta^2*t))

 (2)
> 

NULL
> 

params := {B[1]=1,n=2,delta=1,w=1,k=3 };

{delta = 1, k = 3, n = 2, w = 1, B[1] = 1}

 (3)
> 

NULL

insert numerical values

> 

solnum :=subs(params, T3);

(tanh(x)-1)^(1/4)*exp(I*(-3*x+W(t)))

 (4)
> 

CodeGeneration['Matlab']('(tanh(x)-1)^(1/4)*exp(I*(-3*x+W(t)))')

Warning, the function names {W} are not recognized in the target language

  

cg = ((tanh(x) - 0.1e1) ^ (0.1e1 / 0.4e1)) * exp(i * (-0.3e1 * x + W(t)));

  
> 

N := 100:

use Finance in:
  Wiener := WienerProcess():
  P := PathPlot(Wiener(t), t = 0..10, timesteps = N, replications = 1):
end use:
> 

W__points := plottools:-getdata(P)[1, -1]:
t_grid := convert(W__points[..,1], list):
x_grid := [seq(-2..2, 4/N)]:
> 

T, X := map(mul, [selectremove(has, [op(expand(solnum))], t)])[]:

ST := unapply(eval(T, W(t)=w), w)~(W__points[.., 2]):
SX := evalf(unapply(X, x)~(x_grid)):

STX := Matrix(N$2, (it, ix) -> ST[it]*SX[ix]);

_rtable[36893490640185799852]

 (5)
> 

opts := axis[1]=[tickmarks=[seq(k=nprintf("%1.1f", t_grid[k]), k=1..N, 40)]],
        axis[2]=[tickmarks=[seq(k=nprintf("%1.1f", x_grid[k]), k=1..N, 40)]],
        style=surface:

DocumentTools:-Tabulate(
  [
    plots:-matrixplot(Re~(STX), opts),
    plots:-matrixplot(Im~(STX), opts),
plots:-matrixplot(abs~(STX), opts)
  ]
  , width=60
)

"Tabulate"

 (6)
> 

MatlabFile := cat(currentdir(), "/ST2.txt"); ExportMatrix(MatlabFile, STX, target = MATLAB, format = rectangular, mode = ascii, format = entries)

421796

 (7)

NULL

Download data-analysis.mw

Transform data in maple to Matlab or mak...

salim-barzani 1640 Maple 2024
matlab excel export
December 16 2024
1 7

I have a matrix for data analysis that I want to plot. Ideally, I would like to use Maple, but I’m struggling to create a well-designed plot suitable for submission to journals. Because of this, I’m considering transferring the data to Excel or constructing a 3D graph using MATLAB.

My question is: how can I transfer this data to Excel? The data is currently saved as a Notepad file, but I’m unsure how to convert it into an Excel format. I will upload a figure to show the data structure.

also in last runig program give me error which is (Error, (in ExportMatrix) permission denied

Thank you in advance for any help!

> 

restart;
> 

randomize():
> 

local gamma;

gamma

 (1)
 
> 

T3 := (B[1]*(tanh(2*n^2*(delta^2-w)*k*t/((k*n-1)*(k*n+1))+x)-1))^(1/(2*n))*exp(I*(-k*x+w*t+delta*W(t)-delta^2*t))

(B[1]*(tanh(2*n^2*(delta^2-w)*k*t/((k*n-1)*(k*n+1))+x)-1))^((1/2)/n)*exp(I*(-k*x+w*t+delta*W(t)-delta^2*t))

 (2)
> 

``
> 

params := {B[1]=1,n=2,delta=1,w=1,k=3 };

{delta = 1, k = 3, n = 2, w = 1, B[1] = 1}

 (3)
> 

``

insert numerical values

> 

solnum :=subs(params, T3);

(tanh(x)-1)^(1/4)*exp(I*(-3*x+W(t)))

 (4)
> 

CodeGeneration['Matlab']('(tanh(x)-1)^(1/4)*exp(I*(-3*x+W(t)))')

Warning, the function names {W} are not recognized in the target language

  

cg = ((tanh(x) - 0.1e1) ^ (0.1e1 / 0.4e1)) * exp(i * (-0.3e1 * x + W(t)));

  
> 

N := 100:

use Finance in:
  Wiener := WienerProcess():
  P := PathPlot(Wiener(t), t = 0..10, timesteps = N, replications = 1):
end use:
> 

W__points := plottools:-getdata(P)[1, -1]:
t_grid := convert(W__points[..,1], list):
x_grid := [seq(-2..2, 4/N)]:
> 

T, X := map(mul, [selectremove(has, [op(expand(solnum))], t)])[]:

ST := unapply(eval(T, W(t)=w), w)~(W__points[.., 2]):
SX := evalf(unapply(X, x)~(x_grid)):

STX := Matrix(N$2, (it, ix) -> ST[it]*SX[ix]);

_rtable[36893489786521178348]

 (5)
> 

opts := axis[1]=[tickmarks=[seq(k=nprintf("%1.1f", t_grid[k]), k=1..N, 40)]],
        axis[2]=[tickmarks=[seq(k=nprintf("%1.1f", x_grid[k]), k=1..N, 40)]],
        style=surface:

DocumentTools:-Tabulate(
  [
    plots:-matrixplot(Re~(STX), opts),
    plots:-matrixplot(Im~(STX), opts),
plots:-matrixplot(abs~(STX), opts)
  ]
  , width=60
)

"Tabulate"

 (6)
> 

MatlabFile := cat(currentdir(), "/ST2.txt"); ExportMatrix(MatlabFile, STX, target = MATLAB, format = rectangular, mode = ascii, format = entries)

Error, (in ExportMatrix) permission denied

  
 

 

Download data-analysis.mw

Find parameters for two independent vari...

salim-barzani 1640 Maple
parameters
December 14 2024
0 7

What systematic methods can be used to determine the optimal parameters in a long equation involving two independent variables, and how do techniques like separation of variables, balancing principles, or dimensional analysis aid in simplifying and solving such equations?

parameters_x_t.mw

Better simplification and latex export...

salim-barzani 1640 Maple 2024
collect simplification latex
December 13 2024
2 10

I was rejected because the editor said my equation is too long. My question is: Is there a way to rewrite the equation in a more concise form? Additionally, is there a package in Maple that allows for automatic simplification or collection of terms without using specific commands? Any suggestions for addressing this issue would be appreciated.

> 

restart
> 

``
> 

eq3 := -6*lambda*beta[0]^2*alpha[1]^2*a[3]-2*lambda*beta[0]^2*a[5]*alpha[0]+6*mu*beta[0]*alpha[1]^2*a[2]+3*mu*beta[0]*a[5]*alpha[0]^2+(10*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda))*alpha[1]^2*alpha[0]^3*a[4]+(6*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda))*alpha[1]^2*alpha[0]^2*a[3]+(4*(-(2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda))*lambda+4*mu^2))*alpha[1]^2*a[5]*alpha[0]-12*mu^2*alpha[1]^2*a[5]*alpha[0]+(3*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda))*alpha[1]^2*alpha[0]*a[2]-(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*k^2*a[1]*alpha[1]^2+(1/2)*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*lambda*a[1]+(5*(-(2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda))*lambda+4*mu^2))*alpha[1]^4*alpha[0]*a[4]+(4*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda))*alpha[1]^2*lambda*a[5]*alpha[0]-k^2*a[1]*beta[0]^2+10*beta[0]^2*alpha[0]^3*a[4]+6*beta[0]^2*alpha[0]^2*a[3]+3*beta[0]^2*alpha[0]*a[2]-(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*w*alpha[1]^2-(1/4)*lambda*beta[0]^2*a[1]-9*mu^2*alpha[1]^2*a[1]*(1/4)+3*mu*a[1]*alpha[0]*beta[0]*(1/2)+(1/4)*(3*(-(2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda))*lambda+4*mu^2))*alpha[1]^2*a[1]+(-(2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda))*lambda+4*mu^2)*alpha[1]^4*a[3]-w*beta[0]^2-30*lambda*beta[0]^2*alpha[1]^2*alpha[0]*a[4]-20*mu*beta[0]*lambda*alpha[1]^4*a[4]-7*mu*beta[0]*lambda*a[5]*alpha[1]^2+24*mu*beta[0]*alpha[1]^2*alpha[0]*a[3]+60*mu*beta[0]*alpha[1]^2*alpha[0]^2*a[4] = 0

-k^2*a[1]*beta[0]^2+4*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*lambda*a[5]*alpha[0]-30*lambda*beta[0]^2*alpha[1]^2*alpha[0]*a[4]-20*mu*beta[0]*lambda*alpha[1]^4*a[4]+60*mu*beta[0]*alpha[1]^2*alpha[0]^2*a[4]-7*mu*beta[0]*lambda*a[5]*alpha[1]^2+24*mu*beta[0]*alpha[1]^2*alpha[0]*a[3]-w*beta[0]^2-(9/4)*mu^2*alpha[1]^2*a[1]+6*beta[0]^2*alpha[0]^2*a[3]-(1/4)*lambda*beta[0]^2*a[1]+3*beta[0]^2*alpha[0]*a[2]+(3/4)*(-2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*lambda+4*mu^2)*alpha[1]^2*a[1]-(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*w*alpha[1]^2+10*beta[0]^2*alpha[0]^3*a[4]+(-2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*lambda+4*mu^2)*alpha[1]^4*a[3]+6*mu*beta[0]*alpha[1]^2*a[2]+3*mu*beta[0]*a[5]*alpha[0]^2+(3/2)*mu*a[1]*alpha[0]*beta[0]-6*lambda*beta[0]^2*alpha[1]^2*a[3]-2*lambda*beta[0]^2*a[5]*alpha[0]-12*mu^2*alpha[1]^2*a[5]*alpha[0]+3*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*alpha[0]*a[2]-(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*k^2*a[1]*alpha[1]^2+(1/2)*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*lambda*a[1]+5*(-2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*lambda+4*mu^2)*alpha[1]^4*alpha[0]*a[4]+10*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*alpha[0]^3*a[4]+6*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*alpha[0]^2*a[3]+4*(-2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*lambda+4*mu^2)*alpha[1]^2*a[5]*alpha[0] = 0

 (1)
> 

numer(lhs(3*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*alpha[0]*a[2]+5*(-2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*lambda+4*mu^2)*alpha[1]^4*alpha[0]*a[4]+10*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*alpha[0]^3*a[4]+6*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*alpha[0]^2*a[3]+4*(-2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*lambda+4*mu^2)*alpha[1]^2*a[5]*alpha[0]-6*lambda*beta[0]^2*alpha[1]^2*a[3]-2*lambda*beta[0]^2*a[5]*alpha[0]+6*mu*beta[0]*alpha[1]^2*a[2]+3*mu*beta[0]*a[5]*alpha[0]^2+(3/2)*mu*a[1]*alpha[0]*beta[0]-12*mu^2*alpha[1]^2*a[5]*alpha[0]-(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*k^2*a[1]*alpha[1]^2+(1/2)*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*lambda*a[1]-w*beta[0]^2+4*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*lambda*a[5]*alpha[0]-30*lambda*beta[0]^2*alpha[1]^2*alpha[0]*a[4]-20*mu*beta[0]*lambda*alpha[1]^4*a[4]-7*mu*beta[0]*lambda*a[5]*alpha[1]^2+24*mu*beta[0]*alpha[1]^2*alpha[0]*a[3]+60*mu*beta[0]*alpha[1]^2*alpha[0]^2*a[4]+(-2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*lambda+4*mu^2)*alpha[1]^4*a[3]+(3/4)*(-2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*lambda+4*mu^2)*alpha[1]^2*a[1]-k^2*a[1]*beta[0]^2+10*beta[0]^2*alpha[0]^3*a[4]+6*beta[0]^2*alpha[0]^2*a[3]+3*beta[0]^2*alpha[0]*a[2]-(9/4)*mu^2*alpha[1]^2*a[1]-(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*w*alpha[1]^2-(1/4)*lambda*beta[0]^2*a[1] = 0))*denom(rhs(3*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*alpha[0]*a[2]+5*(-2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*lambda+4*mu^2)*alpha[1]^4*alpha[0]*a[4]+10*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*alpha[0]^3*a[4]+6*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*alpha[0]^2*a[3]+4*(-2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*lambda+4*mu^2)*alpha[1]^2*a[5]*alpha[0]-6*lambda*beta[0]^2*alpha[1]^2*a[3]-2*lambda*beta[0]^2*a[5]*alpha[0]+6*mu*beta[0]*alpha[1]^2*a[2]+3*mu*beta[0]*a[5]*alpha[0]^2+(3/2)*mu*a[1]*alpha[0]*beta[0]-12*mu^2*alpha[1]^2*a[5]*alpha[0]-(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*k^2*a[1]*alpha[1]^2+(1/2)*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*lambda*a[1]-w*beta[0]^2+4*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*lambda*a[5]*alpha[0]-30*lambda*beta[0]^2*alpha[1]^2*alpha[0]*a[4]-20*mu*beta[0]*lambda*alpha[1]^4*a[4]-7*mu*beta[0]*lambda*a[5]*alpha[1]^2+24*mu*beta[0]*alpha[1]^2*alpha[0]*a[3]+60*mu*beta[0]*alpha[1]^2*alpha[0]^2*a[4]+(-2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*lambda+4*mu^2)*alpha[1]^4*a[3]+(3/4)*(-2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*lambda+4*mu^2)*alpha[1]^2*a[1]-k^2*a[1]*beta[0]^2+10*beta[0]^2*alpha[0]^3*a[4]+6*beta[0]^2*alpha[0]^2*a[3]+3*beta[0]^2*alpha[0]*a[2]-(9/4)*mu^2*alpha[1]^2*a[1]-(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*w*alpha[1]^2-(1/4)*lambda*beta[0]^2*a[1] = 0)) = numer(rhs(3*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*alpha[0]*a[2]+5*(-2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*lambda+4*mu^2)*alpha[1]^4*alpha[0]*a[4]+10*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*alpha[0]^3*a[4]+6*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*alpha[0]^2*a[3]+4*(-2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*lambda+4*mu^2)*alpha[1]^2*a[5]*alpha[0]-6*lambda*beta[0]^2*alpha[1]^2*a[3]-2*lambda*beta[0]^2*a[5]*alpha[0]+6*mu*beta[0]*alpha[1]^2*a[2]+3*mu*beta[0]*a[5]*alpha[0]^2+(3/2)*mu*a[1]*alpha[0]*beta[0]-12*mu^2*alpha[1]^2*a[5]*alpha[0]-(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*k^2*a[1]*alpha[1]^2+(1/2)*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*lambda*a[1]-w*beta[0]^2+4*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*lambda*a[5]*alpha[0]-30*lambda*beta[0]^2*alpha[1]^2*alpha[0]*a[4]-20*mu*beta[0]*lambda*alpha[1]^4*a[4]-7*mu*beta[0]*lambda*a[5]*alpha[1]^2+24*mu*beta[0]*alpha[1]^2*alpha[0]*a[3]+60*mu*beta[0]*alpha[1]^2*alpha[0]^2*a[4]+(-2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*lambda+4*mu^2)*alpha[1]^4*a[3]+(3/4)*(-2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*lambda+4*mu^2)*alpha[1]^2*a[1]-k^2*a[1]*beta[0]^2+10*beta[0]^2*alpha[0]^3*a[4]+6*beta[0]^2*alpha[0]^2*a[3]+3*beta[0]^2*alpha[0]*a[2]-(9/4)*mu^2*alpha[1]^2*a[1]-(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*w*alpha[1]^2-(1/4)*lambda*beta[0]^2*a[1] = 0))*denom(lhs(3*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*alpha[0]*a[2]+5*(-2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*lambda+4*mu^2)*alpha[1]^4*alpha[0]*a[4]+10*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*alpha[0]^3*a[4]+6*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*alpha[0]^2*a[3]+4*(-2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*lambda+4*mu^2)*alpha[1]^2*a[5]*alpha[0]-6*lambda*beta[0]^2*alpha[1]^2*a[3]-2*lambda*beta[0]^2*a[5]*alpha[0]+6*mu*beta[0]*alpha[1]^2*a[2]+3*mu*beta[0]*a[5]*alpha[0]^2+(3/2)*mu*a[1]*alpha[0]*beta[0]-12*mu^2*alpha[1]^2*a[5]*alpha[0]-(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*k^2*a[1]*alpha[1]^2+(1/2)*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*lambda*a[1]-w*beta[0]^2+4*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*alpha[1]^2*lambda*a[5]*alpha[0]-30*lambda*beta[0]^2*alpha[1]^2*alpha[0]*a[4]-20*mu*beta[0]*lambda*alpha[1]^4*a[4]-7*mu*beta[0]*lambda*a[5]*alpha[1]^2+24*mu*beta[0]*alpha[1]^2*alpha[0]*a[3]+60*mu*beta[0]*alpha[1]^2*alpha[0]^2*a[4]+(-2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*lambda+4*mu^2)*alpha[1]^4*a[3]+(3/4)*(-2*(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*lambda+4*mu^2)*alpha[1]^2*a[1]-k^2*a[1]*beta[0]^2+10*beta[0]^2*alpha[0]^3*a[4]+6*beta[0]^2*alpha[0]^2*a[3]+3*beta[0]^2*alpha[0]*a[2]-(9/4)*mu^2*alpha[1]^2*a[1]-(lambda*B[1]^2-lambda*B[2]^2-mu^2/lambda)*w*alpha[1]^2-(1/4)*lambda*beta[0]^2*a[1] = 0))

-40*lambda^3*B[1]^2*a[4]*alpha[0]*alpha[1]^4+40*lambda^3*B[2]^2*a[4]*alpha[0]*alpha[1]^4-8*lambda^3*B[1]^2*a[3]*alpha[1]^4+8*lambda^3*B[2]^2*a[3]*alpha[1]^4+40*lambda^2*B[1]^2*a[4]*alpha[0]^3*alpha[1]^2-40*lambda^2*B[2]^2*a[4]*alpha[0]^3*alpha[1]^2-4*k^2*lambda^2*B[1]^2*a[1]*alpha[1]^2+4*k^2*lambda^2*B[2]^2*a[1]*alpha[1]^2-16*lambda^3*B[1]^2*a[5]*alpha[0]*alpha[1]^2+16*lambda^3*B[2]^2*a[5]*alpha[0]*alpha[1]^2-80*lambda^2*mu*a[4]*alpha[1]^4*beta[0]+24*lambda^2*B[1]^2*a[3]*alpha[0]^2*alpha[1]^2-24*lambda^2*B[2]^2*a[3]*alpha[0]^2*alpha[1]^2+120*lambda*mu^2*a[4]*alpha[0]*alpha[1]^4-4*lambda^3*B[1]^2*a[1]*alpha[1]^2+4*lambda^3*B[2]^2*a[1]*alpha[1]^2+12*lambda^2*B[1]^2*a[2]*alpha[0]*alpha[1]^2-12*lambda^2*B[2]^2*a[2]*alpha[0]*alpha[1]^2-120*lambda^2*a[4]*alpha[0]*alpha[1]^2*beta[0]^2+24*lambda*mu^2*a[3]*alpha[1]^4+240*lambda*mu*a[4]*alpha[0]^2*alpha[1]^2*beta[0]-40*mu^2*a[4]*alpha[0]^3*alpha[1]^2+4*k^2*mu^2*a[1]*alpha[1]^2-28*lambda^2*mu*a[5]*alpha[1]^2*beta[0]-4*lambda^2*w*B[1]^2*alpha[1]^2+4*lambda^2*w*B[2]^2*alpha[1]^2-24*lambda^2*a[3]*alpha[1]^2*beta[0]^2+32*lambda*mu^2*a[5]*alpha[0]*alpha[1]^2+96*lambda*mu*a[3]*alpha[0]*alpha[1]^2*beta[0]+40*lambda*a[4]*alpha[0]^3*beta[0]^2-24*mu^2*a[3]*alpha[0]^2*alpha[1]^2-4*k^2*lambda*a[1]*beta[0]^2-8*lambda^2*a[5]*alpha[0]*beta[0]^2+7*lambda*mu^2*a[1]*alpha[1]^2+24*lambda*mu*a[2]*alpha[1]^2*beta[0]+12*lambda*mu*a[5]*alpha[0]^2*beta[0]+24*lambda*a[3]*alpha[0]^2*beta[0]^2-12*mu^2*a[2]*alpha[0]*alpha[1]^2-lambda^2*a[1]*beta[0]^2+6*lambda*mu*a[1]*alpha[0]*beta[0]+12*lambda*a[2]*alpha[0]*beta[0]^2+4*mu^2*w*alpha[1]^2-4*lambda*w*beta[0]^2 = 0

 (2)
> 

simplify(-40*lambda^3*B[1]^2*a[4]*alpha[0]*alpha[1]^4+40*lambda^3*B[2]^2*a[4]*alpha[0]*alpha[1]^4-8*lambda^3*B[1]^2*a[3]*alpha[1]^4+8*lambda^3*B[2]^2*a[3]*alpha[1]^4+40*lambda^2*B[1]^2*a[4]*alpha[0]^3*alpha[1]^2-40*lambda^2*B[2]^2*a[4]*alpha[0]^3*alpha[1]^2-4*k^2*lambda^2*B[1]^2*a[1]*alpha[1]^2+4*k^2*lambda^2*B[2]^2*a[1]*alpha[1]^2-16*lambda^3*B[1]^2*a[5]*alpha[0]*alpha[1]^2+16*lambda^3*B[2]^2*a[5]*alpha[0]*alpha[1]^2-80*lambda^2*mu*a[4]*alpha[1]^4*beta[0]+24*lambda^2*B[1]^2*a[3]*alpha[0]^2*alpha[1]^2-24*lambda^2*B[2]^2*a[3]*alpha[0]^2*alpha[1]^2+120*lambda*mu^2*a[4]*alpha[0]*alpha[1]^4-4*lambda^3*B[1]^2*a[1]*alpha[1]^2+4*lambda^3*B[2]^2*a[1]*alpha[1]^2+12*lambda^2*B[1]^2*a[2]*alpha[0]*alpha[1]^2-12*lambda^2*B[2]^2*a[2]*alpha[0]*alpha[1]^2-120*lambda^2*a[4]*alpha[0]*alpha[1]^2*beta[0]^2+24*lambda*mu^2*a[3]*alpha[1]^4+240*lambda*mu*a[4]*alpha[0]^2*alpha[1]^2*beta[0]-40*mu^2*a[4]*alpha[0]^3*alpha[1]^2+4*k^2*mu^2*a[1]*alpha[1]^2-28*lambda^2*mu*a[5]*alpha[1]^2*beta[0]-4*lambda^2*w*B[1]^2*alpha[1]^2+4*lambda^2*w*B[2]^2*alpha[1]^2-24*lambda^2*a[3]*alpha[1]^2*beta[0]^2+32*lambda*mu^2*a[5]*alpha[0]*alpha[1]^2+96*lambda*mu*a[3]*alpha[0]*alpha[1]^2*beta[0]+40*lambda*a[4]*alpha[0]^3*beta[0]^2-24*mu^2*a[3]*alpha[0]^2*alpha[1]^2-4*k^2*lambda*a[1]*beta[0]^2-8*lambda^2*a[5]*alpha[0]*beta[0]^2+7*lambda*mu^2*a[1]*alpha[1]^2+24*lambda*mu*a[2]*alpha[1]^2*beta[0]+12*lambda*mu*a[5]*alpha[0]^2*beta[0]+24*lambda*a[3]*alpha[0]^2*beta[0]^2-12*mu^2*a[2]*alpha[0]*alpha[1]^2-lambda^2*a[1]*beta[0]^2+6*lambda*mu*a[1]*alpha[0]*beta[0]+12*lambda*a[2]*alpha[0]*beta[0]^2+4*mu^2*w*alpha[1]^2-4*lambda*w*beta[0]^2 = 0, 'symbolic')

-40*(B[1]-B[2])*((a[4]*alpha[0]+(1/5)*a[3])*alpha[1]^2+(2/5)*a[5]*alpha[0]+(1/10)*a[1])*alpha[1]^2*(B[1]+B[2])*lambda^3+4*(-20*a[4]*beta[0]*alpha[1]^4*mu+(10*(B[1]^2-B[2]^2)*a[4]*alpha[0]^3+6*a[3]*(B[1]^2-B[2]^2)*alpha[0]^2+3*(B[1]^2*a[2]-B[2]^2*a[2]-10*a[4]*beta[0]^2)*alpha[0]-6*beta[0]^2*a[3]-7*a[5]*beta[0]*mu-(B[1]-B[2])*(B[1]+B[2])*(k^2*a[1]+w))*alpha[1]^2-2*(a[5]*alpha[0]+(1/8)*a[1])*beta[0]^2)*lambda^2+(120*(a[4]*alpha[0]+(1/5)*a[3])*mu^2*alpha[1]^4+(240*a[4]*beta[0]*alpha[0]^2*mu+32*(mu^2*a[5]+3*mu*a[3]*beta[0])*alpha[0]+24*beta[0]*mu*a[2]+7*mu^2*a[1])*alpha[1]^2-4*(-10*a[4]*beta[0]*alpha[0]^3+3*(-mu*a[5]-2*a[3]*beta[0])*alpha[0]^2+3*(-beta[0]*a[2]-(1/2)*mu*a[1])*alpha[0]+beta[0]*(k^2*a[1]+w))*beta[0])*lambda+4*alpha[1]^2*mu^2*(-10*a[4]*alpha[0]^3+k^2*a[1]-6*a[3]*alpha[0]^2-3*a[2]*alpha[0]+w) = 0

  

 

 

Error, (in collect) invalid input: collect uses a 2nd argument, x, which is missing

  
> 

Q1 := collect(%, {B__1, B__2})

-40*(B[1]-B[2])*((a[4]*alpha[0]+(1/5)*a[3])*alpha[1]^2+(2/5)*a[5]*alpha[0]+(1/10)*a[1])*alpha[1]^2*(B[1]+B[2])*lambda^3+4*(-20*a[4]*beta[0]*alpha[1]^4*mu+(10*(B[1]^2-B[2]^2)*a[4]*alpha[0]^3+6*a[3]*(B[1]^2-B[2]^2)*alpha[0]^2+3*(B[1]^2*a[2]-B[2]^2*a[2]-10*a[4]*beta[0]^2)*alpha[0]-6*beta[0]^2*a[3]-7*a[5]*beta[0]*mu-(B[1]-B[2])*(B[1]+B[2])*(k^2*a[1]+w))*alpha[1]^2-2*(a[5]*alpha[0]+(1/8)*a[1])*beta[0]^2)*lambda^2+(120*(a[4]*alpha[0]+(1/5)*a[3])*mu^2*alpha[1]^4+(240*a[4]*beta[0]*alpha[0]^2*mu+32*(mu^2*a[5]+3*mu*a[3]*beta[0])*alpha[0]+24*beta[0]*mu*a[2]+7*mu^2*a[1])*alpha[1]^2-4*(-10*a[4]*beta[0]*alpha[0]^3+3*(-mu*a[5]-2*a[3]*beta[0])*alpha[0]^2+3*(-beta[0]*a[2]-(1/2)*mu*a[1])*alpha[0]+beta[0]*(k^2*a[1]+w))*beta[0])*lambda+4*alpha[1]^2*mu^2*(-10*a[4]*alpha[0]^3+k^2*a[1]-6*a[3]*alpha[0]^2-3*a[2]*alpha[0]+w) = 0

 (3)
> 

latex(Q1)

-40 \left(B_{1}-B_{2}\right) \left(\left(a_{4} \alpha_{0}+\frac{a_{3}}{5}\right) \alpha_{1}^{2}+\frac{2 a_{5} \alpha_{0}}{5}+\frac{a_{1}}{10}\right) \alpha_{1}^{2} \left(B_{1}+B_{2}\right) \lambda^{3}+4 \left(-20 a_{4} \beta_{0} \alpha_{1}^{4} \mu +\left(10 \left(B_{1}^{2}-B_{2}^{2}\right) a_{4} \alpha_{0}^{3}+6 a_{3} \left(B_{1}^{2}-B_{2}^{2}\right) \alpha_{0}^{2}+3 \left(B_{1}^{2} a_{2}-B_{2}^{2} a_{2}-10 a_{4} \beta_{0}^{2}\right) \alpha_{0}-6 \beta_{0}^{2} a_{3}-7 a_{5} \beta_{0} \mu -\left(B_{1}-B_{2}\right) \left(B_{1}+B_{2}\right) \left(k^{2} a_{1}+w \right)\right) \alpha_{1}^{2}-2 \left(a_{5} \alpha_{0}+\frac{a_{1}}{8}\right) \beta_{0}^{2}\right) \lambda^{2}+\left(120 \left(a_{4} \alpha_{0}+\frac{a_{3}}{5}\right) \mu^{2} \alpha_{1}^{4}+\left(240 a_{4} \beta_{0} \alpha_{0}^{2} \mu +32 \left(\mu^{2} a_{5}+3 \mu  a_{3} \beta_{0}\right) \alpha_{0}+24 \beta_{0} \mu  a_{2}+7 \mu^{2} a_{1}\right) \alpha_{1}^{2}-4 \left(-10 a_{4} \beta_{0} \alpha_{0}^{3}+3 \left(-\mu  a_{5}-2 a_{3} \beta_{0}\right) \alpha_{0}^{2}+3 \left(-\beta_{0} a_{2}-\frac{\mu  a_{1}}{2}\right) \alpha_{0}+\beta_{0} \left(k^{2} a_{1}+w \right)\right) \beta_{0}\right) \lambda +4 \alpha_{1}^{2} \mu^{2} \left(-10 a_{4} \alpha_{0}^{3}+k^{2} a_{1}-6 a_{3} \alpha_{0}^{2}-3 a_{2} \alpha_{0}+w \right)
 = 0

  
 

NULL

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Techniques for Simplifying Differential ...

salim-barzani 1640 Maple 2024
pde differential-equations
December 11 2024
0 0

I'm trying to transform a partial differential equation (PDE) into an ordinary differential equation (ODE) as demonstrated in the paper. However, I find some steps confusing and difficult to follow. The process often feels chaotic, and managing the complexity of the equations is overwhelming. Could you suggest an effective and systematic method to handle such transformations more easily?

> 

restart
> 

with(PDEtools)
> 

with(LinearAlgebra)
> 

NULL
> 

with(SolveTools)
> 

undeclare(prime)

`There is no more prime differentiation variable; all derivatives will be displayed as indexed functions`

 (1)
> 

declare(Omega(x, t)); declare(U(xi))

Omega(x, t)*`will now be displayed as`*Omega

  

U(xi)*`will now be displayed as`*U

 (2)
> 

tr := {t = tau, x = tau*c[0]+xi, Omega(x, t) = U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))}

{t = tau, x = tau*c[0]+xi, Omega(x, t) = U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))}

 (3)
> 

P1 := diff(Omega(x, t)^m, t)

Omega(x, t)^m*m*(diff(Omega(x, t), t))/Omega(x, t)

 (4)
> 

L1 := PDEtools:-dchange(tr, P1, [xi, tau, U])

(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^m*m*(-((diff(U(xi), xi))*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))-I*U(xi)*k*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))*c[0]+I*U(xi)*(-k*c[0]+w+delta*(diff(W(tau), tau))-delta^2)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))/(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))

 (5)
 
> 

pde1 := I*(diff(Omega(x, t)^m, t))+alpha*(diff(Omega(x, t)^m, `$`(x, 2)))+I*beta*(diff(abs(Omega(x, t))^(2*n)*Omega(x, t)^m, x))+m*sigma*Omega(x, t)^m*(diff(W(t), t)) = I*gamma*abs(Omega(x, t))^(2*n)*(diff(Omega(x, t)^m, x))+delta*abs(Omega(x, t))^(4*n)*Omega(x, t)^m

I*Omega(x, t)^m*m*(diff(Omega(x, t), t))/Omega(x, t)+alpha*(Omega(x, t)^m*m^2*(diff(Omega(x, t), x))^2/Omega(x, t)^2+Omega(x, t)^m*m*(diff(diff(Omega(x, t), x), x))/Omega(x, t)-Omega(x, t)^m*m*(diff(Omega(x, t), x))^2/Omega(x, t)^2)+I*beta*(2*abs(Omega(x, t))^(2*n)*n*(diff(Omega(x, t), x))*abs(1, Omega(x, t))*Omega(x, t)^m/abs(Omega(x, t))+abs(Omega(x, t))^(2*n)*Omega(x, t)^m*m*(diff(Omega(x, t), x))/Omega(x, t))+m*sigma*Omega(x, t)^m*(diff(W(t), t)) = I*gamma*abs(Omega(x, t))^(2*n)*Omega(x, t)^m*m*(diff(Omega(x, t), x))/Omega(x, t)+delta*abs(Omega(x, t))^(4*n)*Omega(x, t)^m

 (6)
> 

NULL
> 

L1 := PDEtools:-dchange(tr, pde1, [xi, tau, U])

I*(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^m*m*(-((diff(U(xi), xi))*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))-I*U(xi)*k*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))*c[0]+I*U(xi)*(-k*c[0]+w+delta*(diff(W(tau), tau))-delta^2)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))/(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))+alpha*((U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^m*m^2*((diff(U(xi), xi))*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))-I*U(xi)*k*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^2/(U(xi)^2*(exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^2)+(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^m*m*((diff(diff(U(xi), xi), xi))*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))-(2*I)*(diff(U(xi), xi))*k*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))-U(xi)*k^2*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))/(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))-(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^m*m*((diff(U(xi), xi))*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))-I*U(xi)*k*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^2/(U(xi)^2*(exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^2))+I*beta*(2*(abs(U(xi))*exp(-Im(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^(2*n)*n*((diff(U(xi), xi))*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))-I*U(xi)*k*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))*abs(1, U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))*(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^m/(abs(U(xi))*exp(-Im(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))+(abs(U(xi))*exp(-Im(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^(2*n)*(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^m*m*((diff(U(xi), xi))*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))-I*U(xi)*k*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))/(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))))+m*sigma*(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^m*(diff(W(tau), tau)) = I*gamma*(abs(U(xi))*exp(-Im(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^(2*n)*(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^m*m*((diff(U(xi), xi))*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))-I*U(xi)*k*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))/(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))+delta*(abs(U(xi))*exp(-Im(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^(4*n)*(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^m

 (7)
> 

``

``

 (8)

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