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Difference between 'simplify(expression)...

WA573 80 Maple 2022
simplify size
February 14 2024
1 2

Are the results consistent either we use simplify(expression) or simplify(expression,size)? It seems (2) and (3) are not consistent.

> 

restart
> 

with(PDEtools); with(LinearAlgebra)
> 

b := -(2*I)*exp(2*t*Im(lambda1))*(exp(I*a*x/conjugate(lambda1))*exp((2*I)*a*x/lambda1)*exp((-I*a*x)*(1/conjugate(lambda1)))*exp(I*conjugate(lambda1)*t)+exp(I*lambda1*t)*exp(I*a*x/lambda1)*exp((-I*a*x)*(1/lambda1))*exp((2*I)*a*x/conjugate(lambda1))*(abs(`ε1`)^2+abs(`ε2`)^2))*conjugate(`ε1`)*Im(lambda1)/(exp((2*I)*a*x/lambda1)*abs(`ε1`)^2*exp(-(2*I)*a*x/conjugate(lambda1))*exp(2*t*Im(lambda1))+exp(-(2*I)*a*x/lambda1)*abs(`ε1`)^2*exp((2*I)*a*x/conjugate(lambda1))*exp(2*t*Im(lambda1))+exp(I*a*x/lambda1)*exp((-I*a*x)*(1/lambda1))*exp(I*a*x/conjugate(lambda1))*exp((-I*a*x)*(1/conjugate(lambda1)))*(abs(`ε2`)^4+2*abs(`ε1`)^2*abs(`ε2`)^2+abs(`ε1`)^4+2*abs(`ε2`)^2*exp(2*t*Im(lambda1))+exp(4*t*Im(lambda1))))

-(2*I)*exp(2*t*Im(lambda1))*(exp(I*a*x/conjugate(lambda1))*exp((2*I)*a*x/lambda1)*exp(-I*a*x/conjugate(lambda1))*exp(I*conjugate(lambda1)*t)+exp(I*lambda1*t)*exp(I*a*x/lambda1)*exp(-I*a*x/lambda1)*exp((2*I)*a*x/conjugate(lambda1))*(abs(epsilon1)^2+abs(epsilon2)^2))*conjugate(epsilon1)*Im(lambda1)/(exp((2*I)*a*x/lambda1)*abs(epsilon1)^2*exp(-(2*I)*a*x/conjugate(lambda1))*exp(2*t*Im(lambda1))+exp(-(2*I)*a*x/lambda1)*abs(epsilon1)^2*exp((2*I)*a*x/conjugate(lambda1))*exp(2*t*Im(lambda1))+exp(I*a*x/lambda1)*exp(-I*a*x/lambda1)*exp(I*a*x/conjugate(lambda1))*exp(-I*a*x/conjugate(lambda1))*(abs(epsilon2)^4+2*abs(epsilon1)^2*abs(epsilon2)^2+abs(epsilon1)^4+2*abs(epsilon2)^2*exp(2*t*Im(lambda1))+exp(4*t*Im(lambda1))))

 (1)
> 

bdif := simplify(diff(b, x)); bdifxzero := simplify(subs({x = 0}, bdif))

4*Im(lambda1)*exp(2*t*Im(lambda1))*a*conjugate(epsilon1)*(conjugate(lambda1)*abs(epsilon1)^2*(abs(epsilon1)^2+abs(epsilon2)^2)*exp((t*(I*lambda1+2*Im(lambda1))*abs(lambda1)^2+(4*I)*a*(lambda1-(1/2)*conjugate(lambda1))*x)/abs(lambda1)^2)+exp(((I*conjugate(lambda1)+2*Im(lambda1))*t*abs(lambda1)^2-(2*I)*(lambda1-2*conjugate(lambda1))*a*x)/abs(lambda1)^2)*abs(epsilon1)^2*lambda1+2*abs(epsilon1)^2*(conjugate(lambda1)-(1/2)*lambda1)*exp((I*conjugate(lambda1)^2*t+(2*I)*a*x+2*t*Im(lambda1)*conjugate(lambda1))/conjugate(lambda1))+2*abs(epsilon2)^2*lambda1*(abs(epsilon1)^2+abs(epsilon2)^2)*exp((I*t*abs(lambda1)^2+(2*I)*a*x+2*t*Im(lambda1)*conjugate(lambda1))/conjugate(lambda1))+lambda1*(abs(epsilon1)^2+abs(epsilon2)^2)*exp((I*t*abs(lambda1)^2+(2*I)*a*x+4*t*Im(lambda1)*conjugate(lambda1))/conjugate(lambda1))+2*conjugate(lambda1)*exp((I*t*abs(lambda1)^2+(2*I)*a*x+2*t*Im(lambda1)*lambda1)/lambda1)*abs(epsilon2)^2+conjugate(lambda1)*exp((I*t*abs(lambda1)^2+(2*I)*a*x+4*t*Im(lambda1)*lambda1)/lambda1)+(-abs(epsilon1)^2*(conjugate(lambda1)-2*lambda1)*exp((I*lambda1^2*t+(2*I)*a*x+2*t*Im(lambda1)*lambda1)/lambda1)+(abs(epsilon1)^2+abs(epsilon2)^2)*(lambda1*(abs(epsilon1)^2+abs(epsilon2)^2)*exp(I*(t*abs(lambda1)^2+2*a*x)/conjugate(lambda1))+conjugate(lambda1)*exp(I*(t*abs(lambda1)^2+2*a*x)/lambda1)))*(abs(epsilon1)^2+abs(epsilon2)^2))/((abs(epsilon1)^4+2*abs(epsilon1)^2*abs(epsilon2)^2+abs(epsilon2)^4+2*abs(epsilon2)^2*exp(2*t*Im(lambda1))+exp((2*t*Im(lambda1)*abs(lambda1)^2-(2*I)*(lambda1-conjugate(lambda1))*a*x)/abs(lambda1)^2)*abs(epsilon1)^2+exp((2*t*Im(lambda1)*abs(lambda1)^2+(2*I)*(lambda1-conjugate(lambda1))*a*x)/abs(lambda1)^2)*abs(epsilon1)^2+exp(4*t*Im(lambda1)))^2*abs(lambda1)^2)

  

12*Im(lambda1)*exp(2*t*Im(lambda1))*((2/3)*abs(epsilon2)^2*lambda1*(abs(epsilon1)^2+abs(epsilon2)^2)*exp(t*(I*abs(lambda1)^2+2*Im(lambda1)*conjugate(lambda1))/conjugate(lambda1))+(1/3)*lambda1*(abs(epsilon1)^2+abs(epsilon2)^2)*exp(t*(I*abs(lambda1)^2+4*Im(lambda1)*conjugate(lambda1))/conjugate(lambda1))+(2/3)*conjugate(lambda1)*exp(t*(I*abs(lambda1)^2+2*lambda1*Im(lambda1))/lambda1)*abs(epsilon2)^2+(1/3)*conjugate(lambda1)*exp(t*(I*abs(lambda1)^2+4*lambda1*Im(lambda1))/lambda1)+(1/3)*lambda1*(abs(epsilon1)^2+abs(epsilon2)^2)^3*exp(I*abs(lambda1)^2*t/conjugate(lambda1))+(1/3)*conjugate(lambda1)*(abs(epsilon1)^2+abs(epsilon2)^2)^2*exp(I*abs(lambda1)^2*t/lambda1)+((-I*Im(lambda1)+(1/3)*lambda1+(1/3)*Re(lambda1))*exp((I*conjugate(lambda1)+2*Im(lambda1))*t)+(I*Im(lambda1)+(1/3)*conjugate(lambda1)+(1/3)*Re(lambda1))*(abs(epsilon1)^2+abs(epsilon2)^2)*exp(t*(I*lambda1+2*Im(lambda1))))*abs(epsilon1)^2)*a*conjugate(epsilon1)/(abs(lambda1)^2*((2*abs(epsilon2)^2+2*abs(epsilon1)^2)*exp(2*t*Im(lambda1))+abs(epsilon2)^4+2*abs(epsilon1)^2*abs(epsilon2)^2+abs(epsilon1)^4+exp(4*t*Im(lambda1)))^2)

 (2)
> 

bdif1 := simplify(diff(b, x), size); bdif1xzero := simplify(subs({x = 0}, bdif1), size)

4*Im(lambda1)*(2*exp(I*a*x/lambda1)*exp(I*conjugate(lambda1)*t)*(exp(I*a*x/conjugate(lambda1)))^2*exp(-I*a*x/lambda1)*((1/2)*abs(epsilon1)^4+abs(epsilon1)^2*abs(epsilon2)^2+(1/2)*abs(epsilon2)^4+abs(epsilon2)^2*exp(2*t*Im(lambda1))+(1/2)*exp(4*t*Im(lambda1)))*exp((2*I)*a*x/lambda1)*conjugate(lambda1)*(exp(-I*a*x/conjugate(lambda1)))^2+2*exp(I*a*x/conjugate(lambda1))*(((exp(I*a*x/lambda1))^2*lambda1*exp(I*lambda1*t)*((1/2)*abs(epsilon1)^4+abs(epsilon1)^2*abs(epsilon2)^2+(1/2)*abs(epsilon2)^4+abs(epsilon2)^2*exp(2*t*Im(lambda1))+(1/2)*exp(4*t*Im(lambda1)))*(abs(epsilon1)^2+abs(epsilon2)^2)*(exp(-I*a*x/lambda1))^2+exp(-(2*I)*a*x/lambda1)*exp((2*I)*a*x/lambda1)*exp(2*t*Im(lambda1))*exp(I*conjugate(lambda1)*t)*abs(epsilon1)^2*(conjugate(lambda1)-(1/2)*lambda1))*exp((2*I)*a*x/conjugate(lambda1))+(1/2)*(exp((2*I)*a*x/lambda1))^2*exp(2*t*Im(lambda1))*exp(I*conjugate(lambda1)*t)*exp(-(2*I)*a*x/conjugate(lambda1))*lambda1*abs(epsilon1)^2)*exp(-I*a*x/conjugate(lambda1))+exp(I*a*x/lambda1)*abs(epsilon1)^2*exp(I*lambda1*t)*exp(-I*a*x/lambda1)*exp((2*I)*a*x/conjugate(lambda1))*(conjugate(lambda1)*exp(-(2*I)*a*x/lambda1)*exp((2*I)*a*x/conjugate(lambda1))-exp((2*I)*a*x/lambda1)*exp(-(2*I)*a*x/conjugate(lambda1))*(conjugate(lambda1)-2*lambda1))*exp(2*t*Im(lambda1))*(abs(epsilon1)^2+abs(epsilon2)^2))*conjugate(epsilon1)*a*exp(2*t*Im(lambda1))/((2*exp(I*a*x/lambda1)*exp(I*a*x/conjugate(lambda1))*exp(-I*a*x/lambda1)*((1/2)*abs(epsilon1)^4+abs(epsilon1)^2*abs(epsilon2)^2+(1/2)*abs(epsilon2)^4+abs(epsilon2)^2*exp(2*t*Im(lambda1))+(1/2)*exp(4*t*Im(lambda1)))*exp(-I*a*x/conjugate(lambda1))+exp(2*t*Im(lambda1))*abs(epsilon1)^2*(exp(-(2*I)*a*x/lambda1)*exp((2*I)*a*x/conjugate(lambda1))+exp((2*I)*a*x/lambda1)*exp(-(2*I)*a*x/conjugate(lambda1))))^2*lambda1*conjugate(lambda1))

  

2*Im(lambda1)*exp(0)*conjugate(epsilon1)*exp(2*t*Im(lambda1))*a*(conjugate(lambda1)*exp(I*conjugate(lambda1)*t)+exp(I*lambda1*t)*lambda1*(abs(epsilon1)^2+abs(epsilon2)^2))/((((exp(0))^2*abs(epsilon2)^2+abs(epsilon1)^2)*exp(2*t*Im(lambda1))+(1/2)*(exp(0))^2*(exp(4*t*Im(lambda1))+(abs(epsilon1)^2+abs(epsilon2)^2)^2))*lambda1*conjugate(lambda1))

 (3)
> 

NULL

Download simplisize.mw

expression when t goes to infinity...

WA573 80 Maple 2022
limit asymptotic incomplete-question
January 25 2024
0 11

I am trying to find the value of y4 at t=infinity and t=-infinity when lambda1>lambda2 or lambda1<lambda2. But every time I got the same answer. For example, if we do it by hand then the terms which are responsible for making the indeterminate form can be extracted and canceled (see Fig.). 

But in limit.mw y4 is too lengthy-expression and very difficult to do it manually.

Error, (in plot) incorrect first argumen...

WA573 80 Maple 2022
plotting
January 15 2024
1 3

How to handle these errors in plotting the solutions?compare.mw

'n' and 'm' are real...

WA573 80 Maple 2022
syntax substitution assumptions
January 05 2024
0 7

As I assumed 'n' and 'm' are real, eta is complex. But still, there is a bar on these discrete independent variables. Secondly, the substitution of (8) applies in some terms of 'r2', and the remaining terms remain as is it.

> 

restart
> 

with(LinearAlgebra); with(PDEtools); with(plots); with(LREtools)
> 

setup(mathematicalnotation = true)

setup(mathematicalnotation = true)

 (1)
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assume(n::real); assume(m::real)
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A := proc (n, m) options operator, arrow; Matrix([[eta*phi(n, m), conjugate(eta)*conjugate(psi(n, m))], [phi(n, m), conjugate(psi(n, m))]]) end proc; Adet := Determinant(A(n, m))

eta*phi(n, m)*conjugate(psi(n, m))-conjugate(eta)*conjugate(psi(n, m))*phi(n, m)

 (2)
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B := proc (n, m) options operator, arrow; Matrix([[phi(n, m), conjugate(psi(n, m))], [-psi(n, m), conjugate(phi(n, m))]]) end proc; Bdet := Determinant(B(n, m))

phi(n, m)*conjugate(phi(n, m))+conjugate(psi(n, m))*psi(n, m)

 (3)
> 

r := Adet/Bdet

(eta*phi(n, m)*conjugate(psi(n, m))-conjugate(eta)*conjugate(psi(n, m))*phi(n, m))/(phi(n, m)*conjugate(phi(n, m))+conjugate(psi(n, m))*psi(n, m))

 (4)
> 

p := {eta = 1+I, phi(n, m) = (1+I*a*eta)^n*(1+I*b*eta^2)^m, psi(n, m) = (1-I*a*eta)^n*(1-I*b*eta^2)^m, conjugate(eta) = 1-I, conjugate(phi(n, m)) = (1-I*a*conjugate(eta))^n*(1-I*b*conjugate(eta)^2)^m, conjugate(phi(n, m)) = (1+I*a*conjugate(eta))^n*(1+I*b*conjugate(eta)^2)^m}

{eta = 1+I, phi(n, m) = (1+I*a*eta)^n*(1+I*b*eta^2)^m, psi(n, m) = (1-I*a*eta)^n*(1-I*b*eta^2)^m, conjugate(eta) = 1-I, conjugate(phi(n, m)) = (1-I*a*conjugate(eta))^n*(1-I*b*conjugate(eta)^2)^m, conjugate(phi(n, m)) = (1+I*a*conjugate(eta))^n*(1+I*b*conjugate(eta)^2)^m}

 (5)
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r1 := simplify(subs(p, r))

(2*I)*(1+I*a*eta)^n*(1+I*b*eta^2)^m*conjugate((1-I*a*eta)^n*(1-I*b*eta^2)^m)/((1+I*a*eta)^n*(1+I*b*eta^2)^m*(1-I*a*conjugate(eta))^n*(1-I*b*conjugate(eta)^2)^m+abs(-1+I*a*eta)^(2*n)*abs(I*b*eta^2-1)^(2*m))

 (6)
> 

r2 := 1-I*delta(r1, n)

1-I*((2*I)*(1+I*a*eta)^(n+1)*(1+I*b*eta^2)^m*conjugate((1-I*a*eta)^(n+1)*(1-I*b*eta^2)^m)/((1+I*a*eta)^(n+1)*(1+I*b*eta^2)^m*(1-I*a*conjugate(eta))^(n+1)*(1-I*b*conjugate(eta)^2)^m+abs(-1+I*a*eta)^(2*n+2)*abs(I*b*eta^2-1)^(2*m))-(2*I)*(1+I*a*eta)^n*(1+I*b*eta^2)^m*conjugate((1-I*a*eta)^n*(1-I*b*eta^2)^m)/((1+I*a*eta)^n*(1+I*b*eta^2)^m*(1-I*a*conjugate(eta))^n*(1-I*b*conjugate(eta)^2)^m+abs(-1+I*a*eta)^(2*n)*abs(I*b*eta^2-1)^(2*m)))

 (7)
> 

exp_expr := subs({(1+I*b*eta^2)^m = exp(I*eta^2*t)}, r2)

1-I*((2*I)*(1+I*a*eta)^(n+1)*exp(I*eta^2*t)*conjugate((1-I*a*eta)^(n+1)*(1-I*b*eta^2)^m)/((1+I*a*eta)^(n+1)*exp(I*eta^2*t)*(1-I*a*conjugate(eta))^(n+1)*(1-I*b*conjugate(eta)^2)^m+abs(-1+I*a*eta)^(2*n+2)*abs(I*b*eta^2-1)^(2*m))-(2*I)*(1+I*a*eta)^n*exp(I*eta^2*t)*conjugate((1-I*a*eta)^n*(1-I*b*eta^2)^m)/((1+I*a*eta)^n*exp(I*eta^2*t)*(1-I*a*conjugate(eta))^n*(1-I*b*conjugate(eta)^2)^m+abs(-1+I*a*eta)^(2*n)*abs(I*b*eta^2-1)^(2*m)))

 (8)
> 

``
> 

NULL
> 

NULL
> 

NULL
> 

plot3d(abs(exp_expr), n = -5 .. 5, t = -5 .. 5, eta = 1+I)

Error, (in plot3d) unexpected option: eta = 1+I

  
 

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derivative from finite difference...

WA573 80 Maple 2022
derivative differential-equations finite-difference
November 28 2023
1 2

Since (1/h)[f(i+1,t)-f(i,t)]=f(x,t)_{x} as h goes to zero, 'i' is the discrete index along x-axis. How to do it in Maple? How to reduce Eq. (5) into continuous derivatives?

> 

restart
> 

with(LinearAlgebra)
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with(PDEtools)
> 

with(Physics)
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with(plots)
> 

Setup(mathematicalnotation = true)

[mathematicalnotation = true]

 (1)
> 

``
> 

U := proc (i, t) options operator, arrow; Matrix([[1+I*(q(i+1, t)-q(i, t))/lambda, I*(r(i+1, t)-r(i, t))/lambda], [I*(r(i+1, t)-r(i, t))/lambda, 1-I*(q(i+1, t)-q(i, t))/lambda]]) end proc

proc (i, t) options operator, arrow; Matrix([[1+Physics:-`*`(Physics:-`*`(I, q(i+1, t)-q(i, t)), Physics:-`^`(lambda, -1)), Physics:-`*`(Physics:-`*`(I, r(i+1, t)-r(i, t)), Physics:-`^`(lambda, -1))], [Physics:-`*`(Physics:-`*`(I, r(i+1, t)-r(i, t)), Physics:-`^`(lambda, -1)), 1-Physics:-`*`(Physics:-`*`(I, q(i+1, t)-q(i, t)), Physics:-`^`(lambda, -1))]]) end proc

 (2)
> 

``
> 

V := proc (i, t) options operator, arrow; Matrix([[-((1/2)*I)*lambda, -r(i, t)], [r(i, t), ((1/2)*I)*lambda]]) end proc

proc (i, t) options operator, arrow; Matrix([[Physics:-`*`(Physics:-`*`(Physics:-`*`(Physics:-`*`(1, Physics:-`^`(2, -1)), I), lambda), -1), Physics:-`*`(r(i, t), -1)], [r(i, t), Physics:-`*`(Physics:-`*`(Physics:-`*`(1, Physics:-`^`(2, -1)), I), lambda)]]) end proc

 (3)
> 

NULL
> 

z := diff(U(i, t), t)+U(i, t).V(i, t)-V(i+1, t).U(i, t)

Matrix(%id = 4525182530)

 (4)
> 

z11 := simplify(lambda*z[1, 1]/h, size) = 0

I*(r(i+1, t)^2-r(i, t)^2+(D[2](q))(i+1, t)-(diff(q(i, t), t)))/h = 0

 (5)
> 

NULL

Download limit.mw

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