Ahmed111

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These are replies submitted by Ahmed111

@Rouben Rostamian  When I execute Lineaize.mw on my computer. Maple returns "Error, (in collect) cannot collect 0"

restart

with(LinearAlgebra):

with(PDEtools):

with(Physics):

with(plots):

Setup(mathematicalnotation = true);

[mathematicalnotation = true]

(1)

assume(x::real);

assume(t::real);

assume(xi::real);

assume(tau::real);

interface(showassumed = 0);

0

(2)

alias(phi = phi(x, t), epsilon[1] = epsilon[1](x, t), epsilon[2] = epsilon[2](x, t), rho = rho(x, t), r = r(x, t), q = q(x, t), conjugate(q) = (conjugate(q))(x, t));

phi, epsilon[1], epsilon[2], rho, r, q, conjugate(q)

(3)

Eq := Physics:-`*`(diff(r, x, t), diff(r, x, t))-Physics:-`*`(Physics:-`*`(diff(q, t), conjugate(q))-Physics:-`*`(diff(conjugate(q), t), q), Physics:-`*`(diff(q, t), conjugate(q))-Physics:-`*`(diff(conjugate(q), t), q))-Physics:-`*`(Physics:-`*`(Physics:-`*`(2, I), diff(r, x, t)), Physics:-`*`(diff(q, t), conjugate(q))-Physics:-`*`(diff(conjugate(q), t), q))-Physics:-`*`(Physics:-`*`(4, r^2), 4-Physics:-`*`(diff(r, t), diff(r, t))-Physics:-`*`(Physics:-`*`(4, diff(q, t)), diff(conjugate(q), t))) = 0;

(diff(diff(r, t), x))^2-((diff(q, t))*conjugate(q)-(diff(conjugate(q), t))*q)^2-(2*I)*(diff(diff(r, t), x))*((diff(q, t))*conjugate(q)-(diff(conjugate(q), t))*q)-4*r^2*(4-(diff(r, t))^2-4*(diff(q, t))*(diff(conjugate(q), t))) = 0

(4)

va := {q = Physics:-`*`(Physics:-`*`(Physics:-`*`(sqrt(2), c), exp(Physics:-`*`(Physics:-`*`(I, -1/sqrt(3)), t))), 1+epsilon[2]), r = Physics:-`*`(c, 1+epsilon[1]), conjugate(q) = Physics:-`*`(Physics:-`*`(Physics:-`*`(sqrt(2), c), exp(-Physics:-`*`(Physics:-`*`(I, -1/sqrt(3)), t))), 1+epsilon[2])};

{q = 2^(1/2)*c*exp(-((1/3)*I)*3^(1/2)*t)*(1+epsilon[2]), r = c*(1+epsilon[1]), conjugate(q) = 2^(1/2)*c*exp(((1/3)*I)*3^(1/2)*t)*(1+epsilon[2])}

(5)

Eq1 := simplify(eval(Eq, va), size);

4*c^2*((4/3)*c^2*(exp(((1/3)*I)*3^(1/2)*t))^2*(1+epsilon[2])^4*(exp(-((1/3)*I)*3^(1/2)*t))^2+8*exp(((1/3)*I)*3^(1/2)*t)*c*(-(1/12)*3^(1/2)*(1+epsilon[2])^2*(diff(diff(epsilon[1], t), x))+c*(1+epsilon[1])^2*((diff(epsilon[2], t))^2+(1/3)*(1+epsilon[2])^2))*exp(-((1/3)*I)*3^(1/2)*t)+(1/4)*(diff(diff(epsilon[1], t), x))^2+(1+epsilon[1])^2*(c*(diff(epsilon[1], t))-2)*(c*(diff(epsilon[1], t))+2)) = 0

(6)

indets(Eq1, function);

{epsilon[1], epsilon[2], diff(epsilon[1], t), diff(epsilon[2], t), diff(diff(epsilon[1], t), x), exp(-((1/3)*I)*3^(1/2)*t), exp(((1/3)*I)*3^(1/2)*t)}

(7)

In Eq1 replace diff(`ε`[1], t, x) by the symbol e__1tx, etc., linearize with respect to those symbols,

and then restore the symbols to their original meanings:

subs(diff(`ε`[1], t, x) = e__1tx, diff(`ε`[1], t) = e__1t, `ε`[1] = e__1, diff(`ε`[2], t) = e__2t, `ε`[2] = e__2, lhs(Eq1)); mtaylor(%, [e__1, e__1t, e__1tx, e__2, e__2t], 2); subs(e__1 = `ε`[1], e__2 = `ε`[2], e__1tx = diff(`ε`[1], t, x), %); simplify(%); collect(%, [`ε`[1], `ε`[2], diff(`ε`[1], t, x)]) = 0

Error, (in collect) cannot collect 0

 

 

``

Download Lineaize.mw

@Rouben Rostamian  I got it. But when we use eq (5) into eq (4), maple gives eq (6) after simplification. How can we know the result (eq (6)) is correct?

@Carl Love epsilon derivative raised to power 2 or higher can be removed.

@Carl Love After doing mtaylor, no derivative involve in the final equation. See attachment (where I didn't use mtaylor), there are also derivative terms like diff(epsilon[2], t), etc in equation (6). But I am not clear whether the calculations are going in the right direction or not?

Calculation.mw

@Carl Love as per your suggestion, I got linearize equation. But epsilon[1] and epsilon[2] are functions of and t. How to do it with epsilon[1]=epsilon[1](x,t) and epsilon[2]=epsilon[2](x,t)?

Calculation.mw  

@Kitonum how to get continuous system from dicrete system by sending h (lattice parameter along x-axis) approaches to zero in maple? i.e., in the attached worksheet, if we send h --> 0, then eq (5) reduces to q_{xt} + (2r)r_{x}=0. How to do that in maple?

 

 

CD_new.mw

@Kitonum D2 is the derivative?

@Carl Love very nice explanation. To understand completely I need more details on "order", "digits" and "degree". 

@acer lambda is also complex. I put it on maple and got the result. But one paoint is that maple do not cancel some terms in exponent i.e., "exp((1/2*I)*(-2*lambda*x+conjugate((lambda^2*x+2*t)/lambda)+conjugate((lambda^2*x-2*t)/lambda)))", you can see that "t" part should cancen each other but don't perform it. Also I used "with(PDEtools);
with(LinearAlgebra);
" in the worksheet. But you have used only "kernelopts(version);" in it. Why?simplificationCL18mod.mw

@Carl Love Its work. Further I want to compact and simplify the expression as much as possible. I tried it but can't get the simplest expression. simplification.mw

@Axel Vogt What is MultiSeries? Can you push a bit?

@Christian Wolinski  Please upload maple sheet

@Kitonum Also I want to calculate partial derivative not ordinary derivative

@Kitonum Physics:-diff is different from diff() command?

@acer Its really nice, but I am confuse how do you define R1,R2,R3 and R4. Can you briefly explain it, if possible

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