Maple 2018 Questions and Posts

These are Posts and Questions associated with the product, Maple 2018

Hi,
I have an equation and I want to solve it parametrically to find x , but I couldn't do that with "solve" command. (I know x should be  real and positive). What should I do?
Root_of.mw

Hi. 

I am trying to solve a polynomial equation but the structure leads Maple to return a trivial solution and the other solutions are given as a RootOf expression. The equation involves a single variable, x, that is raised to a power, b and a multiplier, a (both are positive-valued). Please see attached worksheet.

I have not encountered this before and I cannot find a way to get to an explicit solution. Perhaps it is not possible (?).

Does anybody know how to deal with this? 

Thanks in advance ...

Roots_of_a_Polynomial_MaplePrimes.mw 

Good day, all.

I would like to explore the structure of the discrete modified form of the logistic equation.

In particular, I wish to plot the logistic-map to investigate the bifurcations of the system.

Is there a routine available in Maple that I can use?

I would like to consider the standard logistic equation with the inclusion of a shape parameter, m, introduced as a power law.

That is:

f(x) = a*x*(1-x^m)

where a > 0 denotes the growth rate, and m > 0  is a shape parameter. I wish to fix the value of a and take m to be the bifurcation parameter (so the logistic map would show m versus x for any given a).

Please note: The standard logistic equation (in discrete form) is given by f(x) = a*x*(1-x)

I would be grateful for any advice and support you can provide and I thank you for taking the time to read this.

Drear freinds,

I want to simplify f (a long experssion) in the form of f2. How to determine M1^2 and M0^2?

f1.mw

Hi,
How can I remove the mentioned error in attached worksheet?

s1.mw

Dear sir,

In the given problem, eta = 0 to 20, I want the table value of eta for a step size of 1000 (0 to 20 in thousand parts).

i have calculated only for one value, zero

Download Demo_paper_work.mw

Consider the equation  (2^x)*(27^(1/x)) = 24  for which we need to find the exact values ​​of its real roots. This is not difficult to solve by hand if you first take the logarithm of this equation to any base, after which the problem is reduced to solving a quadratic equation. But the  solve  command fails to solve this equation and returns the result in RootOf form. The problem is solved if we first ask Maple to take the logarithm of the equation. I wonder if the latest versions of Maple also do not directly address the problem?

restart;
Eq:=2^x*27^(1/x)=24:
solve(Eq, x, explicit);

map(ln, Eq); # Taking the logarithm of the equation
solve(%, x);
simplify({%}); # The final result

                  

 

How to get the animation graphs for eta =0..10

NULL

restart; with(plots)

``

ga := .2; Gc := .2; n := 2; Sc := .5; Kp := .2; Q := 0.5e-1; Gr := .2

A := .1Pr := 6.2; Nt := .2; alpha := .1; Rd := 1.5; M := .5; E1 := .3; Ec := .3; Thetap := .2; Nb := .2

NULL

a1 := 1.301348831
NULL

a2 := 1.298194584
a3 := .9728927630; a4 := .9161173998

a5 := 1.316893419

a6 := 1.333333333

 

 

OdeSys := a1*((diff(f(eta), eta, eta, eta))*(2*eta*ga+1)+2*(diff(f(eta), eta, eta))*ga)/a2+A^2-(diff(f(eta), eta))^2+f(eta)*(diff(f(eta), eta, eta))-a1*Kp*(diff(f(eta), eta))/a2-a6*M*(diff(f(eta), eta))/a2+a4*(Theta(eta)*Gr+Phi(eta)*Gc)/a2, f(eta)*(diff(Theta(eta), eta))+a5*(1+4*Rd*(1/3))*((diff(Theta(eta), eta, eta))*(2*eta*ga+1)+2*(diff(Theta(eta), eta))*ga)/(a3*Pr)+(diff(f(eta), eta, eta))^2*(2*eta*ga+1)*Ec*a1/a3+Theta(eta)*Q/a3, f(eta)*(diff(Phi(eta), eta))+((diff(Phi(eta), eta, eta))*(2*eta*ga+1)+2*(diff(Phi(eta), eta))*ga)/Sc+Nt*((diff(Theta(eta), eta, eta))*(2*eta*ga+1)+2*(diff(Theta(eta), eta))*ga)/(Nb*Sc)-Kr*(1+Theta(eta)*(Thetap-1))^n*exp(-E1/(1+Theta(eta)*(Thetap-1))); Cond := f(0) = 0, (D(f))(0) = 1, a5*(D(Theta))(0) = -alpha*(Theta(0)-1), Phi(0) = 1, (D(f))(10) = A, Theta(10) = 0, Phi(10) = 0

KrVals := [0.1e-1, .1, .2, .3]

for j to numelems(KrVals) do Ans[j] := dsolve(eval([OdeSys, Cond], Kr = KrVals[j]), numeric, output = listprocedure) end do

``

``

with(plots):
  cols := [red, blue,green, black]:

 plotA:= display
  ( [ seq
      ( odeplot
        ( Ans[k],[eta,D(f)(eta)],
          eta=0..10,
          color=cols[k]
        ),
        k=1..numelems(KrVals)
      )
    ],
    'axes'= 'boxed',labels=[eta,'f(eta)']
  );

 

with(plots):
  cols := [red, blue, green,black]:

plotC:= display( [ seq( odeplot
        ( Ans[k],[eta,Theta(eta)],
          eta=0..10,
          color=cols[k]
        ),
        k=1..numelems(KrVals)
      )
    ],
    'axes'= 'boxed',labels=[eta,'Theta(eta)']
  );

 

 

``

plotA1 := display(seq(plot3d(r*(eval(f(:-eta), Ans[k]))(eta), eta = 0 .. 10, r = -5 .. 5, color = cols[k]), k = 1 .. nops(KrVals)), linestyle = "solid", style = contour, thickness = 1)

 
 

 

Download ode_plots_animation_graphs.mw

i need like this demo plot  

Download Ode_New_TWO_phase.mw

I want like this plots here two phase are there boundary is -2 to0 and 0 to 2

f(±2)=0,g(±2)=0,f(0)=1,h(±2)=1,H(±2)=1

Is it possible to simplify the following relatively simple expression  (10*(5+sqrt(41)))/(sqrt(70+10*sqrt(41))*sqrt(130+10*sqrt(41)))  using 1-2 standard commands  simplify , combine, radnormal  and so on?   I was unable to do this in Maple 2018. Maybe newer versions of Maple will be able to handle this. I managed to simplify it in 3 steps:

expr:=(10*(5+sqrt(41)))/(sqrt(70+10*sqrt(41))*sqrt(130+10*sqrt(41)));
sqrt(simplify(expr^2));

                              

Please Help me to solve this problem 
 

Download ode_Plots_error.mw

exact_solution_error.mw

P1=(D1*ga+C1)/(1+ga) and P2= (D1-C1)/(1+ga), How to substitute in the solution of U_exact and Theta_exact

Please help me to solve

Dear sir ,

I have implemented Dsolve method the code was executed, but i need to apply Kellor Box method to solve the ODES 

Please can any one help how to implement? 

because there is no post regarding the Kellor box method. 

restart; with(plots)

``

S := 1; Rd := .1; delta := .1; Hs := 1; Sc := .1; Pr := 6.8; n := 1; Rc := .1; E := .1; M := 1

NULL

 

OdeSys := a1*(diff(f(eta), eta, eta, eta, eta))/a2-S*(3*(diff(f(eta), eta, eta))+eta*(diff(f(eta), eta, eta, eta))+(diff(f(eta), eta))*(diff(f(eta), eta, eta))-f(eta)*(diff(f(eta), eta, eta, eta)))-a5*M*(diff(f(eta), eta, eta))/a2-a1*Kp*(diff(f(eta), eta, eta))/a2 = 0, (a4+4*Rd)*(diff(Theta(eta), eta, eta))+12*Rd*delta*((diff(Theta(eta), eta))*(diff(Theta(eta), eta))+Theta(eta)*(diff(Theta(eta), eta, eta)))+Hs*Theta(eta)-a3*Pr*S*(diff(Theta(eta), eta))*(eta-f(eta)) = 0, diff(Phi(eta), eta, eta)-S*Sc*(diff(Phi(eta), eta))*(eta-f(eta))-Sc*Rc*(1+delta*Theta(eta))^n*Phi(eta)*exp(-E/(1+delta*Theta(eta))) = 0; Cond := f(0) = 0, ((D@@2)(f))(0) = 0, (D(Theta))(0) = 0, (D(Phi))(0) = 0, f(1) = 1, (D(f))(1) = 0, Theta(1) = 1, Phi(1) = 1

   

KpVals := [1, 2, 3, 4]

for j to numelems(KpVals) do Ans[j] := dsolve(eval([OdeSys, Cond], Kp = KpVals[j]), numeric, output = listprocedure) end do

 

with(plots):
 cols := [red, blue, black,green]:

 plotA:= display
  ( [ seq
      ( odeplot
        ( Ans[k],[eta,(f(eta))],
          eta=0..1,
          color=cols[k]
        ),
        k=1..numelems(KpVals)
      )
    ],linestyle = "solid",
    'axes'= 'boxed',labels=[eta,'f(eta)'],labelfont=[TIMES,BOLD,16]
  );
 

with(plots):
  cols := [red, blue, black,green]:

plotB:= display( [ seq( odeplot
        ( Ans[k],[eta,Theta(eta)],
          eta=0..1,
          color=cols[k]
        ),
        k=1..numelems(KpVals)
      )
    ],linestyle = "solid",
    'axes'= 'boxed',labels=[eta,'Phi(eta)'],labelfont=[TIMES,BOLD,16]
  );

 

 

with(plots):
  cols := [red, blue, black,green]:

plotC:= display( [ seq( odeplot
        ( Ans[k],[eta,Phi(eta)],
          eta=0..1,
          color=cols[k]
        ),
        k=1..numelems(KpVals)
      )
    ],linestyle = "solid",
    'axes'= 'boxed',labels=[eta,'Phi(eta)'],labelfont=[TIMES,BOLD,16]
  );

 

with(plots):
 cols := [red, blue, black,green]:

 plotA:= display
  ( [ seq
      ( odeplot
        ( Ans[k],[eta,(diff(f(eta),eta))],
          eta=0..1,
          color=cols[k]
        ),
        k=1..numelems(KpVals)
      )
    ],linestyle = "solid",
    'axes'= 'boxed',labels=[eta,"f '(eta)"],labelfont=[TIMES,BOLD,16]
  );

 

 

 

Download kellor_box_method.mw

Hi,
I have a problem and I haven't been able to solve it yet. I want to solve an ordinary diffrential equation similar to
                                                                                                   (dphi/dxi)^2+2*V(phi)=0
and plot phi versus xi for a the following conditions:
1) V(phi)=dphi/dxi=0 at (phi=0,phi_m) and
2) dV(phi)/dphi=0 at phi=phi_m and 
3) d^2V(phi)/dphi^2=0 at both phi=0 and phi=phi_m.
How can I do this by Maple?(see the attached file)
w1.mw

If we solve the equation  x^x=1/sqrt(2)  in Maple 2018.2, then Maple returns only one solution  x=1/2 , although this equation has 2 solutions  x=1/2  or  x=1/4 . This can be seen if you plot graphs or , for example, solve by the  Student:-Calculus1:-Roots  command. In this case, the root  x=1/4   is returned only as a numerical approximation. I wonder if this bug has been fixed in the latest versions of Maple?

restart;
solve(x^x=1/sqrt(2));
Student:-Calculus1:-Roots(x^x=1/sqrt(2));

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