Maple Questions and Posts

These are Posts and Questions associated with the product, Maple


 

restart; with(VectorCalculus)

r := `<,>`(sin(t), cos(t), t)

Vector(3, {(1) = 0.2739493386e-115+0.2739493386e-115*I, (2) = 1.0-0.7504824014e-231*I, (3) = t})

(1)

``


what??

Download problem.mw

How I can take Laplace Transform from equation.

Thanks

LAPLACE

The worksheet below rolls an ellipse along the y axis with constant energy.

How can the physics be enhanced to roll the ellipse along a non-linear curve (e.g. a sine curve) with constant energy?

EllipseRoll.mw

Earlier smoothly working generation of normal distribution in v. 2019 unexpectedly shows the error:

RandV  := Statistics[RandomVariable](Normal(0, 1));
Statistics[Sample](RandV, 10);

Error, (in p) unable to convert Float(undefined) to an integer

 

Help create file Excel in ExcelTools, but error row 564?

thu_file.mw

Please help me? 

Hii,

I am using a command -NLPSolve(Ecost1, Q = 10 .. 20, initialpoint = {Q = 10}, assume = nonnegative, maximize = false). I am looking for solution that find the Q value at the minimum value of Ecost1. But Ecost1 should not go below 0. 

and also I am getting an error -Warning, initialpoint option ignored by solver.

Kindly tell how to deal with these issues.

 

Thanks

How I can do ?

Thank you.

 

Substitution of . 5,6,7) into Eqs. 1–(4), gives the new equation as functions of the generalized coordinates,
u_m,n(t);  v_m,n ( t), and w_m,n ( t). These expressions are then inserted in the Lagrange equations (see Eq. 8)) a set of N second-order coupled ordinary differential equations with both quadratic   and cubic nonlinearities.

In Eq (8) q are generalized coordinate such as uvw  and q = {`u__m,n`(t), `v__m,n`(t), `w__m,n`(t)}^T.

\where the elements of the vector,q_i are the time-dependent generalized coordinates.

L_Maple
 

U = (1/2)*(int(int(int(E*(`&PartialD;`(u(x, y, t))/`&PartialD;`(x)+(1/2)*(`&PartialD;`(w(x, y, t))/`&PartialD;`(x))^2+`&PartialD;`(w(x, y, t))/`&PartialD;`(x)*(`&PartialD;`(w__0(x, y, t))/`&PartialD;`(x))-z*(diff(w(x, y, t), x, x))+v(x, y, t)*(`&PartialD;`(v(x, y, t))/`&PartialD;`(y)+(1/2)*(`&PartialD;`(w(x, y, t))/`&PartialD;`(y))^2+`&PartialD;`(w(x, y, t))/`&PartialD;`(y)*(`&PartialD;`(w__0(x, y, t))/`&PartialD;`(y))-z*(diff(w(x, y, t), y, y))))*(`&PartialD;`(u(x, y, t))/`&PartialD;`(x)+(1/2)*(`&PartialD;`(w(x, y, t))/`&PartialD;`(x))^2+`&PartialD;`(w(x, y, t))/`&PartialD;`(x)*(`&PartialD;`(w__0(x, y, t))/`&PartialD;`(x))-z*(diff(w(x, y, t), x, x)))/(-nu^2+1)+E*(`&PartialD;`(nu(x, y, t))/`&PartialD;`(y)+(1/2)*(`&PartialD;`(w(x, y, t))/`&PartialD;`(y))^2+`&PartialD;`(w(x, y, t))/`&PartialD;`(y)*(`&PartialD;`(w__0(x, y, t))/`&PartialD;`(y))-z*(diff(w(x, y, t), y, y))+v(x, y, t)*(`&PartialD;`(u(x, y, t))/`&PartialD;`(x)+(1/2)*(`&PartialD;`(w(x, y, t))/`&PartialD;`(x))^2+`&PartialD;`(w(x, y, t))/`&PartialD;`(x)*(`&PartialD;`(w__0(x, y, t))/`&PartialD;`(x))-z*(diff(w(x, y, t), x, x))))*(`&PartialD;`(v(x, y, t))/`&PartialD;`(y)+(1/2)*(`&PartialD;`(w(x, y, t))/`&PartialD;`(y))^2+`&PartialD;`(w(x, y, t))/`&PartialD;`(y)*(`&PartialD;`(w__0(x, y, t))/`&PartialD;`(y))-z*(diff(w(x, y, t), y, y)))/(-nu^2+1)+E*(`&PartialD;`(u(x, y, t))/`&PartialD;`(y)+`&PartialD;`(v(x, y, t))/`&PartialD;`(x)+`&PartialD;`(w(x, y, t))/`&PartialD;`(x)*(`&PartialD;`(w(x, y, t))/`&PartialD;`(y))+`&PartialD;`(w__0(x, y, t))*`&PartialD;`(w(x, y, t))/(`&PartialD;`(x)*`&PartialD;`(y))+`&PartialD;`(w__0(x, y, t))*`&PartialD;`(w(x, y, t))/(`&PartialD;`(x)*`&PartialD;`(y))-2*z*(diff(w(x, y, t), x, y)))^2/(2*(1+nu))+E*l^2*(diff(w(x, y, t), x, y))^2/(1+nu)+E*l^2*(diff(w(x, y, t), x, y))^2/(1+nu)+E*l^2*(diff(w(x, y, t), y, y)-(diff(w(x, y, t), x, x)))^2/(2*(1+nu))+E*l^2*(diff(v(x, y, t), y, y)-(diff(u(x, y, t), x, x)))^2/(8*(1+nu))+E*l^2*(diff(v(x, y, t), x, y)-(diff(u(x, y, t), y, y)))^2/(8*(1+nu)), z = -(1/2)*h .. (1/2)*h), y = 0 .. b), x = 0 .. a))

U = (1/2)*(int(int((1/12)*(-E*(-v(x, y, t)*(diff(diff(w(x, y, t), y), y))-(diff(diff(w(x, y, t), x), x)))*(diff(diff(w(x, y, t), x), x))/(-nu^2+1)-E*(-v(x, y, t)*(diff(diff(w(x, y, t), x), x))-(diff(diff(w(x, y, t), y), y)))*(diff(diff(w(x, y, t), y), y))/(-nu^2+1)+4*E*(diff(diff(w(x, y, t), x), y))^2/(2+2*nu))*h^3+E*(`&PartialD;`(u(x, y, t))/`&PartialD;`(x)+(1/2)*`&PartialD;`(w(x, y, t))^2/`&PartialD;`(x)^2+`&PartialD;`(w(x, y, t))*`&PartialD;`(w__0(x, y, t))/`&PartialD;`(x)^2+v(x, y, t)*(`&PartialD;`(v(x, y, t))/`&PartialD;`(y)+(1/2)*`&PartialD;`(w(x, y, t))^2/`&PartialD;`(y)^2+`&PartialD;`(w(x, y, t))*`&PartialD;`(w__0(x, y, t))/`&PartialD;`(y)^2))*(`&PartialD;`(u(x, y, t))/`&PartialD;`(x)+(1/2)*`&PartialD;`(w(x, y, t))^2/`&PartialD;`(x)^2+`&PartialD;`(w(x, y, t))*`&PartialD;`(w__0(x, y, t))/`&PartialD;`(x)^2)*h/(-nu^2+1)+E*(`&PartialD;`(nu(x, y, t))/`&PartialD;`(y)+(1/2)*`&PartialD;`(w(x, y, t))^2/`&PartialD;`(y)^2+`&PartialD;`(w(x, y, t))*`&PartialD;`(w__0(x, y, t))/`&PartialD;`(y)^2+v(x, y, t)*(`&PartialD;`(u(x, y, t))/`&PartialD;`(x)+(1/2)*`&PartialD;`(w(x, y, t))^2/`&PartialD;`(x)^2+`&PartialD;`(w(x, y, t))*`&PartialD;`(w__0(x, y, t))/`&PartialD;`(x)^2))*(`&PartialD;`(v(x, y, t))/`&PartialD;`(y)+(1/2)*`&PartialD;`(w(x, y, t))^2/`&PartialD;`(y)^2+`&PartialD;`(w(x, y, t))*`&PartialD;`(w__0(x, y, t))/`&PartialD;`(y)^2)*h/(-nu^2+1)+E*(`&PartialD;`(u(x, y, t))/`&PartialD;`(y)+`&PartialD;`(v(x, y, t))/`&PartialD;`(x)+`&PartialD;`(w(x, y, t))^2/(`&PartialD;`(x)*`&PartialD;`(y))+2*`&PartialD;`(w__0(x, y, t))*`&PartialD;`(w(x, y, t))/(`&PartialD;`(x)*`&PartialD;`(y)))^2*h/(2+2*nu)+2*E*l^2*(diff(diff(w(x, y, t), x), y))^2*h/(1+nu)+E*l^2*(diff(diff(w(x, y, t), y), y)-(diff(diff(w(x, y, t), x), x)))^2*h/(2+2*nu)+E*l^2*(diff(diff(v(x, y, t), y), y)-(diff(diff(u(x, y, t), x), x)))^2*h/(8+8*nu)+E*l^2*(diff(diff(v(x, y, t), x), y)-(diff(diff(u(x, y, t), y), y)))^2*h/(8+8*nu), y = 0 .. b), x = 0 .. a))

(1)

T = rho*h*(int(int((`&PartialD;`(u(x, y, t))/`&PartialD;`(t))^2+(`&PartialD;`(v(x, y, t))/`&PartialD;`(t))^2+(`&PartialD;`(w(x, y, t))/`&PartialD;`(t))^2, y = 0 .. b), x = 0 .. a))

T = rho*h*(int(int(`&PartialD;`(u(x, y, t))^2/`&PartialD;`(t)^2+`&PartialD;`(v(x, y, t))^2/`&PartialD;`(t)^2+`&PartialD;`(w(x, y, t))^2/`&PartialD;`(t)^2, y = 0 .. b), x = 0 .. a))

(2)

F = (1/2)*c*(int(int((`&PartialD;`(u(x, y, t))/`&PartialD;`(t))^2+(`&PartialD;`(v(x, y, t))/`&PartialD;`(t))^2+(`&PartialD;`(w(x, y, t))/`&PartialD;`(t))^2, y = 0 .. b), x = 0 .. a))

F = (1/2)*c*(int(int(`&PartialD;`(u(x, y, t))^2/`&PartialD;`(t)^2+`&PartialD;`(v(x, y, t))^2/`&PartialD;`(t)^2+`&PartialD;`(w(x, y, t))^2/`&PartialD;`(t)^2, y = 0 .. b), x = 0 .. a))

(3)

W = int(int(w(x, y, t)*f__1(x, y, t)*cos(omega*t), y = 0 .. b), x = 0 .. a)

W = int(int(w(x, y, z)*f__1(x, y, z)*cos(omega*t), y = 0 .. b), x = 0 .. a)

(4)

u(x, y, t) = sum(sum(`u__m,n`(t)*sin(m*Pi*x/a)*sin(n*Pi*y/b), n = 1 .. N), m = 1 .. M)

u(x, y, t) = -(1/4)*(cos(Pi*y*N/b)*cos(Pi*x/a)*sin(Pi*y/b)*sin((M+1)*Pi*x/a)-cos(Pi*y*N/b)*cos((M+1)*Pi*x/a)*sin(Pi*x/a)*sin(Pi*y/b)+cos(Pi*x/a)*sin(Pi*y*N/b)*cos(Pi*y/b)*sin((M+1)*Pi*x/a)-cos((M+1)*Pi*x/a)*sin(Pi*x/a)*sin(Pi*y*N/b)*cos(Pi*y/b)-cos(Pi*y*N/b)*sin(Pi*y/b)*sin((M+1)*Pi*x/a)-cos(Pi*x/a)*sin(Pi*y*N/b)*sin((M+1)*Pi*x/a)-cos(Pi*x/a)*sin(Pi*y/b)*sin((M+1)*Pi*x/a)+cos((M+1)*Pi*x/a)*sin(Pi*x/a)*sin(Pi*y*N/b)+sin(Pi*x/a)*sin(Pi*y/b)*cos((M+1)*Pi*x/a)-sin(Pi*y*N/b)*cos(Pi*y/b)*sin((M+1)*Pi*x/a)+sin(Pi*y*N/b)*sin((M+1)*Pi*x/a)+sin(Pi*y/b)*sin((M+1)*Pi*x/a))*`u__m,n`(t)/((cos(Pi*x/a)-1)*(cos(Pi*y/b)-1))+(1/4)*(-cos(Pi*y*N/b)*sin(Pi*y/b)*sin(Pi*x/a)-sin(Pi*y*N/b)*cos(Pi*y/b)*sin(Pi*x/a)+sin(Pi*y*N/b)*sin(Pi*x/a)+sin(Pi*y/b)*sin(Pi*x/a))*`u__m,n`(t)/((cos(Pi*x/a)-1)*(cos(Pi*y/b)-1))

(5)

v(x, y, t) = sum(sum(`v__m,n`(t)*sin(m*Pi*x/a)*sin(n*Pi*y/b), n = 1 .. N), m = 1 .. M)

v(x, y, t) = -(1/4)*(cos(Pi*y*N/b)*cos(Pi*x/a)*sin(Pi*y/b)*sin((M+1)*Pi*x/a)-cos(Pi*y*N/b)*cos((M+1)*Pi*x/a)*sin(Pi*x/a)*sin(Pi*y/b)+cos(Pi*x/a)*sin(Pi*y*N/b)*cos(Pi*y/b)*sin((M+1)*Pi*x/a)-cos((M+1)*Pi*x/a)*sin(Pi*x/a)*sin(Pi*y*N/b)*cos(Pi*y/b)-cos(Pi*y*N/b)*sin(Pi*y/b)*sin((M+1)*Pi*x/a)-cos(Pi*x/a)*sin(Pi*y*N/b)*sin((M+1)*Pi*x/a)-cos(Pi*x/a)*sin(Pi*y/b)*sin((M+1)*Pi*x/a)+cos((M+1)*Pi*x/a)*sin(Pi*x/a)*sin(Pi*y*N/b)+sin(Pi*x/a)*sin(Pi*y/b)*cos((M+1)*Pi*x/a)-sin(Pi*y*N/b)*cos(Pi*y/b)*sin((M+1)*Pi*x/a)+sin(Pi*y*N/b)*sin((M+1)*Pi*x/a)+sin(Pi*y/b)*sin((M+1)*Pi*x/a))*`v__m,n`(t)/((cos(Pi*x/a)-1)*(cos(Pi*y/b)-1))+(1/4)*(-cos(Pi*y*N/b)*sin(Pi*y/b)*sin(Pi*x/a)-sin(Pi*y*N/b)*cos(Pi*y/b)*sin(Pi*x/a)+sin(Pi*y*N/b)*sin(Pi*x/a)+sin(Pi*y/b)*sin(Pi*x/a))*`v__m,n`(t)/((cos(Pi*x/a)-1)*(cos(Pi*y/b)-1))

(6)

w(x, y, t) = sum(sum(`w__m,n`(t)*sin(m*Pi*x/a)*sin(n*Pi*y/b), n = 1 .. N), m = 1 .. M)

w(x, y, t) = -(1/4)*(cos(Pi*y*N/b)*cos(Pi*x/a)*sin(Pi*y/b)*sin((M+1)*Pi*x/a)-cos(Pi*y*N/b)*cos((M+1)*Pi*x/a)*sin(Pi*x/a)*sin(Pi*y/b)+cos(Pi*x/a)*sin(Pi*y*N/b)*cos(Pi*y/b)*sin((M+1)*Pi*x/a)-cos((M+1)*Pi*x/a)*sin(Pi*x/a)*sin(Pi*y*N/b)*cos(Pi*y/b)-cos(Pi*y*N/b)*sin(Pi*y/b)*sin((M+1)*Pi*x/a)-cos(Pi*x/a)*sin(Pi*y*N/b)*sin((M+1)*Pi*x/a)-cos(Pi*x/a)*sin(Pi*y/b)*sin((M+1)*Pi*x/a)+cos((M+1)*Pi*x/a)*sin(Pi*x/a)*sin(Pi*y*N/b)+sin(Pi*x/a)*sin(Pi*y/b)*cos((M+1)*Pi*x/a)-sin(Pi*y*N/b)*cos(Pi*y/b)*sin((M+1)*Pi*x/a)+sin(Pi*y*N/b)*sin((M+1)*Pi*x/a)+sin(Pi*y/b)*sin((M+1)*Pi*x/a))*`w__m,n`(t)/((cos(Pi*x/a)-1)*(cos(Pi*y/b)-1))+(1/4)*(-cos(Pi*y*N/b)*sin(Pi*y/b)*sin(Pi*x/a)-sin(Pi*y*N/b)*cos(Pi*y/b)*sin(Pi*x/a)+sin(Pi*y*N/b)*sin(Pi*x/a)+sin(Pi*y/b)*sin(Pi*x/a))*`w__m,n`(t)/((cos(Pi*x/a)-1)*(cos(Pi*y/b)-1))

(7)

diff(`&PartialD;`(T(x, y, t))/`&PartialD;`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("&period;"),none(),none())`), t)-`&PartialD;`(T(x, y, t))/`&PartialD;`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("&period;"),none(),none())`)+`&PartialD;`(U(x, y, t))/`&PartialD;`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("&period;"),none(),none())`)+`&PartialD;`(U(x, y, t))/`&PartialD;`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("&period;"),none(),none())`)+`&PartialD;`(F(x, y, t))/`&PartialD;`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("&period;"),none(),none())`) = `&PartialD;`(W(x, y, t))/`&PartialD;`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("&period;"),none(),none())`), j = 1, () .. (), N

(D(`&PartialD;`))(T(x, y, t))*(diff(T(x, y, t), t))/`&PartialD;`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("&period;"),none(),none())`)-`&PartialD;`(T(x, y, t))/`&PartialD;`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("&period;"),none(),none())`)+2*`&PartialD;`(U(x, y, t))/`&PartialD;`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("&period;"),none(),none())`)+`&PartialD;`(F(x, y, t))/`&PartialD;`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("&period;"),none(),none())`) = `&PartialD;`(W(x, y, t))/`&PartialD;`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("&period;"),none(),none())`), j = 1, () .. (), N

(8)

NULL


 

Download L_Maple

 

 

I want to made a comparison via plots of RK-4, NSFD and LWM.

I have noticed a few times now with Maple 2019. It looses kernel connection when it is sitting there idly. This time I observed it. Had saved a document after an intensive calculation. The memory used was about 30Gig. shortly after saving the cpu fan was running hard. I checked task manager and cpu was cycling to 100%, it was mserever. Then the memory usage droped to about 6gig and message as shown. During this time Maple screen down in the LH corner displayed "Ready", so it didn't think it was doing anything.
 

Hello,

How I can take variation from left-hand side of  5, and reach to right-hand side of  5. After by using integral by part obtained  7?

Thank you

Maple pdsolve supports periodic boundary conditions. So I was hoping it will be able to solve the heat PDE inside disk with periodic boundary conditions. But I am not able to make it work. 

Is there a trick to make Maple solve this, is there something I need to add or adjust something else? or it is just the functionality is not currently implemented?

This is what I tried

restart;

pde := diff(u(r,theta,t),t)=diff(u(r,theta,t),r$2) + 1/r*diff(u(r,theta,t),r)+1/r^2*diff(u(r,theta,t),theta$2);
bc1 := u(a,theta,t)=0;
bc2 := eval(diff(u(r,theta,t),theta),theta=-Pi)=eval(diff(u(r,theta,t),theta),theta=Pi);
bc3 := u(r,-Pi,t)=u(r,Pi,t);
ic  := u(r,theta,0)=f(r,theta);
sol := pdsolve([pde, bc1,bc2,bc3, ic], u(r, theta, t), HINT = boundedseries(r = 0)) assuming a>0,r>0

I solved this analytically by hand using standard separation of variables method. The issue of telling Maple the solution is bounded at center of disk, I assume is being handled automatically by the HINT=boundedseries(r = 0).

If I remove the hint, it also does not solve it. 

Maple 2019, Physics package 338

For this problem

I'd like to see if Maple can give, or simplify the solution it now gives to look like this solution 

The one it currently gives is

restart;

pde:=diff(w(x,t),t)+c*diff(w(x,t),x)=0; 
ic:=w(x,0)=f(x);
bc:=w(0,t)=h(t);
sol:=pdsolve([pde,ic,bc],w(x,t))  assuming t>0,x>0,c>0

 

And I did not know how to simplify it or obtain the simpler one. I tried strip and TWS hints.  I also do not understand why Maple gives an integral with 0 as upper limit there (the second integral).

Using Physics package cloud version 338 and Maple 2019. On windows 10.

Thank you

Is there a way to take the laplacian of 1/r and get the "physics" answer of -4*pi*delta(\vec{r})?

with(plots):R := 5; alpha := (1/9)*Pi;
C1 := plot([R*cos(t), R*sin(t), t = 0 .. 2*Pi], color = blue);
A := [R*cos(alpha), R*sin(alpha)]; B := [R*cos(alpha+Pi), R*sin(alpha+Pi)]; AB := plot([A, B], scaling = constrained);
display({AB, C1}, scaling = constrained);# bad drawing

 

https://aws.amazon.com/getting-started/projects/deploy-elastic-hpc-cluster/

is it possible to use maple on high performance clusters?

i can only think to use c program to call cmaple with MPI in linux to use high performance clusters.

is there any other official method to do this?

if i upload my maple 2015 version to amazon for this computing, will it used up all license in this first chance of installation leading to that i can not install maple 2015 linux version to other machine?

https://docs.aws.amazon.com/AWSEC2/latest/WindowsGuide/ConfigWindowsHPC.html#ComputeNode

which virtual machine should i install the maple 12? on one virtual machine or all compute nodes?

 

how many compute nodes are need to compute dsolve 100,000 systems which may or may not have solution in maple 12?

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