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dharr

Dr. David Harrington

8205 Reputation

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20 years, 337 days
University of Victoria
Professor or university staff
Victoria, British Columbia, Canada

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Contact Dr. David Harrington
I am a retired professor of chemistry at the University of Victoria, BC, Canada. My research areas are electrochemistry and surface science. I have been a user of Maple since about 1990.

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These are answers submitted by dharr

missing * and extra spaces...

dharr 8205
October 18 2019
0 1

Changes in red.
 

> 

 RK2skritt:=proc(FR::procedure,xo,yo,vxo,vyo,h)
 local x, y, vx, vy, r, kx1, kx2, kx3, kx4, ky1, ky2, ky3, ky4, lx1, lx2, lx3, lx4, ly1, ly2, ly3, ly4, tmp;
 r:=sqrt(xo*xo+yo*yo);
 tmp:=-(h*FR(r))/(r);
 lx1:=h*vxo;
 ly1:=h*vyo;
 kx1:=tmp*xo;
 ky1:=tmp*yo;
 lx2:=h*(vxo+0.5*kx1);
 ly2:=h*(vyo+0.5*ky1);
 r:=sqrt((xo+0.5*lx1)^(2)+(yo+0.5*ly1)^(2));
 tmp:=-(h*FR(r))/(r);
 kx2:=tmp*(xo+0.5*lx1);
 ky2:=tmp*(yo+0.5*ly1);
 lx3:=h*(vxo+0.5*kx2);
 ly3:=h*(vyo+0.5*ky2);
 r:=sqrt((xo+0.5*lx2)^(2)+(yo+0.5*ly2)^(2));
 tmp:=-(h*FR(r))/(r);
 kx3:=tmp*(xo+0.5*lx2);
 ky3:=tmp*(yo+0.5*ly2);
 lx4:=h*(vxo+kx3);
 ly4:=h*(vyo+ky3);
 r:=sqrt((xo+lx3)^(2)+(yo+ly3)^(2));
 tmp:=-(h*FR(r))/(r);
 kx4:=tmp*(xo+lx3);
 ky4:=tmp*(yo+ly3);  #edit - one more here
 x:=xo+(lx1+2*lx2+2*lx3+lx4)/(6);
 y:=yo+(ly1+2*ly2+2*ly3+ly4)/(6);
 vx:=vxo+(kx1+2*kx2+2*kx3+kx4)/(6);
 vy:=vyo+(ky1+2*ky2+2*ky3+ky4)/(6);
 [x,y,vx,vy]; end proc:
> 

 

 

Download proc.mw

Works...

dharr 8205
October 17 2019
1 11

There is a missing *. The differentiation works in Maple 2017.
 

Download Diff.mw

 

 

a start...

dharr 8205
October 02 2019
2 1
> 

restart

Case 1

> 

de1:=diff(U(x),x,x)=0;
bc1:=U(0)=10,U(L)=20;
ans1:=dsolve({de1,bc1},U(x));

diff(diff(U(x), x), x) = 0

U(0) = 10, U(L) = 20

U(x) = 10*x/L+10

Case 2

> 

de2:=diff(U(x),x,x)=-x^2;
bc2:=U(0)=T,D(U)(L)=0;
ans2:=dsolve({de2,bc2},U(x));

diff(diff(U(x), x), x) = -x^2

U(0) = T, (D(U))(L) = 0

U(x) = -(1/12)*x^4+(1/3)*L^3*x+T

> 

 

> 

 


 

Download rod.mw

use Vectors...

dharr 8205
October 01 2019
0 2

Like this?

NULL

> 

restart
> 

n := 4; C := proc (K) options operator, arrow; Vector([seq(cos(2*Pi*K*l/n), l = 0 .. n-1)]) end proc; S := proc (K) options operator, arrow; Vector([seq(sin(2*Pi*K*l/n), l = 0 .. n-1)]) end proc

4

proc (K) options operator, arrow; Vector([seq(cos(2*Pi*K*l/n), l = 0 .. n-1)]) end proc

proc (K) options operator, arrow; Vector([seq(sin(2*Pi*K*l/n), l = 0 .. n-1)]) end proc

> 

C(2); S(3)

Vector[column]([[1], [-1], [1], [-1]])

Vector[column]([[0], [-1], [0], [1]])

These two are orthogonal

> 

C(2).S(3)

0

> 

C(2).C(2)

4

> 

 

NULL


 

Download orthogonal.mw

laplace and solve...

dharr 8205
September 22 2019
1 0

I did G11; the others are similar

Transfer_function.mw

Document properties...

dharr 8205
September 18 2019
2 1

In the File Menu under Document Properties, you should have an attribute named Active which is false for a regular help page and true for an active worksheet.

2-D math...

dharr 8205
September 15 2019
2 0

in 1D input:

ode:=diff(y(x),x,x)=2*y(x)+1;
dsolve(ode);
 

does work, so this is probably incorrect entry of the derivative in 2-D math. [Edit: actually the error message confirms this.] Use the right-click menu 2-D math/convert to 1D math input and see if it looks like the above. If not, use 1-D input (ctrl-M) before the input, or use the expression palette for the derivative and use "^" to raise the powers of 2.

symmetry...

dharr 8205
September 13 2019
0 0

I don't understand exactly what you mean - if you want real-only frequency domain, you need the time domain array to have the real part  symmetric about the centre and the imaginary part antisymmetric about the centre.

Edit: And here is a complex example that works pretty much as I would expect:

> 

restart;
> 

with(DiscreteTransforms):
> 

N:=2^10;df:=.2;dt:=1/df;T1:=500.;

1024

.2

5.000000000

500.

> 

tvec:=Vector(N,i->i*dt):
fvec:=Vector(N,i->i*df):
> 

fidr:=Vector(N,i->cos(2*Pi*df*i)*exp(-dt*i/T1)):
fidi:=Vector(N,i->sin(2*Pi*df*i)*exp(-dt*i/T1)):
fid:=fidr+I*fidi:
> 

plot(tvec,fidr);

> 

spec:=FourierTransform(fid):
> 

plot(fvec,map(abs,spec)); #magnitude

> 

plot(fvec,map(Im,spec));  #imaginary part

> 

plot(fvec,map(Re,spec));  #real part

> 

 


 

Download fid.mw

works in Maple 2017...

dharr 8205
September 12 2019
2 1


 

> 

restart;
with(ThermophysicalData);
with(ThermophysicalData[CoolProp]);
fluid := "R717"; T__in := 310; p__in := 11*10^5; v__in := 10;  A := 0.005;
h__in := Property(massspecificenthalpy, T = T__in, P = p__in, fluid);
rho__in := Property(density, T = T__in, P = p__in, fluid);
m := A*v__in*rho__in;
p__out := 2*10^5;
eq1 := h__out = Property("massspecificenthalpy", "temperature" = T__out, "P" = p__out, fluid);
eq2 := rho__out = Property("D", "temperature" = T__out, "P" = p__out, fluid);
eq3 := h__in + v__in^2/2 = h__out + v__out^2/2;
eq4 := m = A*v__out*rho__out;

res := fsolve({eq1, eq2, eq3, eq4});

[CoolProp, PHTChart, Property, PsychrometricChart, TemperatureEntropyChart]

[HAPropsSI, PhaseSI, Props1SI, PropsSI]

"R717"

310

1100000

10

0.5e-2

1655590.94731116970

8.15132757401520891

.4075663787

200000

h__out = ThermophysicalData:-Property("massspecificenthalpy", "temperature" = T__out, "P" = 200000, "R717")

rho__out = ThermophysicalData:-Property("D", "temperature" = T__out, "P" = 200000, "R717")

1655640.947 = h__out+(1/2)*v__out^2

.4075663787 = 0.5e-2*v__out*rho__out

{T__out = 285.0179004, h__out = 1654113.289, v__out = 55.27490598, rho__out = 1.474688637}

> 

 


 

Download CoolProp.mw

Open file...

dharr 8205
September 11 2019
1 0

Under the file/open menu, you can open a Mathematica notebook. I haven't tried this, but presumably it automates what can be done by the commands that @tomleslie pointed out.

No simple way for longest path...

dharr 8205
September 09 2019
2 0

See the help for the GraphTheory package, ?GraphTheory. This is a hard problem (see the Wikipedia page), and as far as I can see, Maple has no direct commands for this. Some of the commands there may help you, but if you want an efficient routine you would probably need to write something for the class of graphs you are interested in.

parameter sequence op...

dharr 8205
September 06 2019
2 0


 

> 

f := (a,b,c)->a*b*c;

proc (a, b, c) options operator, arrow; a*b*c end proc

> 

F:=proc(f,val) local fargs;
fargs:=op(1,eval(f)); #parameter sequence
f(val,fargs[2..]);
end proc;

proc (f, val) local fargs; fargs := op(1, eval(f)); f(val, fargs[2 .. ()]) end proc

> 

F(f,3);

3*b*c

> 

 


 

Download fargs.mw

division by zero...

dharr 8205
August 27 2019
2 2

If you don't have tau = 0, then Maple will carry out operations that are generally valid, but may not be for tau=0. So if you then set tau=0, you may get an invalid result. For example here, the 2,2 entry has a denominator that becomes zero if tau=0, and so a division by zero problem arises, which would not have done if Maple had manipulated knowing tau=0.

assuming positive...

dharr 8205
August 20 2019
0 1

To assume everything is positive works here
 

f := (Pi*x+2*c+2*m)/(`μ__c`*S)+2*`ε`/(`μ__a`*S)+(Pi*x+2*c)/(`μ__s`*S)

(Pi*x+2*c+2*m)/(mu__c*S)+2*epsilon/(mu__a*S)+(Pi*x+2*c)/(mu__s*S)

g := simplify(f, symbolic)

(((Pi*x+2*c+2*m)*mu__s+2*mu__c*((1/2)*Pi*x+c))*mu__a+2*epsilon*mu__c*mu__s)/(mu__c*S*mu__a*mu__s)

`assuming`([int(1/g, x = 0 .. w, AllSolutions)], [positive]) =
S*mu__c*mu__s*(ln(Pi*mu__a*mu__c*w+Pi*mu__a*mu__s*w+2*c*mu__a*mu__c+2*c*mu__a*mu__s+2*epsilon*mu__c*mu__s+2*m*mu__a*mu__s)-ln(2)-ln(c*mu__a*mu__c+c*mu__a*mu__s+epsilon*mu__c*mu__s+m*mu__a*mu__s))/(Pi*(mu__c+mu__s))
NULL

``


 

Download IntAssPositive.mw

 

Dirac(0)?...

dharr 8205
August 13 2019
1 1

Download Dirac2.mw
 

> 

restart;
> 

with(IntegrationTools):
> 

int(Dirac(x),x=-infinity..infinity);

1

> 

q:=Int(Dirac(x)^2,x=-infinity..infinity);
value(q);

Int(Dirac(x)^2, x = -infinity .. infinity)

int(Dirac(x)^2, x = -infinity .. infinity)

Now generally,

> 

int(f(x)*Dirac(x-a),x=-infinity..infinity);

f(a)

So we may expect if a=0 and f(x)=Dirac(x), then the integral is Dirac(0). But we can't get it directly

> 

int(Dirac(x)*Dirac(x),x=-infinity..infinity);

int(Dirac(x)^2, x = -infinity .. infinity)

Though integration by parts will do it.

> 

Parts(q,Dirac(x));
value(%);

-(Int(Heaviside(x)*Dirac(1, x), x = -infinity .. infinity))

Dirac(0)

But it doesn't really help becasue Dirac(0) is not defined...


 

Download Dirac2.mw

 

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