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Rouben Rostamian

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18 years, 268 days
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These are questions asked by Rouben Rostamian

How to differentiate a vector-valued fun...

Rouben Rostamian 8249 Maple 2022
vector differentiation
December 26 2022
1 8

Given a vector-valued function z(u,v), I want to calculate the derivative of z with respect to its first argument by applying the D operator to it. I don't see how.  Any suggestions?

> 

restart;
> 

z := (u,v) -> < a(u,v), b(u,v) >;

proc (u, v) options operator, arrow; `<,>`(a(u, v), b(u, v)) end proc

Calculate the derivative of z with respect to its first argument:

> 

P := diff(z(u,v), u);

Vector(2, {(1) = diff(a(u, v), u), (2) = diff(b(u, v), u)})

Express P through the D operator:

> 

Q := convert(P, D);

Vector(2, {(1) = (D[1](a))(u, v), (2) = (D[1](b))(u, v)})

Question:  How do we obtain Q directly by applying the D operator to z

without the help of diff?  This one doesn't work:

> 

D[1](z)(u,v);
type(%, Vector);

(D[1](`<,>`))(a(u, v), b(u, v))*(D[1](a))(u, v)+(D[2](`<,>`))(a(u, v), b(u, v))*(D[1](b))(u, v)

false

Download diff.mw

Unexpected results from pdsolve...

Rouben Rostamian 8249 Maple 2022
pde pdsolve differential-equations
September 10 2022
2 2
> 

restart;
> 

pde := diff(u(x,t),t) + u(x,t)*diff(u(x,t),x) = 0;

diff(u(x, t), t)+u(x, t)*(diff(u(x, t), x)) = 0

These are all wrong:

> 

pdsolve({pde,u(x,0)=f(x)});
pdsolve({pde,u(x,0)=sin(x)});
pdsolve({pde,u(x,0)=erf(x)});

u(x, t) = 0

u(x, t) = 0

u(x, t) = 0

But these ones are correct:

> 

pdsolve({pde,u(x,0)=exp(x)});
pdsolve({pde,u(x,0)=x});

u(x, t) = LambertW(t*exp(x))/t

u(x, t) = x/(t+1)

Download mw.mw

Physics[Vectors] vector type...

Rouben Rostamian 8249 Maple 2022
vector type physics
July 25 2022
1 7

Is there  a specific type name for vectors in Physics[Vectors]?  Specifically, Let's say we want to write a proc whose argument is expected to be a (Physics) Vector, as in  these (trivial) demos:

with(Physics[Vectors]);

f := proc(a_::???)
    return a_ . a_;
end proc:

g := proc(a_::???, b_::???)
    return a_ &x b_;
end proc:

What do we put in place of "???".

Tubeplot colors...

Rouben Rostamian 8249 Maple 2022
color linux plotting
July 19 2022
0 12

When displaying two tubeplots together, we may specify their colors at will, as long as they are different colors!  For instance, specifying red and green works correctly, but specifying red and red results in red and black!

See the attached worksheet.  Interestingly, when displaying the contents of the worksheet on this website, the colors are rendered correctly!  So don't go with what you see on this web page; look inside the worksheet instead.

> 

restart;
> 

kernelopts(version);

`Maple 2022.1, X86 64 LINUX, May 26 2022, Build ID 1619613`

> 

with(plots):

Two intersecting tori colored red and green -- works as expected:

> 

display(
        tubeplot([cos(t), 0, sin(t)], t=-Pi..Pi, radius=0.2),
        tubeplot([cos(t), sin(t), 0], t=-Pi..Pi, radius=0.2),
style=surface, color=[red,green]);

When we set both colors to red, one of the surfaces is painted black!  Why?

Please note: This website displays the colors corectly as red and red.  But

within the worksheet the colors are read and black.

> 

display(
        tubeplot([cos(t), 0, sin(t)], t=-Pi..Pi, radius=0.2),
        tubeplot([cos(t), sin(t), 0], t=-Pi..Pi, radius=0.2),
style=surface, color=[red,red]);

Specifying colors as red/red within the tubeplots still produces red/black!

> 

display(
        tubeplot([cos(t), 0, sin(t)], t=-Pi..Pi, radius=0.2, color=red),
        tubeplot([cos(t), sin(t), 0], t=-Pi..Pi, radius=0.2, color=red),
style=surface);

Download mw.mw

PS: As a workaround, we may replace the red & red specification with
COLOR(RGB, 1, 0, 0) and
COLOR(RGB, 1, 0, 0.01)
which are different enough to make Maple happy, but produce essentially the same red color.

How to have Maple solve this simple ODE?...

Rouben Rostamian 8249 Maple 2021
ode dsolve differential-equations
June 27 2022
4 5

The worksheet here shows a couple of failed attempts at coaxing Maple to calculate the general solution of a pretty simple second order ODE.  I have also included the expected solution which I  have calculated by hand.  Perhaps I am missing a key trick.  Any ideas?

The ODE that I am actually interested in is significantly more complex. The one in the worksheet is a much simplified "bare bones" specimen that exhibits the issue that I am facing.

Attempt to solve with Heaviside

> 

restart;
> 

de := diff(u(x),x$2) = Heaviside(x - a)*u(x);

diff(diff(u(x), x), x) = Heaviside(x-a)*u(x)

dsolve fails:

> 

dsolve(de);

u(x) = DESol({diff(diff(_Y(x), x), x)-Heaviside(x-a)*_Y(x)}, {_Y(x)})

Attempt to solve with piecewise

> 

restart;
> 

de := diff(u(x),x$2) = piecewise(x < a, 0, 1)*u(x);

de := diff(u(x), x, x) = piecewise(x < a, 0, 1)*u(x)

> 

dsolve(de);

Error, (in dsolve) give the main variable as a second argument

> 

dsolve(de, u(x));

Error, (in dsolve) give the main variable as a second argument

> 

 

> 

 

The solution is easy to calculate by hand

We just solve the (quite trivial) DE over the intervals x < a and x>a

separately, and patch the two solutions by requiring the continuity

of u(x) and diff(u(x), x) at x = a.  We get

> 

sol := piecewise(x < a,
        x*c[1] + c[2],
        ((a*c[1] + c[1] + c[2])*exp(x))/(2*exp(a)) + ((a*c[1] - c[1] + c[2])*exp(-x))/(2*exp(-a)));

sol := piecewise(x < a, x*c[1]+c[2], (a*c[1]+c[1]+c[2])*exp(x)/(2*exp(a))+(a*c[1]-c[1]+c[2])*exp(-x)/(2*exp(-a)))

Download solving-an-ode.mw

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