nm

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9 years, 146 days

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These are questions asked by nm

I can't get pdsolve to solve this pde. Here are my tries below. One of them works, but the analytical solution Maple gives is wrong. So I am not sure if it needs some additional hints or some other help to make it give the correct solution.

This is Laplace pde but with non-zero on RHS. On disk centered at origin and with radius 1, with given BC on edge of disk.

restart;
the_rhs := r^2*cos(theta)*sin(theta);
pde := VectorCalculus:-Laplacian(u(r,theta),'polar'[r,theta])=the_rhs;
bc := u(1, theta) = cos(theta)*sin(theta);
sol:=pdsolve([pde, bc], u(r, theta));

Maple gives

But this does not look right. There is a complex exponential on its own there. The solution should be real. 

I tried to solve it using pdsolve numerically, but could not make it work. It kepts complaining about missing BC. I am not good with numerical solvers in Maple. May be someone could try to do it.

evalf(eval(rhs(sol),[r= sqrt(1/4 + 1/4),theta=Pi/4]))

gives 0.5951891582 which is wrong,. It should be  0.23958

It tried to give it the symmetry conditions on theta, but now it did not solve it

restart;
the_rhs := r^2*cos(theta)*sin(theta);
pde := VectorCalculus:-Laplacian(u(r,theta),'polar'[r,theta])=the_rhs;
bc := u(1, theta) = cos(theta)*sin(theta),u(r,0) = u(r,2*Pi),(D[2](u))(r, 0) = (D[2](u))(r, 2*Pi);
sol:=pdsolve([pde, bc], u(r, theta));

No solution. I tried adding HINT = boundedseries(r=0) but that also did not help. No solution.

Could someone solve this PDE numerically in Maple and check the solution at the above location? It should be 0.23958 which shows the analytical solution given is not correct. Can Maple solve this numerically?

Maple 2021.1

ps. Analytical solution was obtained by Mathematica. I did not solve this by hand.

I was expanding an expressions, and found that Maple expands trig also. Then found there is a way to turn that off. Which is great. I tested it in my worksheet and it works. But when I put the same code inside my module, I get an error.  It seems like some scoping issue which I do not understand.

Here is a MWE

restart;
expr:=(1+x)*sin(3*x+k);
forget(expand);
expand(expandoff());
expandoff(cos,sin,exp);
F:=expand(expr);
forget(expand);

which gives 

Which does what I want. But inside a proc, the code fails

foo:=proc(expr)
local F;

forget(expand);
expand(expandoff());
expandoff(cos,sin,exp);
F:=expand(expr);
forget(expand);

end proc;

And now when calling it as 

expr:=x*sin(3*x+k);
foo(expr)

What Am I doing wrong? And how to make it work? I need to prevent expand() from expanding these functions. Help says that expand has memory table, and I want to clear that before and after doing this, so it does not affect later code.

Maple 2021.1

Update

I just found out, if I add a restart before defining my function, then the error goes away

restart;
expr:=(1+x)*sin(3*x+k);
forget(expand);
expand(expandoff());
expandoff(cos,sin,exp);
F:=expand(expr);
forget(expand);

restart;  # had to add this. But why??
foo:=proc(expr)
local F;

forget(expand);
expand(expandoff());
expandoff(cos,sin,exp);
F:=expand(expr);
forget(expand);
end proc;

#
expr:=(1+x)*sin(3*x+k);
foo(expr)

#no error. now it works

I need this to work repeatedly inside a module, so I can't do restart each time ofcourse like the above.

Even if I do restart before defining the function, next time I call it, it will fail:

 

restart; 
foo:=proc(expr)
local F;

forget(expand);
expand(expandoff());
expandoff(cos,sin,exp);
F:=expand(expr);
forget(expand);
end proc;

#
expr:=(1+x)*sin(3*x+k);
foo(expr); #OK
foo(expr); #FAILED

Very strange. 

I can't find how to use Maple structured type, to check for say sin(x) and say sin(x)^2 without having to duplicate the code and make a structured type for each of the two cases. An example will explain.

I need to check if expression is of this form   m*sin(anything)^n*cos(anything)^r  Where in this example below, m,n,r can be integer or rational.     

The problem is that 'specfunc'(sin)^Or(integer,rational) will not match sin(x) but will only match if sin(x) is raised to an actual power different from default 1.

So I have to duplicate the check below, by writing Or('specfunc'(sin),'specfunc'(sin)^Or(integer,rational) and the same for cos(x). I am trying to avoid all this duplication, because as the structured type becomes more involved, it will hard to read and work with like this.

Is there a way to eliminate this? In Mathematica for example, this is done using the pattern   x_. where the little dot at the end there, allows for zero of more. So for the above, it will be  Sin[x_]^n_. and this will match Sin[anything] and also match Sin[anything]^2 

I know it is possible to use patmatch in Maple to do this, but is it possible without using pathmatch and just using structured types? as I am trying to do everything without having to use patmatch now.

restart;
my_type:=''`*`'( { Or('specfunc'(sin),'specfunc'(sin)^Or(integer,rational)),
                   Or('specfunc'(cos),'specfunc'(cos)^Or(integer,rational)),
                   Or(integer,rational)})';
type(3*sin(x)^2*cos(x)^3,my_type);
type(sin(x)^2*99*cos(x),my_type);
type(sin(x)*cos(x),my_type);
type(cos(x)*sin(x)^(1/2),my_type);

gives

btw, the help page for structured type mentiones zero or more occurances of but do not know how to do this for the above example

 

Ok, I think I am starting to get the hang of this. So in Maple, structured types is like a pattern in Mathematica. To find some subexpression one needs to first define a structured type which match that subexpression, and then use  select(type,.....).  

This works well so far (but it is not as easy as setting up a pattern).

But one small problem is that select() starts looking at the operands of the expression to look for a match.

So if the whole expression itself matches the structured type, it will not detect the whole expression, since select goes over each operand, missing that the whole thing actually matches the type.

May be an example helps shows what I mean. I want to find if an expression has cos(anything)*sin(anything) so I made a structured type for this 

mytype_3 :=  ''`*`'( {'specfunc'(cos),'specfunc'(sin)})';

btw, I do not know if I should use `&*` or '`*`' but that is a side point, I'll try to find this out.   

Now I want to use the above to check if expression has the same "structured type", or "pattern". The expression is expr:=cos(x)*sin(x); clearly it matches it. But since select looks at the operands, it will only see cos(x) by itself, and then sin(x) by itself and hence not find the structured type. 

restart;
mytype_3 :=  ''`*`'( {'specfunc'(cos),'specfunc'(sin)})';
expr:=cos(x)*sin(x);
type(expr,mytype_3);  # true
select(type, expr, mytype_3); # does not work, does not find it.

Since I am doing this in code, and I do not know what the expression is, I think I have to now do the following 

restart;
mytype_3 :=  ''`*`'( {'specfunc'(cos),'specfunc'(sin)})';
expr:=cos(x)*sin(x);
if type(expr,mytype_3) then
   print("The whole expression is a match! nothing to do. no need to use select");
else
   select(type, expr, mytype_3);
fi;

Which is OK. I can do this, But it will be nice if there was a way to have select (or another function like it) starts at the top level before looking at the operands?  

How do others handle such cases? does the above sound like an OK solution to this?


 

I am learning how to use select with my own types defined, to find parts of expression. This is instead of using patmach.

For example, given   3+x^2*sin(x) and then I want to find any POLYNIMAL*sin function, if present. So I did the following

restart;
expr_1:=3+x^2*sin(x):
mytype_1 := `&*`( polynom(And(algebraic, satisfies(u -> not has(u, I))),x),specfunc('sin')):
select( z->type(z,mytype_1),expr_1);

Which works. Maple returned 

The problem is that if I change the order of multiplication, and also at same time change the polynomial by adding one more term, it no longer works!

I have no idea why. It seems Maple rememebrs something.  Here is a screen shot, followed by plain text code.

 

code

restart;
expr_1:=3+(1+x)*sin(x):
mytype_1 := `&*`( polynom(And(algebraic, satisfies(u -> not has(u, I))),x),specfunc('sin')):
select( z->type(z,mytype_1),expr_1);
expr_1:=3+sin(x)*(1+x):
select( z->type(z,mytype_1),expr_1);

#change polynomial but keep same order, it works
expr_2:=3+(1+x+x^2)*sin(x):
select( z->type(z,mytype_1),expr_2);

#change order BUT keep same polynomial, it works
expr_3:=3+sin(x)*(1+x+x^2):
select(z->type(z, mytype_1),expr_3);

#keep same order as above, but change polynomial, now it does not works
expr_4:=3+sin(x)*(1+x+x^2+x^4):
select(z->type(z, mytype_1),expr_4);

#keep same order as first one  but change polynomial, it does not work
expr_5:=3+(1+x+x^2+x^4)*sin(x):
select(z->type(z, mytype_1),expr_5);

#keep same order as first one but change polynomial back to what it was, now it works
expr_6:=3+(1+x)*sin(x):
select(z->type(z, mytype_1),expr_6);

What Am I doing wrong?

 

Maple 2021.1

 

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