Dear all

I have a PDE and its analytical solution. I want to find the numerical solution by Finite Difference Method.

I duscratize the PDE and boundary condition and Could not able to solve them togethe.

Here is the file FEM-Nu.mw

of the implicit function sin(x+y)+sin(x) = y at x = Pi , y=0 of order 15? Here is one of the difficulties

restart; eval(implicitdiff(sin(x+y)+sin(x) = y, y, x$15), [x = Pi, y = 0]);
Error, (in expand/bigprod) Maple was unable to allocate enough memory to complete this computation. Please see ?alloc

Hi everybody,

I solve an ODE system numerically with the command

Tend := ….. :
sol := dsolve(MySys, numeric, method=rosenbrock, range=0..Tend):    # or rkf45

One of my colleagues wants the solution in array form for using it in a spreadsheet (namely Excel). 
Here is an example of what I do :
 
data := plottools:-getdata( plots:-odeplot(sol, […..], 0..Tend, refine = Q) ):  # Q rational
result := data[3]:
ExcelTools:-Export (result, …)

To understand the following, you need to know that the time step varies by several decades (at least 3 or 4). and that  numfun is typically in the order of 10^5
 
When I ask Maple to construct  data , Maple seems to sample the solution solaccording to the local time step dsolve has selected (and probably some other parameters ?)
Because this time step varies in very large proportions, the result is sometimes a concentration of points in regions where the solution is rapidly changing (which is desirable) … with some lacunarity in slow varying regions.

To obviate this, sometimes  poor appearance, I used to force Maple to compute the solution by regularly sampling the range ]0, Tend[  and combining this array with the original one (result).
But this not satisfactory for I do not avoid lacunarity region (unless I proceed in some ad hoc way)

Here is my question :
Could you give me some elements about  the way odeplot adapts its sampling strategy according to the local time step, the budget N (the Q value I guess) and maybe numfun ?

Thank you in advance

I have excuted a loop for i from.....by...while.....do l2:=NLPSolve(.......) with variables, t1,th,tl, etc. I need get the numerical result of theses variables and assign them to t1,th, etc. After execute a function with them. I have tried with op, map and assign, but the program don't understand the numerical value of variables. Why?. How can i assign theses numerical vaules?. Thanks! 

hi

please help to me for solve this equation via pdsolve?

thanks

dsove2.mw

Download dsove2.mw

hi

in attached file below not answer found for dsolve?

please help me

thanks..

dsove.mw

Download dsove.mw

I'm trying to compute the tensor product of two column vectors as

with(LinearAlgebra):

A:=Matrix([[1/sqrt(2)],[0],[0],[1/sqrt(2)]]);

KroneckerProduct(A,A);

And the output is a column vector with entries: "16 x 1 Matrix", "Data Type: Anything", "Storage: rectangular", "Order: Fortran_order"

The Maple documentation indicates that this function should output the result of the kronecker tensor product of the input matrices, and I've followed the same form as the examples in the documentation... Does anyone know why this isn't working as it should?

> restart;
> libname = [shootlib, libname];
> with(shoot);
Error, invalid input: with expects its 1st argument, pname, to be of type {`module`, package}, but received shoot
> with(plots);
Pr := 10; s = -.1; lambda := 0; Gr := 1.0; Gm := 1.0; beta := -1.20;
10
s = -0.1
0
1.0
1.0
-1.20
> M := 0.; z := .1; Xi := .5; Nt := .5; Nb := .2; l := 5; Nr := .5; epsilon1 := .2; epsilon2 := .2;
0.
0.1
0.5
0.5
0.2
5
0.5
0.2
0.2
> Prff := Pr/(1+4.*N*(1/3));
10
-----------------
1 + 1.333333333 N
> FNS := {f(eta), h(eta), r(eta), u(eta), v(eta), theta(eta), `ϕ`(eta)};
{f(eta), h(eta), r(eta), u(eta), v(eta), theta(eta), ϕ(eta)}
> ODE := {diff(h(eta), eta)+.75*l*f(eta)*h(eta)-(1/4)*l*u(eta)*epsilon2-Nt*(.75*f(eta)*r(eta)-(1/4)*u(eta)*epsilon1+Nb*r(eta)*h(eta)+Nt*r(eta)*r(eta))/Nb = 0, .75*f(eta)*r(eta)+diff(r(eta), eta)-(1/4)*u(eta)*epsilon1+Nb*r(eta)*h(eta)+Nt*r(eta)*r(eta) = 0, diff(v(eta), eta)+3*(f(eta)*v(eta)-u(eta)*u(eta))/(4*Pr)-(M+lambda)*u(eta)+theta(eta)-Nr*`ϕ`(eta) = 0, diff(f(eta), eta) = u(eta), diff(u(eta), eta) = v(eta), diff(theta(eta), eta) = r(eta), diff(`ϕ`(eta), eta) = h(eta)};
/ / d \
{ 0.75 f(eta) r(eta) + |----- r(eta)| - 0.05000000000 u(eta)
\ \ deta /

2 / d \ 3
+ 0.2 r(eta) h(eta) + 0.5 r(eta) = 0, |----- v(eta)| + -- f(eta) v(eta)
\ deta / 40

3 2 / d \
- -- u(eta) + theta(eta) - 0.5 ϕ(eta) = 0, |----- h(eta)|
40 \ deta /

+ 3.75 f(eta) h(eta) - 0.1250000000 u(eta) - 1.875000000 f(eta) r(eta)

2
- 0.5000000000 r(eta) h(eta) - 1.250000000 r(eta) = 0,

d d d
----- f(eta) = u(eta), ----- u(eta) = v(eta), ----- theta(eta) = r(eta),
deta deta deta

d \
----- ϕ(eta) = h(eta) }
deta /
> IC := {f(0) = s, h(0) = xi, r(0) = tau, u(0) = 0, v(0) = alpha(0), theta(0) = 1-(1/4)*epsilon1, `ϕ`(0) = (1/4)*epsilon2};
{f(0) = s, h(0) = xi, r(0) = tau, u(0) = 0, v(0) = alpha(0),

theta(0) = 0.9500000000, ϕ(0) = 0.05000000000}
> L := 2;
2
> BC = {u(L) = 0, theta(L) = 0, `ϕ`(L) = 0};
BC = {u(2) = 0, theta(2) = 0, ϕ(2) = 0}
> S := Shoot(ODE, IC, BC, FNS, [alpha = .42453091564332, tau = -.21166705749821127, xi = -.4944583739651814]);
/ / / d \
Shoot|{ 0.75 f(eta) r(eta) + |----- r(eta)| - 0.05000000000 u(eta)
\ \ \ deta /

2 / d \ 3
+ 0.2 r(eta) h(eta) + 0.5 r(eta) = 0, |----- v(eta)| + -- f(eta) v(eta)
\ deta / 40

3 2 / d \
- -- u(eta) + theta(eta) - 0.5 ϕ(eta) = 0, |----- h(eta)|
40 \ deta /

+ 3.75 f(eta) h(eta) - 0.1250000000 u(eta) - 1.875000000 f(eta) r(eta)

2
- 0.5000000000 r(eta) h(eta) - 1.250000000 r(eta) = 0,

d d d
----- f(eta) = u(eta), ----- u(eta) = v(eta), ----- theta(eta) = r(eta),
deta deta deta

d \
----- ϕ(eta) = h(eta) }, {f(0) = s, h(0) = xi, r(0) = tau, u(0) = 0,
deta /

v(0) = alpha(0), theta(0) = 0.9500000000, ϕ(0) = 0.05000000000}, BC,

{f(eta), h(eta), r(eta), u(eta), v(eta), theta(eta), ϕ(eta)}, [

alpha = 0.42453091564332, tau = -0.21166705749821127,

\
xi = -0.4944583739651814]|
/
RungeKutta(ODE, BC, alpha = .42453091564332, tau = -.21166705749821127, xi = -.4944583739651814, output=plot);
/ / / d \
RungeKutta|{ 0.75 f(eta) r(eta) + |----- r(eta)| - 0.05000000000 u(eta)
\ \ \ deta /

2 / d \ 3
+ 0.2 r(eta) h(eta) + 0.5 r(eta) = 0, |----- v(eta)| + -- f(eta) v(eta)
\ deta / 40

3 2 / d \
- -- u(eta) + theta(eta) - 0.5 ϕ(eta) = 0, |----- h(eta)|
40 \ deta /

+ 3.75 f(eta) h(eta) - 0.1250000000 u(eta) - 1.875000000 f(eta) r(eta)

2
- 0.5000000000 r(eta) h(eta) - 1.250000000 r(eta) = 0,

d d d
----- f(eta) = u(eta), ----- u(eta) = v(eta), ----- theta(eta) = r(eta),
deta deta deta

d \
----- ϕ(eta) = h(eta) }, BC, alpha = 0.42453091564332,
deta /

\
tau = -0.21166705749821127, xi = -0.4944583739651814, output = plot|
/
>

Dear sir 

in the above problem im geiitng the problem with , with(shoot) command and even it is not executing at

S := Shoot(ODE, IC, BC, FNS, [alpha = .42453091564332, tau = -.21166705749821127, xi = -.4944583739651814]) this command, here alpha,tau and zi variable should change.

> restart;
> with(plots);
> Eql := diff(f(eta), eta, eta, eta)+.5*f(eta)*(diff(f(eta), eta, eta)) = 0;
/ d / d / d \\\ / d / d \\
|----- |----- |----- f(eta)||| + 0.5 f(eta) |----- |----- f(eta)|| = 0
\ deta \ deta \ deta /// \ deta \ deta //
> blt := 10;
10
> bcs1 := f(0) = f0, (D(f))(0) = 0, (D(f))(blt) = 1;
f(0) = f0, D(f)(0) = 0, D(f)(10) = 1
> L := [0];
[0]
> for k to 1 do R := dsolve(eval({Eql, bcs1}, f0 = L[k]), f(eta), numeric, output = listprocedure); X1 || k := rhs(R[3]); X2 || k := rhs(R[4]) end do;
[
[eta = proc(eta) ... end;, f(eta) = proc(eta) ... end;,
[

d
----- f(eta) = proc(eta) ... end;,
deta

d / d \ ]
----- |----- f(eta)| = proc(eta) ... end;]
deta \ deta / ]
proc(eta) ... end;
proc(eta) ... end;
> print([X2], [1 .. 1, 0]);

dear sir/madam

in the above problem i should get the asnser (at print line) but its not getting so please can you tell me why it is not getting.

Let us denote the cardinality of the subsets of {1,..,n} without two consequent numbers
(e.g. {..,4,5,..} is not allowed) by A[n]. What is the asymptotics of A[n] as n approaches infinity?
The same question for the case of three consequent numbers.
Here is my math experiment.
restart; L := combinat:-powerset({seq(i, i = 1 .. 11)}):#n = 11
nops(%);
2048
M := selectremove(c-> min([seq(c[k+1]-c[k], k = 1 .. nops(c)-1)]) = 1, L)[2]:
nops(M);
233
The other results are [11, 233], [15, 1597], [20, 17711], [21, 28657], [22, 46368].
These points are very close to some straight line in logarithmic scale as
plot([[11, 233], [15, 1597], [20, 17711], [21, 28657], [22, 46368]], axis[2] = [mode = log]);
shows. However, the ones do not exactly belong to a straight line:
evalf(ln(46368)-ln(28657), 15);
0.4812118247230
evalf(ln(28657)-ln(17711), 15);
0.48121182594077
eval(exp(.4812118247230*n), n = 15);
1364.000725  .
These results suggest that A[n] is asymptotically equal to exp(c*n) with c about 0.481.
I have not succeeded to find out the nature of the constant c.

question_on_asymptotics.mw

I need to solve an ode of the type ay''+by'+cy=f(x) using cubic b spline.

can any one help me with the code or algorithm. Thanks

N := 4;
print(`output redirected...`); # input placeholder
4
y := sum(A[2*n].cos(2.*n.x), n = 0 .. N);

eq1 := diff(y, `$`(x, 2))+(a+2*q*cos(2*x))*y

eq2 := map(combine, eq1, trig)

for i from 0 to 4 do eq4[i] := coeff(eq2, cos(2*n*x)) end do

From these I want to extract the co-ffficients of cos(0x),cos(2x),cos(4x)..

and form a simultaneous linear equation containg A0,A2,A4

The solution is 

aA0+qA2=0

2q*A0+(a-4)*A2+q*A4=0

Can anybody tell me how to do it

so I'm trying this:

restart;

sigma := 0.143e-18;

l_0 := 1.87;

l0 := 1.87;

roll := rand(0 .. 25.0);

f_gauss := proc (x) options operator, arrow; exp(-(1/2)*x^2/`σ_x`^2)/sqrt(2*Pi*`σ_x`^2) end proc;

f_norm := proc (dx) options operator, arrow; int(f_gauss(x), x = -(1/2)*dx .. (1/2)*dx) end proc;

sol_gauss := proc (mix) options operator, arrow; evalf(eval(-ln((int(f_gauss(x)*exp(-2*sigma*N2O*sqrt((1/4)*l_0^2-x^2)), x = -(1/2)*dx .. (1/2)*dx))/f_norm(dx))/(sigma*N2O), [N2O = 0.25e20*mix/100])) end proc;

for ii to 10 do

a := roll();

eval(sol_gauss(a), [dx = l_0, `σ_x` = l0])

end do

Hello people in mapleprimes,

I want to ask you about how to make a function of function which makes a logarithmic derivative of a function.

For example, x^3 is mapped as h(x^3)=3h(x), h(x+y)=(x/(x+y))*h(x)+(y/(x+y))*h(y),

h(g(x)*k(x))=h(g(x))+h(k(x)).

I hope someone give me a hing to create h.

Best wishes.

taro