# Items tagged with numericnumeric Tagged Items Feed

### Recursive integral equation...

July 17 2015
0 10

I want to find numerically the limit lim(y[m](t),m = infinity), do you have an idea how to do implement it in maple?

 >
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### IVP Dsolve Error...

July 11 2015
1 1

Hello,

I am a student doing some self study over the summer trying to work through some of the John Taylor computer problems from his classcial mechanics book. Currently I hit a snag that most likely comes from the fact I am not well acquinted with Maple for solving IVP and DE's (we used Matlab in my DE class). I just need to know how I remove the following error:

Error, (in dsolve/numeric/SC/IVPsetup) initial conditions must be numeric

Here is a copy of my code:

R := 5;
5
g := 9.8;
9.8
deq1 := {diff(x(t), [\$(t, 2)]) = -g*sin(x(t))/R, x(0) = 20};
/ d / d \ \
{ --- |--- x(t)| = -1.960000000 sin(x(t)), x(0) = 20 }
\ dt \ dt / /
dsol1 := dsolve(deq1, numeric);
Error, (in dsolve/numeric/SC/IVPsetup) initial conditions must be numeric

My hunch is that I need to set x'(0)=0 or something like I do not have enough intial values to solve the problem, but I could be wrong. Anyway anyone who can point out my mistake feel free to do so! Thank you!

### Solving PDE numerically...

July 08 2015
0 5

I have the following PDE system to solve numerically and I am not sure how to use maple to solve it.

v_t = v_{xx} for 0<x<1 , t>0

v(x,0)=1

v_x(1,t)=-hv^4(1,t) (where h is some numerical number);

v_x(0,t)=0

To solve this pde numerically I need to use the following condition on v(1,t):

v(1,t) = 1-h*\int_{0}^t \theta_3(\tau)v(1,t-\tau)^4d\tau

this is the numerical boundary condition, where \theta_3 is Jacobi theta3 function.

I don't see how can I use maple for this numerical pde problem.

Here's my attempt at solution:

[code]

PDE := diff(v(x, t), t) = diff(v(x, t), x, x);

JACOBIINTEGRAL := int(JacobiTheta3(0, exp(-Pi^2*s))*v(1, t-s)^4, s = 0 .. t);

IBC := {&PartialD;(v(0, t))/&PartialD;(x) = 0, &PartialD;(v(1, t))/&PartialD;(x) = -0.65e-4*v(1, t)^4, v(x, 0) = 1};

pds := pdsolve(PDE, IBC, numeric, time = t, range = 0 .. 1, spacestep = 0.1e-2, timestep = 0.1e-2, numericalbcs = {v(1, t) = 1-0.65e-4*JACOBIINTEGRAL}, method = ForwardTimeCenteredSpace)

[/code]

But I get the next error message:

Error, (in pdsolve/numeric/process_IBCs) improper op or subscript selector

How to fix this or suggest me a better way to solve this pde numerically?

### Very hard integral...

June 28 2015
1 10

How to calculate the integral

(symbolically or/and numerically) with Maple?

### the system of differential equations and the selec...

June 14 2015
0 5

Hello, dear experts.
I have a question...
solve the system of differential equations,where one of the initial conditions need to be chosen so thatcondition is metat the end of integration.
The task is not difficult, but I'm having trouble with the syntax.

1.I can't "pull"the desired function from the solution and find its value at a certain point.
I try to do so:
r_ravn:=s->subs(F,r(s));
evalf(r_ravn(s_end));
evalf(r_ravn(0));
but there is no result

2.In this case,instead of"for"it is better to use a while loop, but again the problem arises 1.
Tell me, please,how to implemen my program.

restart:
R:=0.3:
theta_min:=Pi/6:
theta_max:=Pi/2:
betta_max:=evalf(Pi/180*80);
p:=2*10^5:

theta0:=s->Pi/3/s_end*s+Pi/6:
r0:=s->R*sin(theta0(s)):
s_end:=evalf(R*(theta_max-theta_min)):

sol1:=solve({sin(betta_max)=c/r0(0)},{c});
const1:=0.1477211630;

betta0:=s->arcsin(const1/r0(s)):
betta:=s->arcsin(r(s)/r0(s)*sin(betta0(s))):
A:=s->cos(betta(s))/cos(betta0(s)):
T1:=s->rT1(s)/r(s):
T2:=s->T1(s)*tan(betta(s))^2:

step:=0.001:
delta:=0.001:
for i from 1 to 3000 do
r_min:=0.3-step:
rT1_n:=p*Pi*r_min^2/2/Pi/sin(theta_min):

sys := diff(rT1(s), s)-A(s)*T2(s)*cos(theta(s)),diff(theta(s), s)-A(s)/T1(s)*(p-T2(s)*sin(theta(s)/r(s))),diff(r(s),s)-A(s)*cos(theta(s)),diff(z(s),s)-A(s)*sin(theta(s));
fcns := {rT1(s),theta(s),r(s), z(s)};
F := dsolve({sys,rT1(0)=rT1_n, theta(0)=theta_min,r(0) = r_min, z(0) = 0}, fcns, numeric,output=listprocedure):
r_ravn:=s->subs(F,r(s)):
if abs(evalf(r_ravn(s_end))-R)=delta then break:
print(r_min):
end if:
end do:

r_ravn:=s->subs(F,r(s));
evalf(r_ravn(s_end));
evalf(r_ravn(0));
plot([r_ravn(s),r(s)],s=0..s_end);

### Derivative from pdsolve/numeric solution?...

June 07 2015
1 12

Is it possible to somehow extract a derivative from numeric solution of partial differential equation?

I know there is a command that does it for dsolve but i couldn't find the same thing for pdsolve.

The actual problem i have is that i have to take a numeric solution, calculate a derivative from it and later use it somewhere else, but the solution that i have is just a set of numbers an array of some sort and i can't really do that because obviously i will get a zero each time.

Perhaps there is a way to interpolate this numeric solution somehow?

I found that someone asked a similar question earlier but i couldn't find an answer for it.

### Error, (in plot) two lists or Vectors of numerical...

June 03 2015
1 2

Hey there,

I've a numerical solved system of differential equations, which depend on one argument and one index. I can solve it, but when I try plot it I have this error: Error, (in plot) two lists or Vectors of numerical values expected.

Could anyone help me figure out what I'm doing wrong?

> restart;
> A := 115.1558549; B := .3050464658; n := 3; f0 := 0.5e-4;

>f:=theta->f0*(cos(arcsin(sin(theta)/n)))^2;
I0:=Ir(z)+sum(Is[k](z),k=1..20);

> alpha := [0, 1, 2, 3, 4, 5, 6];

Theta := [3*Pi*(1/180), 6*Pi*(1/180), 9*Pi*(1/180), 12*Pi*(1/180), 15*Pi*(1/180), 18*Pi*(1/180), 21*Pi*(1/180), 24*Pi*(1/180), 27*Pi*(1/180), 30*Pi*(1/180), 33*Pi*(1/180), 36*Pi*(1/180), 39*Pi*(1/180), 42*Pi*(1/180), 45*Pi*(1/180), 48*Pi*(1/180), 51*Pi*(1/180), 54*Pi*(1/180), 57*Pi*(1/180), 60*Pi*(1/180)];

>G:= theta->A*sin(theta)*cos(2*arcsin((sin(theta)/n)))/((1+sin(theta)^2/B^2)*cos(arcsin(sin(theta)/n)));

>for j from 1 to 7 do
d1 := diff(Ir(z), z) = -sum(G(Theta[k])*Ir(z)*Is[k](z)/I0,k=1..20)-alpha[j]*Ir(z)-sum(f(Theta[k])*Ir(z),k=1..20):
d2 := diff(Is[1](z), z) = G(Theta[1])*Ir(z)*Is[1](z)/I0-alpha[j]*Is[1](z)+f(Theta[1])*Ir(z):
d3 := diff(Is[2](z), z) = G(Theta[2])*Ir(z)*Is[2](z)/I0-alpha[j]*Is[2](z)+f(Theta[2])*Ir(z):
d4 := diff(Is[3](z), z) = G(Theta[3])*Ir(z)*Is[3](z)/I0-alpha[j]*Is[3](z)+f(Theta[3])*Ir(z):
d5 := diff(Is[4](z), z) = G(Theta[4])*Ir(z)*Is[4](z)/I0-alpha[j]*Is[4](z)+f(Theta[4])*Ir(z):
d6 := diff(Is[5](z), z) = G(Theta[5])*Ir(z)*Is[5](z)/I0-alpha[j]*Is[5](z)+f(Theta[5])*Ir(z):
d7 := diff(Is[6](z), z) = G(Theta[6])*Ir(z)*Is[6](z)/I0-alpha[j]*Is[6](z)+f(Theta[6])*Ir(z):
d8 := diff(Is[7](z), z) = G(Theta[7])*Ir(z)*Is[7](z)/I0-alpha[j]*Is[7](z)+f(Theta[7])*Ir(z):
d9 := diff(Is[8](z), z) = G(Theta[8])*Ir(z)*Is[8](z)/I0-alpha[j]*Is[8](z)+f(Theta[8])*Ir(z):
d10 := diff(Is[9](z), z) = G(Theta[9])*Ir(z)*Is[9](z)/I0-alpha[j]*Is[9](z)+f(Theta[9])*Ir(z):
d11 := diff(Is[10](z), z) = G(Theta[10])*Ir(z)*Is[10](z)/I0-alpha[j]*Is[10](z)+f(Theta[10])*Ir(z):
d12 := diff(Is[11](z), z) = G(Theta[11])*Ir(z)*Is[11](z)/I0-alpha[j]*Is[11](z)+f(Theta[11])*Ir(z):
d13 := diff(Is[12](z), z) = G(Theta[12])*Ir(z)*Is[12](z)/I0-alpha[j]*Is[12](z)+f(Theta[12])*Ir(z):
d14 := diff(Is[13](z), z) = G(Theta[13])*Ir(z)*Is[13](z)/I0-alpha[j]*Is[13](z)+f(Theta[13])*Ir(z):
d15 := diff(Is[14](z), z) = G(Theta[14])*Ir(z)*Is[14](z)/I0-alpha[j]*Is[14](z)+f(Theta[14])*Ir(z):
d16 := diff(Is[15](z), z) = G(Theta[15])*Ir(z)*Is[15](z)/I0-alpha[j]*Is[15](z)+f(Theta[15])*Ir(z):
d17 := diff(Is[16](z), z) = G(Theta[16])*Ir(z)*Is[16](z)/I0-alpha[j]*Is[16](z)+f(Theta[16])*Ir(z):
d18 := diff(Is[17](z), z) = G(Theta[17])*Ir(z)*Is[17](z)/I0-alpha[j]*Is[17](z)+f(Theta[17])*Ir(z):
d19 := diff(Is[18](z), z) = G(Theta[18])*Ir(z)*Is[18](z)/I0-alpha[j]*Is[18](z)+f(Theta[18])*Ir(z):
d20 := diff(Is[19](z), z) = G(Theta[19])*Ir(z)*Is[19](z)/I0-alpha[j]*Is[19](z)+f(Theta[19])*Ir(z):
d21 := diff(Is[20](z), z) = G(Theta[20])*Ir(z)*Is[20](z)/I0-alpha[j]*Is[20](z)+f(Theta[20])*Ir(z):
dsys := {d1, d10, d11, d12, d13, d14, d15, d16, d17, d18, d19, d2, d20, d21, d3, d4, d5, d6, d7, d8, d9}:
dSol[j] := dsolve({op(dsys), Ir(0) = 1, Is[1](0) = 0.1e-1, Is[2](0) = 0.1e-1, Is[3](0) = 0.1e-1, Is[4](0) = 0.1e-1, Is[5](0) = 0.1e-1, Is[6](0) = 0.1e-1, Is[7](0) = 0.1e-1, Is[8](0) = 0.1e-1, Is[9](0) = 0.1e-1, Is[10](0) = 0.1e-1, Is[11](0) = 0.1e-1, Is[12](0) = 0.1e-1, Is[13](0) = 0.1e-1, Is[14](0) = 0.1e-1, Is[15](0) = 0.1e-1, Is[16](0) = 0.1e-1, Is[17](0) = 0.1e-1, Is[18](0) = 0.1e-1, Is[19](0) = 0.1e-1, Is[20](0) = 0.1e-1}, [Ir(z), Is[1](z), Is[2](z), Is[3](z), Is[4](z), Is[5](z), Is[6](z), Is[7](z), Is[8](z), Is[9](z), Is[10](z), Is[11](z), Is[12](z), Is[13](z), Is[14](z), Is[15](z), Is[16](z), Is[17](z), Is[18](z), Is[19](z), Is[20](z)], numeric);
end do:

>for j from 1 to 7 do
dSol[j](0.4);
as:='as':
for l from 1 to 20 do
as[l]:=[Theta[l],rhs(dSol[j](0.4)[2+l])];
od:
plo[j]:=convert(as,listlist);
od:

>plot(plo[2],plo[1]);
Error, (in plot) two lists or Vectors of numerical values expected

### How to speed up numerical integration...

June 02 2015
0 2

Hello all.

Need help.

My project are too slow calculate integrals. What can i do to speed up numerical integration?

Thanks all.
MP_1.mw

### conversion to an explicit first-order system; Help...

May 31 2015
2 5

I'm currently having some difficulties in solving a system of differential equations numerically.

This is my code.

### How to calculate that integral?...

May 28 2015
1 11

Let us consider the definite integral

J:=int(abs(x-(-x^5+1)^(1/5)), x = 0 .. 1);

Maple fails with it, Mathematica 10.1 finds it in terms of  special functions. Let us look at the integrand:
plot(x-(-x^5+1)^(1/5), x = 0 .. 1);

We see the expression under the modulus changes its sign at the unique point of RealRange(0,1). Therefore

solve(x-(-x^5+1)^(1/5));

Then

J:= int(-x+(-x^5+1)^(1/5), x = 0 .. (1/2)*2^(4/5))+int(x-(-x^5+1)^(1/5), x = (1/2)*2^(4/5) .. 1);

which outputs a complicated expression

(1/8)*2^(4/5)*(4*hypergeom([-1/5, 1/5], [6/5], 1/2)-2^(4/5))+(1/2)*2^(4/5)*((1/2)*2^(1/5)-(1/4)*2^(4/5))-(1/25)*Pi*csc((1/5)*Pi)*(-(25/2)*sin((1/5)*Pi)*GAMMA(4/5)*2^(4/5)*hypergeom([-1/5, 1/5], [6/5], 1/2)/Pi+(5/4)*sec((3/10)*Pi)*cos((1/10)*Pi)*2^(3/5)*Pi^(1/2)*csc((3/10)*Pi)/GAMMA(7/10))/GAMMA(4/5).

At the same time we have

int(abs(x-(-x^5+1)^(1/5)), x = 0 .. 1, numeric);

0.5000000000

How to obtain 1/2 symbolically?

### Warning, unable to evaluate the function to numeri...

May 27 2015
1 2

Hi guys,

I'm studying a system of six differential equations. Given the fact that the system cannot be solved symbolically, I've tried the numeric procedure, and it works. I proceeded like this :

soleqd:=dsolve(sysd2,numeric,var);

then i checked if maple could calculate the solutions for given values of t. It works for t=0, t=0.5, t=1,t=2,...,t=5. The solutions are all real numbers.

But when i try to draw a graphic representation of the solutions, it doesn't work. I do :

ff1:=t->subs(soleqd(t),u[1](t));
gg1:=t->subs(soleqd(t),nu[1](t));

Then :

plot(['ff1(t)','gg1(t)',t=0..5],u[1]=0...1,nu[1]=0...1);

(The square brackets are indices)

Now maple answers that it is "unable to evaluate the function to numeric values in the region". I went to the help page but no solution seems to work. I can't figure it out by myself. Does anybody notice something wrong with my code ?

Best regards,

Louis

### how to find skin friction coefficient Cf/Re^1/2=f'...

May 22 2015
0 20

restart;

with(plots);

Eq1 := diff(f(eta), eta, eta, eta)+f(eta)*(diff(f(eta), eta, eta))-(diff(f(eta), eta))^2-M^2*(diff(f(eta), eta))+B(f(eta)*(diff(f(eta), eta, eta))*(diff(f(eta), eta))-f(eta)^2*(diff(f(eta), eta, eta, eta))) = 0;

Eq2 := (diff(theta(eta), eta, eta))/Pr+f(eta)*(diff(theta(eta), eta))-2*(diff(f(eta), eta))*theta(eta) = 0;

Pr := 1

M := 1

S := 0

epsilon := 1

blt := 10

bcs1 := f(0) = S, (D(f))(0) = epsilon, (D(f))(blt) = 0;

bcs2 := theta(0) = 1, theta(blt) = 0;

L := [0, .2, .4, .6, .8, 1.2];

for k to 6 do R := dsolve(eval({Eq1, Eq2, bcs1, bcs2}, B = L[k]), [f(eta), theta(eta)], numeric, output = listprocedure); X1 || k := rhs(R[3]); X2 || k := rhs(R[4]); Y1 || k := rhs(R[5]); Y2 || k := -rhs(R[6]) end do:

print([(X2 || (1 .. 6))(0)])

### Issue with testeq?...

May 22 2015
0 4

Consider the equation

eqn1:=5^(2*abs(y-1)+2) = (1/15625)*625^abs(y-1);

and note that

eval( eqn1, y=-3);
eval( eqn1, y=5);

both return valid solutions. So why does

testeq(eqn1);

throw a numeric exception.

Given the manual description of testeq(), I would expect it to return either true or FAIL - but not a numeric exceprion!!!!

Any ideas?

### ODE:NUMERICAL SOLUTION...

May 21 2015
1 4

Please, i need help USING ODE1 and ODE2 with given BCS and Pr=0.714

it is needed to to generates

[-0.2], [0.51553], [0.4000]
[-0.1], [0.57000], [0.4371]
[0.], [0.62756], [0.4764]
[0.1], [0.68811], [0.5176]
[0.2], [0.75153], [0.5609]

but it is generarting

[-0.2], [0.51553], [0.42342]
[-0.1], [0.57000], [0.46114]
[0.], [0.62756], [0.50088]
[0.1], [0.68811], [0.54261]
[0.2], [0.75153], [0.58628]

the values of D(theata)(0) is wrong. Please i need HELP. this the code below that i use:
>restart;
>with (plots):ode1:=diff(f(eta),eta,eta,eta)+f(eta)*diff(f(eta),eta,eta)-M*diff(f(eta),eta)=0:

>ode2:=diff(theta(eta),eta,eta)+Pr*f(eta)*diff(theta(eta),eta)=0:

>bcs1:= f(0)=w,D(f)(0)=1,D(f)(10)=0:
>bcs2theta(10)=0,theta(0)=1:
>fixedparameter1:=[M=0.0]:
>ode3:=eval(ode1,fixedparameter1):
>fixedparameter2:=[Pr=0.714]:
>ode4:=eval(ode2,fixedparameter2):

>G:=[-0.2,-0.1,0.0,0.1,0.2]:
>for ode3 and ode4:
for k from 1 to 5 do
sol_All:=dsolve(eval({ode3,ode4,bcs1},w=G[k]),    [f(eta),theta(eta)],numeric,output=listprocedure);
Y_sol||k:= -rhs(sol_All[4]);
YP_sol||k:=-rhs(sol_All[6]);
end do:
>Digits:=5:

>for k from 1 to 5 do
evalf([G[k]]),evalf([(Y_sol||k(0))]),evalf([YP_sol||k(0)]);
od;
[-0.2], [0.51553], [0.42342]
[-0.1], [0.57000], [0.46114]
[0.], [0.62756], [0.50088]
[0.1], [0.68811], [0.54261]
[0.2], [0.75153], [0.58628]

May 20 2015
0 9

restart;
pp:=-55471918776960000*tanh((1/3220)*sqrt(10368400-cp^2)*Pi*x/cp)+5350094400*tanh((1/3220)*sqrt(10368400-cp^2)*Pi*x/cp)*cp^2-129*tanh((1/3220)*sqrt(10368400-cp^2)*Pi*x/cp)*cp^4+2670899840*tanh((1/6450)*sqrt(41602500-cp^2)*Pi*x/cp)*sqrt(41602500-cp^2)*sqrt(10368400-cp^2);
Student[Calculus1]:-Roots(subs(x=8000,pp),cp=1..3220,numeric);
p1:=proc(v)
option hfloat;
local a;
a:=Student[Calculus1]:-Roots(subs(x=v,pp),cp=1..3220,numeric);
if nops(a)>=1 then seq([v,a[i]],i=1..nops(a));
end if;
end proc:
SS1:=[seq(p1(i),i=3500..20000,200)]:
plot(SS1,style=point,gridlines);

The final figure is different between maple12 and maple17.

On 17, unwanted points apprear.

is it a bug?

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