Items tagged with integral

Dear all,

I would like to evaluate a double integral numerically. The integrand is a complicated function of the variables beta and s, with complex values. The computation lasts for decades without obtaining a result.

I was wondering whether there exists subroutines / methods / tricks that could be helpful to accelerate the integration process. I have attached a Maple script of the double integral of interest. Rough precision would be fine (4 or 5 digits).

Any help would be highly appreciated.

Thanks

Federiko

Question.mw

I am tryng to change variables in multiple integral as below, but receive error. Help me to do so.
 

``

restart

Error, invalid input: with expects its 1st argument, pname, to be of type {`module`, package}, but received shareman

 

with(IntegrationTools):

V := Int(Physics:-`*`(f(Physics:-`^`(x, 2)), g(y)), [x = a .. b, y = c .. d])

Int(f(x^2)*g(y), [x = a .. b, y = c .. d])

(1)

``

Change(V, {x = u-W, y = v-Q})

Error, (in IntegrationTools:-Change) missing a list with the new variables

 

``


 

Download inttt.mw

The maple I used at school is a much older version and when I do Definite Integrals there and copy it to Word as part of the project, it just copies perfectly.

Now I have Maple 16 at home and when I have a definite integral, I have to copy it using copy special and take it as an image to MS Word. That's not the problem. The problem is the limits sometimes seem to be cut off. The left hand-side image is the older version of Maple. Limits look perfect and even the integral sign is darker and so on. The right hand-side image is the Maple 2016. Can anyone help me change the style on Maple 2016 so it's like the old one so the integrals look better on my project. Thanks alot

 

I am trying to calculate the integral

where

Maple cannot calculate the integral. I tried to expand theta in the series form and substitute in the integral, still cannot calculate it.

any suggestion to tackle this problem whould be helpful.

Thank you

 

hi.

please help me for remover this problem.

''''

-Float(infinity)*signum((5.*A3*A1-24.*A2^2)*A1/A2^2)''''

ReducedCantiler.mw
 

restart

f := -(2/3)*eta^3+(1/2)*eta^2+eta; -1; g := -eta^2+1; -1; h := -eta^2+1; 1; F := proc (eta) options operator, arrow; A1*f end proc; 1; G := proc (eta) options operator, arrow; A2*g end proc; 1; H := proc (eta) options operator, arrow; A3*h end proc

proc (eta) options operator, arrow; A3*h end proc

(1)

Q1 := diff(F(eta), eta, eta, eta)+.5*H(eta)*((diff(F(eta), eta))^2+F(eta)*(diff(F(eta), eta, eta)))/G(eta)^2+2*(diff(G(eta), eta))*(diff(F(eta), eta, eta))/G(eta)-(diff(H(eta), eta))*(diff(F(eta), eta, eta))/H(eta); 1; Q2 := diff(G(eta), eta, eta)+H(eta)*((diff(F(eta), eta))*G(eta)+.5*F(eta)*(diff(eta, eta)))/G(eta)^2+2*(diff(G(eta), eta))^2/G(eta)-((diff(H(eta), eta))*(diff(H(eta), eta)))/H(eta)+(diff(F(eta), eta, eta))^2-(H(eta)/G(eta))^2; 1; Q3 := diff(H(eta), eta, eta)+(.5*1.3)*H(eta)*(5*(diff(F(eta), eta))*H(eta)+F(eta)*(diff(H(eta), eta)))/G(eta)^2+2*(diff(G(eta), eta))*(diff(H(eta), eta))/G(eta)-(diff(H(eta), eta))^2/H(eta)+(1.3*1.44)*H(eta)*(diff(F(eta), eta, eta))/G(eta)-(1.3*1.92)*(H(eta)/G(eta))^3

-2*A3+.65*A3*(5*A1*(-2*eta^2+eta+1)*A3*(-eta^2+1)-2*A1*(-(2/3)*eta^3+(1/2)*eta^2+eta)*A3*eta)/((-eta^2+1)*A2^2)+4*A3*eta^2/(-eta^2+1)+1.872*A3*A1*(-4*eta+1)/A2-2.496*A3^3/A2^3

(2)

Eq1 := int(Q1*f, eta = 0 .. 1);

-0.2600000000e-1*A3*(24.*A1*A2^2-65.*A1*A2*A3+64.*A3^2)/A2^3

(3)

sol := solve({Eq1 = 0, Eq2 = 0, Eq3 = 0}, {A1, A2, A3}); J := min(select(`>`, sol, 0))

Error, invalid input: `>` expects 2 arguments, but received 1

 

A11 := evalf(simplify(sol[1, 1])); A22 := evalf(simplify(sol[1, 2])); A33 := evalf(simplify(sol[1, 3]))

Error, invalid subscript selector

 

``


 

Download ReducedCantiler.mw

 

Hi,

I have a second order, linear, non-homogeneous differential equation and for the solution Maple takes the particular solution under a indefinite integral form. After I substitute the values of the coefficients I want Maple to perform the integration. The integration is possible because I individually integrated one small part of the expression. The full expression has a lenghty sumation of different indefinite integrals so it would be cumbersome to perform each integration by hand.

Can somebody help me force Maple to perform these integrations?

I already tried eval, evalf, simplfy and it doesn't work.

Thanks a lot.

Let:

f:=x->1/sqrt(2*Pi)*exp(-x^2/2);

I.e. f is a standard Gaussian PDF.

Then (in Maple 2016.1):

Int(convert(f(x)*f(y)*x*x*abs(x+y),piecewise,x),x=-infinity..infinity,y=-infinity..infinity):
evalf(%);

Returns:

1.692568751

However (again in Maple 2016.1):

int(convert(f(x)*f(y)*x*x*abs(x+y),piecewise,x),x=-infinity..infinity,y=-infinity..infinity):
evalf(%);

Returns:

-0.5641895835

This is clearly incorrect, as the integral of a positive function must be positive.

This also seems to be a problem in which ever version of Maple is used behind the scenes on this forum.

int(convert(1/sqrt(2*Pi)*exp(-x^2/2)*1/sqrt(2*Pi)*exp(-y^2/2)*x*x*abs(x+y),piecewise,x),x=-infinity..infinity,y=-infinity..infinity)

gives:

int(convert(1/sqrt(2*Pi)*exp(-x^2/2)*1/sqrt(2*Pi)*exp(-y^2/2)*x*x*abs(x+y),piecewise,x),x=-infinity..infinity,y=-infinity..infinity)

Hi all,

How to calculate this integral:

for k>0,m>0

Int(exp(-(1/2)*v/k)*v^3*exp((1/2)*v/m)*Ei(1, -(1000*I)*v+(1/2)*v/m), v = 0 .. infinity)

I'm  tried to take advantage of with(IntegrationTools) but I failed

and and I got a strange result ,like this:

Integral.mw

how to find the integration of z(x) form 0 to x with the given condition...
 

diff(z(x), x) = x*Typesetting:-delayDotProduct(b, 1+3*y(x))/(a^2*(1-(x/a)^2));

diff(z(x), x) = x*(b.(1+3*y(x)))/(a^2*(1-x^2/a^2))

(1)

`%%where%`, y(x) = b*((1-(x/a)^2)^(1/2)-(1-(R/a)^2)^(1/2))/(3*(1-(R/a)^2)^(1/2)-(1-(x/a)^2)^(1/2));

`%%where%`, y(x) = b*((1-x^2/a^2)^(1/2)-(1-R^2/a^2)^(1/2))/(3*(1-R^2/a^2)^(1/2)-(1-x^2/a^2)^(1/2))

(2)

with*condition; -1; z(R) = ln(1-(R/a)^2)

z(R) = ln(1-R^2/a^2)

(3)

``


 

Download integration.mw

How get answer of this integral

int(1/u.t.exp(-t/u), t = 0 .. infinity)

i want to have plot int.  but i get error 

"Error, (in type/EvalfableProp) too many levels of recursion"

how can i draw this plot

please help me . thank you

There seems to be a bug in determining the folowing integral analytically:

integrate(-(3/2*(exp(-(1/4)*x)*x-sqrt(Pi)*erf((1/2)*sqrt(x))*sqrt(x)))/(sqrt(x)*sqrt(Pi)*erf((1/2)*sqrt(x))), x = 0..1)

Maple gives as a result

3/2

However, numerically integrating it

integrate(-(3/2*(exp(-(1/4)*x)*x-sqrt(Pi)*erf((1/2)*sqrt(x))*sqrt(x)))/(sqrt(x)*sqrt(Pi)*erf((1/2)*sqrt(x))), x=0..1,numeric)

gives

0.1195461293

In fact, integrating it from a to b,

integrate(-(3/2*(exp(-(1/4)*x)*x-sqrt(Pi)*erf((1/2)*sqrt(x))*sqrt(x)))/(sqrt(x)*sqrt(Pi)*erf((1/2)*sqrt(x))), x=a..b)

gives

-3/2 a + 3/2 b

suggesting that Maple thinks the integrand is just 3/2. If one plots it, then it becomes obvious that this is not the case.

I dont understand how to approach this question, can anyone explain what it means by bessel and myJ?? Ive tried but i cant get the integral to work with the dy?

 

I  encountered a non-integrable integral in the process of solving the following process, . How to achieve its numerical solution? Such as in a looping   code:

#######
pa[i] := pa[i-1]-(Int(subs(t = tau, Lpa[i-1]+Na1[i-1]-Na2[i-1]), tau = 0 .. t)); 

pw[i] := pw[i-1]-(Int(subs(t = tau, Lpw[i-1]+Nw1[i-1]-Nw2[i-1]), tau = 0 .. t)); u[i] := u[i-1]-(Int(subs(t = tau, Lu[i-1]+Nu1[i-1]+Nu2[i-1]), tau = 0 .. t));

######
Detailed code see annexBC2.mw

Correct computatiton for

for reasonable expressions f(x,y), g(x,y) would be very useful in double integrals.

For the moment this is not possible. Too many bugs:

int(Heaviside(1-x^2-y^2), x=-infinity..infinity, y=-infinity..infinity); #should be Pi
                           undefined
int(Heaviside(1-x^2-y^2), x=-1..1, y=-1..1); #should be Pi
                               0
int(Heaviside(y-x^2), x=-1..1, y=-1..1); #should be 4/3
                               -2

int(Heaviside(y-x^2), y=-1..1, x=-1..1); #This one is OK!
                              4/3

 

 

 

 

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