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Hi All,

I would like to request information the representation of the following result from Mathematica :

Mathematica result:    MeijerG[{{0, 1/2}, {}}, {{0, 1}, {-1, -1}}, a, 1/2]

Maple is able to take: MeijerG[{{0, 1/2}, {}}, {{0, 1}, {-1, -1}}, a] which is represented as
                                MeijerG([[0, 1/2], []], [[0, 1], [-1, -1]], a)

Could you please advise me on how to implement this function in maple. I would be grateful if you also include matlab in the discussion.

The matiematica output is a result of the following integration:

Integrate[r*(BesselI[1, al*r]* BesselK[1, al*r]), r]

 

Looking forward to your reply. Thanks in advance.

Regards

Raj

Dear all,

I am trying to use Maple for Finite Element calculations. I have a 2d setup with linear basis functions and a 2d gaussian kernel that can rotate with respect to the axes. Attached please find the work sheet I am using.

Basis_function:

B := (x1,y1,x,y) -> max(0, 1-abs(x-x1))*max(0, 1-abs(y-y1))

transmissibility function:

t_hat:= (x1,y1,x,y) -> A*exp(-a*(x-x1)^2-2*b*(x-x1)*(y-y1)-c*(y-y1)^2)

where A  and a,b,c are positive constants. a,b,c are calculated based on an angle phi and the two variances of the gaussian function.

I want to calculate the following function for different points (x1,y1) , (x2,y2):

trans := (x1, y1, x2, y2) -> int(int(B(x1, y1, xz, yz)*(int(int(t_hat(xz, yz, xp, yp)*(B(x2, y2, xz, yz)-B(x2, y2, xp, yp)), xp = x2-10*sigma1 .. x2+10*sigma1), yp = y2-10*sigma2 .. y2+10*sigma2)), xz = x1-hx .. x1+hx), yz = y1-hy .. y1+hy);

this integral in the form that is in the work sheet, works well for phi=0 and the results are what I want (numbers that go to zero as we move points 1 and 2 away from each other). for other values for phi it either gives an error (too many levels of recursion) or it returns expressions that seem unreasonable when I evaluate them (they don't go to zero).

for example, it doesn't work for phi = 0.5 at all. for phi = Pi/4 it will calculate some expression,

but as you move away from a point (e.g. trans(0,0,100,100)) the value does not become smaller than a certain value, but they should go to zero.

It seems that what I am trying to do is very sensitive to a,b,c, but actually it shouldn't be so different. I like to avoid exact integration, and just get a number, but I have no idea how to do this numerically. and I don't know how to write the problem in a way that would work for every angle phi.

any ideas?

thanks in advace,2d_maple_primes.mw

with(plots); with(LinearAlgebra); with(ArrayTools)

``

Transmissibility function specifications

alpha := 1;

1

 

4

 

1

 

(1/4)*Pi

(1)

 

a := (1/2)*(cos(phi)/sigma1)^2+(1/2)*(sin(phi)/sigma2)^2;

17/64

 

15/64

 

17/64

 

A = (1/8)/Pi

(2)

 

 

Transmissibility*Kernel

t_hat := proc (x1, y1, x, y) options operator, arrow; A*exp(-a*(x-x1)^2-2*b*(x-x1)*(y-y1)-c*(y-y1)^2) end proc

proc (x1, y1, x, y) options operator, arrow; A*exp(-a*(x-x1)^2-2*b*(x-x1)*(y-y1)-c*(y-y1)^2) end proc

(3)

B := proc (a, b, x, y) options operator, arrow; max(0, 1-abs(x-a))*max(0, 1-abs(y-b)) end proc

proc (a, b, x, y) options operator, arrow; max(0, 1-abs(x-a))*max(0, 1-abs(y-b)) end proc

(4)

trans := proc (x1, y1, x2, y2) options operator, arrow; int(int(B(x1, y1, xz, yz)*(int(int(t_hat(xz, yz, xp, yp)*(B(x2, y2, xz, yz)-B(x2, y2, xp, yp)), xp = x2-10*sigma1 .. x2+10*sigma1), yp = y2-10*sigma2 .. y2+10*sigma2)), xz = x1-hx .. x1+hx), yz = y1-hy .. y1+hy) end proc

proc (x1, y1, x2, y2) options operator, arrow; int(int(B(x1, y1, xz, yz)*(int(int(t_hat(xz, yz, xp, yp)*(B(x2, y2, xz, yz)-B(x2, y2, xp, yp)), xp = x2-10*sigma1 .. x2+10*sigma1), yp = y2-10*sigma2 .. y2+10*sigma2)), xz = x1-hx .. x1+hx), yz = y1-hy .. y1+hy) end proc

(5)

 

#####testing here######

#for phi == 0 the results are what i want, numbers that go to zero as the points go far from each other. for phi != 0 trans returns an expression and the evaluation of that expression doesn't go to zero as we move the points far apart.

NULL

trans(0, 0, 0, 1)

trans(0, 0, 1, 0)

trans(0, 0, 5, 5)

``

This should be zero for any angle

trans(0, 0, 50, 50)

 

 

 

``


Download 2d_maple_primes.mw

Dear All,

I have problems connected with a double integration in Maple.

The simplified task with its description is in the uploaded file.

Any help will be appreciated.

Thank You! 

 

mp_qn_double_integra.mw

Dear All,

 

I would like perform a symbolic integration to the following integrand with Heaviside function:

 

The integral I defined is as follows:

The output I got is

 

Could any one explain me how to avoid the undefined in the output. I tried to define the assumtions to constrain the solution. However, I failed to supress the undefined.

I expect a result of the integration should also be piecewise function.

 

Appreciate you constructive inputs.

 

Thanks!

 

 

hi all.
i have a system of ODE's including 9 set of coupled OED's . 

i have  converted second deravaties to dd2 , in other words : diff(a[i](t),t,t)=dd2[i](t) . i =1..9 :

and i have set these 9 equations in form of vibrational equations such :  (M.V22)[i]+(K(t).V(t))[i]+P(t)[i] = eq[i] , where M is coefficient Matrix of second  derivatives , V22 is Vector of second derivaties , for example V22[1] = diff(a[1](t),t,t) , and  P(t) is the numeric part of equations ( they are pure number and do not contain any symbolic function ) and K(t).V(t) is the remaining part of equations such that : (K(t).V(t))[i] = eq[i] - (M.V22)[i] - P(t)[i]  , and V(t) are vector of a[i](t)'s which V(t)[1] = a[1](t) ,

i have used step by step time integration method (of an ebook which i have attachted that part of ebook here), when i set time step of solving process to h=0.01 , i can solve this system up to time one second or more, but when i choose h=0.001 or smaller, the answer diverges after 350 steps . i do not know whether the problem is in my ODS system, or maple can not handle this ?the answer about the time t=0.3 are the same in both steps, but after that, the one with stpe time h=0.001 diverges. my friend has solved this in mathematica without any problem, could any body help me ?! it is urgent for me to solve this problem,thnx everybody.


ebook.pdf  step_=_0.001.mw  step_=_0.01.mw 

Please advise as to the proper coding entries needed in the triple integration palette  to transform from the Cartesian placeholders x;y;z to spherical coordinates rho; theta; phi so that the triple integration palette can be used in spherical coordinmates. Dr. Lopez alreadyb has a standalone template which does this but I would like to set a palette option for spherical  calculations.

 

Thanx

Hi. 

 

I have this expresssion:

 

int(sqrt(1+(diff(f(x), x))^2), x = 0 .. 17)

I know the result is:

30.75115129-1.376953493*10^(-8)*I

 

But it takes 13 minutes for maple to get to that solution, and i do not need the part with the I, complex number.

 

Is there any way to make maple evaluate this expression without computeing the complex number? A setting in generel, or some function to put the expression into?

 

Thanks in advance!

 

Emil Kristensen

I'm trying to numerically calculate the following:

int(e^(-1.5*t)/sqrt(t*(t+1)), t = 1 .. infinity)

But Maple can't do it.

I then made it a lot simpler and tried to calculate the following:

evalf(int ((e^(-t)), t=1..infinity));

 

Thats just e^(-t), integrated from t=1 to t=infinity.

 

Maple just gives me back the original equation in the first case, and in the second gives me a limit that I can see has a numerical answer, but Maple can't. Is this something Maple should be able to do and I'm just now pushing the right buttons? I'm using Maple 12, which I know is old and limited, but these really aren't very complicated numerical integrations.

 

 

 

after integration,
result is x^2

f := x^2

we know in the range 5..-5 = 5^2 - (-5)^2 = 0

this example is not good enough, as i can be any number x, -x

if in another example, i use a and b to represent that would like to find

3*a^2 - 3*b^2 = 0, how to find possible a and b?

my example is not good, is there example that a and b are different number, not only in sign

Mapleprimes_Integral.mw

I have a question regarding following problem:

assume(a > 0, a < 1, t > 0, Z0 > 0, z > 0)

f1 := proc (z) options operator, arrow; 1/z end proc

proc (z) options operator, arrow; 1/z end proc

(1)

I_1 := int(f1(z)*ln((a*z+1)/(1+z/a)), z = 0 .. Z0); 1; MultiSeries:-asympt(%, Z0, 3)

2*ln(Z0)*ln(a)+(a^2-1)/(a*Z0)-(1/4)*(a^4-1)/(a^2*Z0^2)+O(1/Z0^3)

(2)

Using the representation which should hold for all a>0 and z>0

int(z*exp(t)*(a^2-1)/((exp(t)+a*z)*(exp(t)*a+z)), t = 0 .. infinity); 1; combine(%)

ln((a*z+1)*a/(z+a))

(3)

I'm calculating the result the other way around

int(z*exp(t)*(a^2-1)*f1(z)/((exp(t)+a*z)*(exp(t)*a+z)), z = 0 .. Z0); 1; I_2 := int(%, t = 0 .. infinity); 1; MultiSeries:-asympt(%, Z0, 3)

2*ln(Z0)*ln(a)+(a^2-1)/(a*Z0)-(1/4)*(a^4-1)/(a^2*Z0^2)+O(1/Z0^3)

(4)

plot(eval([I_1, I_2], a = 1/2), Z0 = 0 .. 10)

 

So the results are the same.

But if I calculate this with another function

f2 := proc (z) options operator, arrow; 1/(z*(z+a)) end proc

proc (z) options operator, arrow; 1/(z*(z+a)) end proc

(5)

I_3 := int(f2(z)*ln((a*z+1)/(1+z/a)), z = 0 .. Z0); 1; MultiSeries:-asympt(%, Z0, 3)

-dilog(a^2)/a-2*ln(a)/Z0+(1/2)*(2*ln(a)*a^2+a^2-1)/(a*Z0^2)+O(1/Z0^3)

(6)

int(z*exp(t)*(a^2-1)*f2(z)/((exp(t)+z*a)*(exp(t)*a+z)), z = 0 .. Z0); 1; I_4 := IntegrationTools:-Change(int(%, t = 0 .. infinity), t = ln(z)); 1; MultiSeries:-asympt(%, Z0, 3); 1; simplify(convert(convert(MultiSeries:-series(I_4, Z0, 1), polynom), polynom))

ln(a)*(ln(Z0)+ln(a))/a

(7)

I get another result :-/ The Integral doesn't even vanish in the limit Z0 -> 0

Though if I take the limit prior:

int(z*exp(t)*(a^2-1)*f2(z)/((exp(t)+z*a)*(exp(t)*a+z)), z = 0 .. infinity);

-dilog(a^2)/a

(8)

the result is correct. What is the problem here?



Download Mapleprimes_Integral.mw

 

 

main_screened_Poisso.mw

In the attached Maple program I construct a matrix of which the entries involve numerical integration. Nothing difficult, I think, yet the ReducedRowEchelonForm is taking forever. What am I doing wrong?

Dear All,

 

I would like to perform a numerical integration of the following. Please let me know the method.

 

The integration consits of one variable essentially. Since the symbolic integration is not possibl and hence I have to do it numerically.

Appreciate you reply. Thanks

 

> restart; with(plots); with(LinearAlgebra); with(IntegrationTools); pde := rho(x)*(Diff(u(t, x), t, t))+lambda(x)*(Diff(u(t, x), t))-(Diff(P(x)*(Diff(u(t, x), x)), x))-Q(x)*u(t, x)+R(x);

> eq1 := int(pde*phi[i](x), x = -1 .. 1);

eq2 := expand(eq1);

> eq3 := applyop(u -> Parts(u, phi[i](x)) end proc, 3, eq2);

i get an error

Error, (in applyop) improper op or subscript selector

how to correct it??

Dear Experts,

 

Analytical integration is not a choice for the integrals listed here. Hence, Maple is not able to find the numerical integration for the following oscillatory functions. However mathematica can. However, before I take the mathematica results, just need to check with you.

Following link shows that for a diverging series, the (Numerical) integral is finite.

http://books.google.com.sg/books?id=vntDnyh0gacC&pg=PA664&lpg=PA664&dq=nintegrate++seqlim&source=bl&ots=iNYR1o6kVd&sig=VuiQuiUMDEEGOBnguSwfcPPSHQA&hl=en&sa=X&ei=k_J4UofXJYjNkwW79IDYBw&ved=0CE4Q6AEwBA#v=onepage&q=nintegrate%20%20seqlim&f=false

 

1)

 Maple: 

eval(int(r^2 BesselJ(1,r)* BesselJ(0,r), r = 0..infinity)) 

Float(undefined)

Mathematica: 

NumberForm[ NIntegrate[BesselJ[0, x]*BesselJ[1, x]*x^2, {x, 0, Infinity},   AccuracyGoal -> 20], 15]

SequenceLimit::seqlim: The general form of the sequence could not be determined, and the result may be incorrect. >>

-95982.37707206068

 

2)  Maple:
       eval(int(r* BesselJ(0,r), r = 10..infinity)) 

                         Float(undefined)

Mathematica: 

NumberForm[ NIntegrate[BesselJ[0, x]*x, {x, 10, Infinity},   AccuracyGoal -> 20], 15]

SequenceLimit::seqlim: The general form of the sequence could not be determined, and the result may be incorrect. >>

-0.434727

 

Any conclusions on the result.

a) Why maple not able to evaluate the integrals.  b) Are the result of the Mathematica can be considered as appropriate.

 

Attached is the maple file for your consideration.

Integrations.mw 

 

Lookinf forward to your reply.

 

Thanks.

Consider:

> restart:
> int(exp(I*x)/x,x=-infinity..infinity,CauchyPrincipalValue)
                                                                                                   Pi I

> int(exp(I*3*x)/x,x=-infinity..infinity,CauchyPrincipalValue)
                                                                                                   Pi I

> int(exp(I*k*x)/x,x=-infinity..infinity,CauchyPrincipalValue) assuming k::positive;
                                                                                                  2 I Pi

I would expect the last line to also give me Pi I

Mathematica gives a Pi I via:

FullSimplify[Integrate[Exp[I k x]/x, {x, -\[Infinity], \[Infinity]}, PrincipalValue->True, Assumptions->{k>0}],  Assumptions->{k>0}]


Any idea what this could be?

thanks.

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