Items tagged with numericalnumerical Tagged Items Feed

how to evaluate numerical inverse Laplace transfor...

4 hours ago
1 0

I would like to apply inverse Laplace transform to U(x,p), which is defined by

For simplicity with my calculations, I assumed p:=i*beta^2. That is why I have the following equation after applying Laplace transform

(beta=0 is not a pole, that is why I removed the last term in my calculations later. Because there is no contribution) where

Here p and beta are complex values, we can write Re(p)=-2*Re(beta)*Im(beta), Im(p)=(Re(beta))^2-(Im(beta))^2 due to p:=i*beta^2. I numerically compute the roots of h(beta), you can find the numerical values of beta (I assumed digits are 50 due to accuracy ) betap.mw

Finally, I would like to plot U(x,t) with the values t=0.8, lambda=1, L=10, k=1. For checking the figure give t=0 and observe that U(x,0)=0.

I am expecting the plot is more or less like the following figure

PS: I already tried to solve and plot the problem, but I could not find where I make a mistake. I did not share the worksheet, because I know it is hard to find the mistake easily from someone's worksheet. Thank you!

Iterative Gauss-Seidel...

July 09 2016
1 9

I am trying to solve a matrix system to find the relative arrival rates of a queueing network using Gauss-Seidel.The maple commands are below:

restart;
with(Student[NumericalAnalysis]); with(LinearAlgebra);

A := Matrix([[1, -.333, -.333, -.333], [0, 1, -.333, -.333], [0, -.333, 1, -.333], [0, -.333, -.333, 1]]);

IsDefinite(A, 'query' = 'positive_semidefinite');

true

b := Vector([1, 1, 1, 1]);

IterativeApproximate(A, initialapprox = Vector([1, 1, 1, 1]), tolerance = 10^(-3), maxiterations = 20, stoppingcriterion = relative(infinity), method = gaussseidel);

Error, (in Student:-NumericalAnalysis:-IterativeApproximate) check that the augmented matrix has the correct dimensions

I do not understand this error as the matrix is 4x4 as shown. Can anyone see where I went wrong?

Numerical integration hangs...

July 04 2016
0 2

Hello Mapleprime users

I am having an issue with a numerical integration calculation. I have a large(ish) polynomial integrand and need to apply two integrations which I am using the _cuhre method for. See attached the minimum working example file.

Numerical_integration_HF.mw

Firstly some simplifications are done to the integrand and then a basic for loop which calculates the integration for r from 0 to 20 in steps of 0.1. The issue occurs when the loop hits r~2.5 (on my machine, Maple 2015, i5, 16GB ram). Up to that point the calculation is steady with the following stats per point calculated:

memory used=87.59MiB, alloc change=0 bytes, cpu time=3.45s, real time=3.15s, gc time=439.16ms

Then when the calculation gets to ~2.5 it just hangs and will not calculate past it. Any ideas as to what is going on here?

Any help would be appreciated with this issue.

Yeti

3D Numerical Integration...

July 03 2016
0 2

Hi, Maple community

I am trying to integrate a 3D data numerical set. I managed to execute an array interpolation algorithm and apply it to plot a smooth surface but I do not know how to integrate it numerically. I want to be able to integrate the data not only in its domain but also in sub-domains for multiple purposes.

I've found here a topic about it and tried to apply suggestions made with no success.

Here I copy the worksheet steps:

restart:
with(LinearAlgebra):
with(CurveFitting):
with(plots):

Initial Parameters

Nx:=8;    Points in the x direction
Nz:=6;    Points in the z direction
Lx:=3;    This means x goes from 0 to 3
Lz:=2;    This means z goes from 0 to 2
Delta[ze]:=evalf(Lz/(Nz-1));
Delta[xe]:=evalf(Lx/(Nx-1));

DataPn:=Matrix(Nx,Nz,0):         Matrix to store values in the 3rd direction

The following step is a function that I used to simulate data that I will later on will get by measurings.

for i from 1 to Nx do
for j from 1 to Nz do
DataPn[i,j]:=evalf(sin(Pi*(i-1)*Delta[xe]/Lx)*sin(Pi*(j-1)*Delta[ze]/Lz)):
end do;
end do;

Setting data to use ArrayInterpolation

datax:=Array([seq((i-1)*Delta[xe],i=1..Nx)]);
dataz:=Array([seq((j-1)*Delta[ze],j=1..Nz)]);

In this step here, I don't really understand what this argument [[a],[b]] means but ArrayInterpolation cannot work without it.

MI:=(a,b)->ArrayInterpolation([datax, dataz],Array(DataPn),[[a],[b]],'method' = 'spline')[1,1];

Plot and display the interpolation and the real function to compare the quality of the interpolation, actually it does an excellent job, at least with this data.

G1:=plot3d(MI, datax[1]..datax[Nx], dataz[1]..dataz[Nz],labels=[x,z,Pn]);
G2:=plot3d(evalf(sin(Pi*x/Lx)*sin(Pi*z/Lz)),x=0..3,z=0..2,color=red);
display(G1,G2);

And finally I found this integration method here in mapleprimes

evalf(Int(x->evalf(Int( y->MI(x,y), 0..3,method=_d01akc)),0..2,method=_d01akc));

And the result is 1.102279199. I thought maybe I was setting wrongly the integration limits or the order in the command but in any case the result was near the analytic case which was:

int(evalf(sin(Pi*x/Lx)*sin(Pi*z/Lz)),x=0..3,z=0..2);

2.431708408

In this scenario I know the result is wrong because I can compute the exact solution, but later on I won't be able to do so. Also, later I'm interested in the integration, let's say, x from 0..0.5 and later from 0.5..1 and so on (keeping z always from 0 to 2) So, the main questions are:

1. Am I setting something wrong?
2. If I delete the method option in the integral, it takes a lot of time for the program to compute any result. Why?
3. Is there any other way to integrate this? I mean, any other way to write the command to compute the integral.
4. What does the [[a],[b]] argument stands for in ArrayInterpolation?

Regards and thanks for your help.

Numerically solving parametric integral...

April 13 2016
0 1

Hello everyone,

I am trying to solve numerically int( f(t,z) , t=0..T ) = 0 , in z for a cumbersome f.

I tried z1=fsolve( int( f(t,z) , t=0..T ) = 0 , z). But then I tried int( f(t,z1) , t=0..T ) and the result is clearly not zero nor anything small.

It looks like Maple evaluates analytically the integral, and does it wrong (check this for more details) so fsolve uses the wrong equations.

Anyone knows how I can force Maple to evaluate numerically the integral at each step of the fsolve function?

Thank you!

April 06 2016
0 2

I am solving a complicated ODE and I would like to know if there is a way for Maple to output the ODEs without doing any numerical substitutions for known parameters. Say one of my parameters, call it P, is initialized (there are many more but Ill just simplify and consider one here) to a value of 10. In order to chekc that I have coded the ODEs correctly it would help me if Maple does not substitue with the numerical value 10 for P when displayng teh ODEs, but rather keeps P, as a parameter. Is there a way to achieve this?

Numerical evaluation of integrals...

March 31 2016
1 4

Hi all,

I'm trying to perform some calculations containing exponential integrals. Here is a snippet of my code:

restart;

H:=0.5:

q[0]:=10^5:
Es := 4*10^9:

p[0]:=10^6:
q[s]=10^10:

C := (q) -> q^(H+1.5):

G := (q) -> (int(q^3*C(q), q = q[0] .. q)):
P := (q) -> 1/sqrt(G(q)):

p := (xi) -> p[0]/P(xi*q[0]):
w:= (q,xi) -> 1/(Pi*(int(C(qs)*qs^3, qs = xi*q[0] .. q)))^(1/2):

U := (xi) -> (int(q*C(q)*w(q,xi)*(Int(exp(-(w(q, xi)*ps/Es)^2)/ps,ps=p(xi)..infinity)),q=xi*q[0]..q[s])):

My objective is to evaluate U for a set of discrete values of xi for further processing e.g. visualisation via plots. Neither value(U(xi)) nor evalf(U(xi)) produces a numerical result so I keep searching for solution. Does anybody have an advice how to solve U?

Regards, lassa

Evaluate sum ...

March 16 2016
2 10

Hi!

I wonder how is it possible to numerically evaluate two-dimensional sum, something like this:

Problem with numerical multiple integration...

January 19 2016
1 7

Hello everyone !

I have a problem when I want to calculate the following multiple integration numerically:

>evalf(Int(exp(sum(x[i],i=1..6)^2),[seq(x[i]=-1..1,i=1..6)]));
value(%);

It doesn't work. But when I replace sum(x[i],i=1..6)^2 with sum(x[i],i=1..6), it works. Is there any feasible solution to my problem ?

Need Numerical Solution...

December 17 2015
0 8

Dear Friends

Hope everything going fine with you. I want the numerical solution of nonlinear system of ordinary differential equations using RK method. The system of ODEs and their required results are present in attached file. I am waiting your quick response.

Help.docx

With my best regards and sincerely.

Mob #: 0086-13001903838

How to achieve a piecewise function?...

November 08 2015
0 6

My goal is to plot the integral J with respect to t and as you can see J is a piecewise function.

This is my code.   hw2_numerical_2.mw

www.mapleprimes.com/questions/206645-How-To--Numerically--Solve-It-

Problem in plotting ...

October 13 2015
0 19

I have the solution to a heat PDE, v and the error esitmate u + cos(x+t) = v

I want to plot log v(1,t) as function of log u(1,t) in maple, but I seem to get an error:

Error, (in plot) unexpected option: ln(u(1, t))

I am attaching my code below.

How to fix this problem?

PDE+cos.mw

How to use function in boundary condition? ...

October 09 2015
0 1

Hi !

I am trying to solve a pde with initial/boundary conditions, in the numeric mode. It works very well when I provide 3 numerical conditions : 1 initial condition (uniform distribution at first), and two boundary conditions.

Now I want to switch the initial condition to a function of r. It's a polynome I obtained after interpolation of another result. I tested it and the function interp20(r) works. But the pdsolve doesn't seem to evaluate the function, when it comes to start the procedure and pdsolve doesn't return a module as it should, although it doesn't send any message error.

You can see the code following, with the error messages I get.

Is it strictly impossible to use a function as an initial condition ?
Is it just a problem I could solve by converting the function or its result to something else ?
Is float the right type of input ? For example, if I write a:=283.15, is it a float ?
Have you already had similar problems ? How did you solve them ? Where could I find working code examples on this ?
Could I solve this problem with non-uniform initial condition with another Maple function ?

Then you very much for your help !

How to pdsolve heat transfer equation?...

October 06 2015
1 2

Hi everyone,

I am trying to solve the equation of heat tranfer, time dependent, with particular Initial and boundary conditions but I am stuck by technical problems both in getting an analytical solution and a numerical one.

The equation

the equation.

I defined a and b numerically. domain is : and I defined surf_power numerically.

The initial condition is : , T0 defined numerically

The boundary condition is : , because it has a shperical symetry.

To me, it looks like a well posed problem. Does it look fine ?

Problem in analytical solution :

It doesn't accept the boundary condition so I only input the initial condition and it actually gives me back an expression that can be evaluated but it never does : I can't reduce it more than an expression of fourier which I can't eval. The solution :
The solution calculated in (0,0). I was hoping T0...

Are you familiar with these problems ? What would be the perfect syntax you would use to solve this ?

The numerical solution problems :

Sometimes it tells me that my boundary condition is equivalent  to 0 = 0, and I don't see why. Some other times it tells me I only gave 1 boundary/initial condition even if I wrote both. Here is what I wrote for example :

(because it kept asking me to add these two options : 'time' and 'range')

Are you familiar with these problems ? What would be the perfect syntax you would use to solve this ? I must at least have syntax problems because even if I keep reading the Help, it's been a long time since I used Maple.

Thank very much for any indication you could give me !

Simon

find root method...

September 19 2015
1 5

hi every one..

how i solve numerically  couple equations which attached below .in solve this equation we must  starting from a very small value of V(voltage) with initial guesses for x1 and x3

near zero and using find root method.it is noted that  the solution at this voltage step are used as initial guesses

for the next voltage step, and the process is repeated..

thanks

2DOF.mw

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