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Laplace Transform of integral

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Laplace Transform of integral

I got some homework to do and cant figure out how to solve an equation.

I have an idea but I'm not sure if I can do it the way I want.

As you can see in the document my problem is that the formula I use has f(t)dt in the integer but the equation I have to solve uses tau in in the integer but it wants me to take the integer from 0 to t.

I don't feel like i explained the problem very well but it should be easier to understand if you read through the document.

There might be other errors in the document but I have been wrestling with that part of the problem for what feels like hours now so I just finished it and threw it up here for you to have a look at.

Anton

1 ACCEPTED SOLUTION

Accepted Solutions
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Re: Laplace Transform of integral

Tay is a dummy variable, this is, some letter but can be any other. The primitive (this is, the integral transformation, not just a number, a function) of y(tau)dtau must to be applied following Barrow rule, this is, substracting the values from the end to the initial point. So, tay dissapear, and the integral is well writed.

In the attached the solution. See the tables for the laplace transform properties to see how it applies to a derivative and to an integral. In mathcad 11 it can be automated to handle the entire process in only one function, but in mathcad 14 is ... better use copy, paste and editing.

Regards. Alvaro.

View solution in original post

14 REPLIES 14
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Re: Laplace Transform of integral

It might be as simple as that. I didn't read the document, only your work sheet. The way the integral is shown is for continuous plotting . But in term of the numerical evaluation it is done between limits, between o and and the "end" in the Odesolve as well. That's my interpretation but wait more collabs agree on that interpretation.

MCADsolve_8.gif

jmG

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Re: Laplace Transform of integral

What I meant by document was Worksheet

I dont think this is the answer I need, since i cant solve it numericaly. I would have no problem solving this equation if all the varibales had a value or if i had to find the laplace for the integer between 0 and t for y(t)*d*t

But how would you find the laplace for the integer between 0 and t for y(tau)*d*tau. I cant find any examples thatt are simmilar, and I cant find any thing in my notes that help me here...

I am beginning to think that it could be typing error on my teachers behalf.

Anton

Highlighted

Re: Laplace Transform of integral

Tay is a dummy variable, this is, some letter but can be any other. The primitive (this is, the integral transformation, not just a number, a function) of y(tau)dtau must to be applied following Barrow rule, this is, substracting the values from the end to the initial point. So, tay dissapear, and the integral is well writed.

In the attached the solution. See the tables for the laplace transform properties to see how it applies to a derivative and to an integral. In mathcad 11 it can be automated to handle the entire process in only one function, but in mathcad 14 is ... better use copy, paste and editing.

Regards. Alvaro.

View solution in original post

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Re: Laplace Transform of integral

Thank you

I looks about right. I got the same solution in another worksheet.

I'll find out if it is correct tomorrow

Anton

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Re: Laplace Transform of integral

Alvaro,

Your approach is not solvable toward an explicit solution. Your Laplace is undetermined naturally and incorrect in the sense of the next DE solver. The problem is not ill posed but "quizly posed" wrt the integrand. The integrand is not a function and will therefore have only a numerical running solution . The integrand and the differential operator is correct but these representation is only to isolate the solution for the running limit, which limit maybe simply linear or any nonlinear function itself. But for the case it is a member of a DE. the upper limit is the "end" of the DE . This type of homework quiz is typical of a "Mathcad classroom" It may have to be entered much differently in other CAS.

MCADlaplace_3.gif

The other point is what is the point to transform something in the Laplace domain if no Laplace algebra is applied, good question indeed ! That problem reminds me the Melanie Turenne ITER DE. It was like abandoned though I had given the final Odesolve. About the ITER solution, it is obvious that there was a discontinuity in the solution and the final solution exhibits this discontinuity about the voltage surge in the cabling system, surge under opening the circuit. The discontinuity is in literature "courant de rupture" on what the breakers are designed wrt the energy it is desired to protect the grid.

Cheers Alvaro & collabs: Jean

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Re: Laplace Transform of integral

The proposed symbolic soltion verify the original equation. By theorems of unicity, given that a solution is proposed (even can be determined if it exists, is hard to handle, needs Lipschitz conditions or other hardest, and are not easy to prove) the solution is also unique. For this read the Check label in the file submited.

The correct interpretation for a definite integral as a function of the upper parameter is the given also. Notice that a "definite integral" is a real number, but a "primitive" is a function, or more preciselly, a familly of functions, if you take in consideration de usual "integration constant" C.

To avoid problems in the interpretations is why the integral parameter (tau in our case) is used, and not the upper limit (t). This is, a dummy variable (variable muda in spanish)

The proposed equation isn't an ODE. Is an integrodiferencial -in spanish- equation, and have a lot of common applications, not just for classroom.

In your gif, or the t in the upper limit of the integral isn't the same as the argument of laplace, or is a bug in mcad11. The property to apply is:

laplace(int(y(tau),tau=0..t),t,s) = laplace(y(t),t,s)/s

Check laplace tables for it.

In the attached the symbolic and the numerical soltuions, and the plots, showing that are basically the same.

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Re: Laplace Transform of integral

Tanks Alvaro,

Nice explanation and setup.

It would be interesting to put it in the Laplace DE solver.

RemToDo for spare time .

Jean

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Re: Laplace Transform of integral

You explanation and the one i got from my teacher to day was almost the same. So my big "problem" was that I didn't understand the function of t and tau but hey I got it done.

for your wieving pleasures here is the final woorksheet. Did it manualy too just get the traning.

Anton

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Re: Laplace Transform of integral

Anton Jelle wrote:

You explanation and the one i got from my teacher to day was almost the same. So my big "problem" was that I didn't understand the function of t and tau but hey I got it done.

for your wieving pleasures here is the final woorksheet. Did it manualy too just get the traning.

Anton

A more formal solution is needed. What you got is intuitively not correct.

MCADintegral_0.gif

jmG

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