inverse Laplace transforms - well basic algebra actually

[Simon]

I learn best by understanding the application of it. Yes calculus made no sense to me at school, but when the electronics teacher finished off the math for us so that he could explain Fourier it made more sense.

attached is all that I have from my course on inverse Laplace transforms.

[rstofer]

I learn best by understanding the application of it. Yes calculus made no sense to me at school, but when the electronics teacher finished off the math for us so that he could explain Fourier it made more sense.

attached is all that I have from my course on inverse Laplace transforms.

The meat of it is in Worked Examples 10..13.  In fact, WE 13 has a bit of everything.  Before my grandson even started his Differential Equations course last semester, I told him to spend time on partial fraction expansion.  It's covered in Algebra II but not like it's covered in DEs.  We were laughing about it the other day.  Well, I was laughing, he was groaning.

MATLAB, the only way to do this work:

Script file InverseLaplace:

Code: [Select]

syms a s
F = 1/(s-a)^2;
ilaplace(F)

[/font]

Command line:

Code: [Select]

>> InverseLaplace

ans =
 
t*exp(a*t)

[/font]

Where my grandson attends university, they do all of their work with MATLAB.  There is a mandatory 1 semester course on using MATLAB the very first semester.

[Simon]

This is why university education is no longer that useful. Yes ideally I should be taught to use these tools. So I have a math module that now has turned into two plus if only i just know how to use what I gather is non trivial software. I am supposed to be doing this around a day job but they don't care, take the money and run.

[rstofer]

It is exponentially more difficult to do a degree program while working.  I did it, I hated it but I gutted it out,  I do NOT recommend it.  College is supposed to be fun.  This Covid thing has taken out all of the social interaction and has really ruined the experience.

[Simon]

The problem is that it is not until I get to the assignment that I find that the poorly written crap they charged £500 for has very little to do with what I am doing and that having struggled through it with the aid of books I am now starting again to do the actual learning.

[rstofer]

Your program sounds less than optimal and it may be a result of having so few options.  No doubt you would pick a different program if it were available.

Thanks to Covid, colleges and universities are opening up their courses to online learning and this requires a huge shift in thinking by the staff.  When the class sizes are small (like a dozen students), Zoom is one way to interact.  I think it gets out of hand when there are 100 students or more.  I don't know how the bigger schools are doing it.

I think distance learning is going to be with us forever.

[Simon]

It is less than optimal. It's a sideline of a university that like all universities has to be a business. I don't know how it compares to on campus learning but it certainly does not deliver. It can take me days to get an answer from the tutor who can't even be bothered to include in her out of office email when she will be back. The material often is deficient in English never mind math.

It is the only university that offers online HNC and HND courses in electrical engineering (years one and two of uni for people who won't do a degree). Everyone else seems to be in love with mechanical engineering. Even the well respected and now crap open university has stopped doing electrical.

[rstofer]

Last week, in another thread, I posted links to the Bureau of Labor Statistics (www.bls.gov) and they point out that growth in Electrical Engineering employment doesn't come close to the growth in Computer Science.  I didn't link Mechanical Engineering but it is marginally better at 12,000 new jobs over the next 10 years versus 10,000 for EEs.  Those aren't very promising numbers when you divide by 10 for the annual number.

Software Developers have fantastic job growth (300k+) versus Computer Programmers with a loss of 20,000 jobs over the next 10 years.

https://www.bls.gov/ooh/computer-and-information-technology/software-developers.htm

My grandson changed his major from ME to Applied Mathematics.  Job growth is abysmal but that may be a side effect of the way the job is categorized.  Large data is the growth area.  Machine learning, AI, all that stuff is driving industry.  The job pays better than engineers (Computer and Information Research Scientists) but 10 year growth is pegged at just 5000 new jobs.

In the meantime, Applied Math applies everywhere so his job prospects should be pretty decent.  We'll see...

I pursued EE as a hobby, never intending to work in the field.  I don't think I would recommend it today.  It's easy to get one of 300k+ Software Developer jobs, not so easy to get 1 of 10,000 EE jobs.  Remember, those numbers are over 10 years.  You can expect just 1,000 new jobs in EE per year - pretty slim.

Of course those are all US numbers.  I have no idea what is going on in the UK.

[coppice]

Last week, in another thread, I posted links to the Bureau of Labor Statistics (www.bls.gov) and they point out that growth in Electrical Engineering employment doesn't come close to the growth in Computer Science.  I didn't link Mechanical Engineering but it is marginally better at 12,000 new jobs over the next 10 years versus 10,000 for EEs.  Those aren't very promising numbers when you divide by 10 for the annual number.

Software Developers have fantastic job growth (300k+) versus Computer Programmers with a loss of 20,000 jobs over the next 10 years.

https://www.bls.gov/ooh/computer-and-information-technology/software-developers.htm

My grandson changed his major from ME to Applied Mathematics.  Job growth is abysmal but that may be a side effect of the way the job is categorized.  Large data is the growth area.  Machine learning, AI, all that stuff is driving industry.  The job pays better than engineers (Computer and Information Research Scientists) but 10 year growth is pegged at just 5000 new jobs.

In the meantime, Applied Math applies everywhere so his job prospects should be pretty decent.  We'll see...

I pursued EE as a hobby, never intending to work in the field.  I don't think I would recommend it today.  It's easy to get one of 300k+ Software Developer jobs, not so easy to get 1 of 10,000 EE jobs.  Remember, those numbers are over 10 years.  You can expect just 1,000 new jobs in EE per year - pretty slim.

Of course those are all US numbers.  I have no idea what is going on in the UK.

The Bureau of Labor Statistics of labour statistics haven't a clue what the demand for labour will be more than a year or two into the future. Nobody has. In 10 years most advanced companies will be making a large percentage of their income from products which don't exist right now. Historically this has been the case for the last 100 years, and the need to keep moving to the next big thing doesn't seem to be slowing down.

[rstofer]

The Bureau of Labor Statistics of labour statistics haven't a clue what the demand for labour will be more than a year or two into the future. Nobody has. In 10 years most advanced companies will be making a large percentage of their income from products which don't exist right now. Historically this has been the case for the last 100 years, and the need to keep moving to the next big thing doesn't seem to be slowing down.

Much of their information comes from industry.  What does the industry project?

Everybody is just guessing but some guesses are better than others.  The worst guessers are those signing up for classes.  They have no clue!

I was working on a project when I worked for a major chip manufacturer.  We were going to build a plant in southern California to build mainframes.  Somebody asked about internal heat gain and I suggested that it would remain what it was today.   To which a VP of manufacturing told me "never bet against technology!".  Boy, was he right!  The PCs of today are orders of magnitude more powerful than even the largest mainframe of the late '70s and uses 3 or 4 orders of magnitude less power.

If you choose not to believe the absolute value of the numbers, it might still be reasonable to accept the ratios.  CS > ME > EE  Even with this scheme, EE doesn't look too impressive.

We may not have a decent crystal ball but, one way or another, CS is going to play a huge role.

[IanB]

With regards to EE today, wouldn't much of the high value area come under "systems engineering"?

For example, who designs smartphones? Is that EE, or software development, or is it a highly complex system of interacting parts containing hardware, firmware and software?

[Mechatrommer]

Where my grandson attends university, they do all of their work with MATLAB.  There is a mandatory 1 semester course on using MATLAB the very first semester.

because differential equations are not easily solvable by hand of engineering level people. people higher up in the intellectual hirarchy have to developed numerical method (close approximation) for it to be implemented practically using computer, that is your grandson's MATLAB. and then there are boundary conditions whether we aware of it or behind the screen, that are nonsense from reality perspective. there is place where MATLAB is banned, is when you attend pure mathematics course where you need to understand the inverted triangle and those convoluted sort of musical notes signs and do some abstract mental manipulations is a must.

This is why university education is no longer that useful.

its funny why i have to hear this from time to time? maybe its just for you, why you attend this Laplace course anyway? if you ask a farmer, they will tell you all those diodes and transistors and what you are working on your bench are all nonsense, they are all not needed, crops is what makes you not hungry, money is what matters. are you attending just to get engineering grade salary? the people you are paying £500 to are a bunch of people that need to feed just trying to get the community top notched and "mentally fit". there are areas where Laplace Transform is super important, ie that involves controls and stable feedback loop. most of us dont really appreciate it because we are not in the business, we just use opamp for granted, happy to use our car everyday and enjoy watching rocket science and exhibition jet fighter flying around without knowing behind those are heavy Laplace transform. if its not our business, there is no need for us to dig into it, just as farmers dont need any electrical parts nonsense, they just enjoying to use their cropping machineries. imho your statement is more dangerous than anyone promoting slopiness while working with mains. your statement is like telling kids they dont have to go to higher schools, its ok just to become a farmer, its ok just to aim as a janitor, you wont be hungry.

btw i myself feeling "crippled" when dealing with these weird transform, when all they can give me is a table for limited number of Inverse Transforms. sometime we derived a weird equation from our designed system that is not in the table (like you experienced in OP). i think i have to go back to first principle, there must be reason why they dont teach first principle very well.. well.. this is the first principle of LT for your hate pet...



https://en.wikipedia.org/wiki/Laplace_transform

and this for inverse...


now we can start from there... is it not needed? is like saying all those people working on this since industrial age back to Al-Jabar is for nothing... i bought and downloaded some more advanced Controls books because i want to build a hobby plane, man my brain got dizzy, eyeballs spin and got sleepy very quick when trying to follow them, luckily my hobby plane project is postponed indefinitely, but i know i will have to one day, when i get more serious, when it involves life or death matter. if it tastes bitter then i need to make it tastes better whether i like it or not. if we are not doing it, there is no need for higher education, i agree, its just for us. its ok to be here on low side admit that, let the man up there do their things. but dont say they are not needed or not that useful. fwiw ymmv.

[rstofer]

Some universities, like the University of Florida, have separate math tracks for math majors and engineering majors.  The math majors are going to enjoy the elegance of the math itself; engineers want to know how to apply it.  Completely different focus.

In the end, engineering is nothing by Applied Mathematics.  Once you have the math skills, the rest of the program (EE, ME, CE, etc) is pretty easy.  A few principles, a bunch of math and you're there.

I have the (dis) advantage of graduating in the early '70s when slide rules were king.  We did a lot of stuff by approximation, it was all we had.  We certainly didn't have the MATLAB meshgrid or surf functionality.  We had to work through the plot by hand and then analyze it with a Spirule and the Root Locus method.  Control Systems was a difficult course.  I remember very little...

https://www.nzeldes.com/HOC/Spirule.htm

[Nominal Animal]

Not just Matlab, but Octave (free), Maxima (free), SageMath (free), too.

In particular, if you install Maxima on your machine (it's available for all OSes), you can verify if your inverse Laplace transform of (3-2s)/(s²-4s+9) is correct by telling it
    ilt( (3-2*s)/(s^2-4*s+9), s, t);
    expand(ratsimp(%));
That second line first tries to simplify the expression, then expands the result.  It says
    -e^2t sin(t√5) / √5 - 2 e^2t cos(t√5)
To check that that answer makes sense, we can do the forward Laplace transform to that (and simplify it):
    ratsimp(laplace(%, t, s));

It is useful as an exploration tool, but only if you control yourself so you don't use it to skip the exercises.  See more (and a better inverse Laplace transform) at the maximalist.  In particular, if you write your own notes, it can be useful to verify you don't make silly typos or such errors.

[rstofer]

Not just Matlab, but Octave (free), Maxima (free), SageMath (free), too.

I have Octave and would have used it for the example above but, for some reason, the 'symbolic' package wasn't working.  It's an error on my part, I'm sure.  I often port my MATLAB scripts to Octave just to verify that they are (or aren't) portable.

I have used wxMaxima (Maxima with a GUI frontend) and I like it a lot.  It seems to be about 1 step up from Fortran and I really like Fortran.

I'll have to take a look at SageMath.

I use symbolab.com for solving and desmos.com for graphing.  Both are excellent.

[coppice]

I often port my MATLAB scripts to Octave just to verify that they are (or aren't) portable.

Its frustrating how often something written for MATLAB doesn't work properly with Octave. In most cases the code runs, but gives the wrong answer. So, you need MATLAB installed anyway, so you can run some test cases on both platforms and compare the results.

[rstofer]

I just pay for the MATLAB Home license and call it good.  It costs me a couple of hundred dollars per year (I have several optional packages) but that's the way it goes.  I whine and snivel at renewal time but I get over it pretty quick.

[coppice]

I just pay for the MATLAB Home license and call it good.  It costs me a couple of hundred dollars per year (I have several optional packages) but that's the way it goes.  I whine and snivel at renewal time but I get over it pretty quick.

If its just for occasional testing, just find the students in your family.

[Nominal Animal]

Me, I'm seriously addicted to Maple.  It does a much better job at simplifying complex algebraic expressions than Maxima or SageMath.
I have a license through my Uni till the end of this year, but after that, I may need to buy my own (for Linux).

I very often examine classical atomic potential models (as used in molecular-level material physics with lots of atoms, and in chemistry and biology in examining biomolecules and their interactions via force-field and similar potential models).  Not only does one need the forces, F(r) = -dV(r)/dr, for simulations – so even reading a paper I like to ask Maple what the derivative of the potential looks like, as that is what really matters from simulation point of view –; but models like embedded atom models can have their potential defined via numerical models, and then using a distance function f(r) instead of the distance r (especially f(r)=r² or f(r)=r^-2 in particular, so one can avoid the slow square root operation completely), and examining the theoretical forms, helps understand the corner cases one will encounter when implementing one in a simulator.
So, on average, each week my subconscious tends to bubble up a reformatting/convolution/idea that is interesting enough to experiment on, to see if it might help with these potential models and VERY large distributed simulations (as in millions to billions of atoms)...  And this is not "for work", this is basically what my mind does to me.  Call it a hobby, I guess?

What I can tell, is that by doing this, no equation or mathematical problem really scares me anymore.  Indeed, answering maths questions whose answers I don't know before I find out, has become a nice hobby; much like playing with physical puzzle toys.  (I like those too.)  And I really have started to value theorems and transforms; not for their internal detail, but their power of manipulation.

[rstofer]

I have developed a fascination with analog computing and I have 3 computers, 2 of which are ComDyna 6's  These things are fun to play with.

I can do the same thing with MATLAB's Simulink.  I can plop down integrators, wire them through some attenuators, adders and even multipliers, perhaps add in some adjustment knobs and I'm off to the races.

Here's a more or less silent video on the Predator Prey problem:



I attached a Simulink design for the Predator Prey along with the output

The problem gives a wide open territory where, were it not for foxes, rabbits would grow without bound.  But the more foxes there are, the more fox/rabbit interactions occur and fox population increases but rabbit population decreases until the fox population starts to die off and the rabbit population recovers.

[rstofer]

I just pay for the MATLAB Home license and call it good.  It costs me a couple of hundred dollars per year (I have several optional packages) but that's the way it goes.  I whine and snivel at renewal time but I get over it pretty quick.

If its just for occasional testing, just find the students in your family.

My grandson has the Student license for MATLAB but he uses it every single day.  Moving the license from his machine to mine doesn't seem to be workable.

At first I didn't see the Individual license for Maple but I finally found it: $239  That's entirely reasonable if it isn't a subscription and it doesn't seem to be.

OTOH, MATLAB itself is just the base software.  It's the packages that build on top that are important.  It's a really complete product if you keep adding packages.  Simulink is a great add-on.

[coppice]

My grandson has the Student license for MATLAB but he uses it every single day.  Moving the license from his machine to mine doesn't seem to be workable.

Students are usually able to install MATLAB on multiple machines using the academic licences. Quite a few seem to have it on both a desktop PC and a laptop.