Items tagged with lambertw

Is there a way to force the branch choice with the LambertW?

If I turn on all _EnvAllSolutions:=true:

I get a placeholder for the branch. Unfortunately the name of this placeholder changes every time I re-evaluate.

Is there a way to force this to take a certain value?

 

Regards.

I want to solve the following equation for E[k]:

fr = 1.64*10^6*E[k]^2*exp(-8.5/E[k]);

Maple 17 gives

solve(fr = 1640000.00*E[k]^2*exp(-8.5/E[k]), E[k]);


          4.25000000000000             4.25000000000000     
    ----------------------------, --------------------------
            /  5442.65560181792\          /5442.65560181792\
    LambertW|- ----------------|  LambertW|----------------|
            |        (1/2)     |          |      (1/2)     |
            \      fr          /          \    fr          /

Maple 2015 gives:

solve(fr = 1.64*10^6*E[k]^2*exp(-8.5/E[k]),E[k]);
Error, (in solve) invalid input: `evalf/LambertW` uses a 2nd argument, zz, which is missing

This is Maple 2015.2 on OS X 10.10 Yosemite.

I can copy-paste Maple 17's solution into Maple2015 and it evaluates correctly (for a given value of fr).

Is this a bug or am I missing something? Help on LambertW looks identical for both versions of Maple.

M.D.

(Maple 2015)

For the simple ODE with initial condition
dsolve({ diff(y(t),t) = y(t)^2 - y(t)^3, y(0) = 1/10 }, y(t));

dsolve produces two different answers, almost randomly (even after restart or after closing Maple and reloading the worksheet). Namely:

(1)

(2)

or

(2')

but this simplifies to (2), so it's not a "true" bug.

Notice that (2) is correct but (1) is incorrect even for t=0 (the initial condition!):

evalf(eval(RootOf(-ln(_Z)*_Z+ln(_Z-1)*_Z-ln(10)*_Z-ln(9/10)*_Z-I*Pi*_Z+_Z*t-10*_Z+1),t=0))=1/10;

Maple seems to prefer the wrong solution (1) but occasionally produces (2) e.g. in a new whorksheet!
In earlier versions it seems that only (1)  appears.

The same ODE with another IC

dsolve({ diff(y(t),t) = y(t)^2 - y(t)^3, y(0) = 1/100 }, y(t));

 

evalf(eval(%,t=0));

 

is always incorrect. It should be

 

but Lambert's function never shows up!
Let me mention that only the exact solutions are affected, numeric is ok.

Without an initial condition, dsolve always uses LambertW:

dsolve({ diff(y(t),t) = y(t)^2 - y(t)^3}, y(t));

 

 

Can you explain this behavior?

 

 

Maple is providing answers with the gamma function and LambertW function.  I am not a math major.  Although I did take advanced engineering mathematics in grad school, I am not familiar with transcendental functions.  

I did look at some of the help.  I tried to use the convert command to convert to elementary functions, however it did not help.  I looked at the definition for the Gamma function, GAMMA(z), and the upper incomplete Gamma function, GAMMA(a,z) that uses the generalized hypergeometric function 1F1.  Looking at the generalized hypergeometric function 1F1 defintion, hypergeom(n, d, z), it references the general pochhammer function, pochhammer(z, a), and product functions, product(f,k=m..n).  This all seems way too complicated.  

 

eqn1:= R = exp(t/MTBF) * (t/MTBF) * sum(1/(i!), i=0..n);

exp(t/MTBF) * (t/MTBF) * Sum(1/(i!), i=0..n)

exp(t/MTBF) * (t/MTBF) * sum(1/(i!), i=0..n)

 

eqn2:= MTBF = solve(eqn1,MTBF);

solve(R = exp(t/MTBF) * (t/MTBF) * sum(1/(i!), i=0..n),MTBF)

 

Is there any easy way to convert to use basic functions like sum, int, etc.??

I am wanting to use the answer in Excel.  I have found some custom Excel VBA code that I could use to implement these non-standard function in Excel, but that seems like a lot of effort.

Lambert W function for Excel, work on real and complex number (VBForums)

Excel Add-Ins Incomplete Gamma Function (PC Review)

 

Update: Some more background on the problem is as follows. R is the expected reliability say 90%. MTBF is mean time between failures in hours. t is the time period of operation excluding allowable downtime. n is the allowed number of failures. I am looking to find the MTBF needed to achieve a certain required reliability, with the number of failures allowed, and the up-time operational period, being defined by the requirements as well.

To clarify the simplification does not necessarily have to be closed form, and an approximation would be acceptable. (in Excel I currently am using iteration over a number of rows, then using the Goal Seek tool to find the MTBF, however this is less than desirable to have to do everytime when looking at different trade studies) I am not that familiar with how to use assumptions in Maple, and do not know what would be the appropriate assumptions to make when simplifying. Thanks for your help.

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