Finding a function based on some of its data values

Questions
,
1

Dear Wolfram Community

The function f is sought. The function takes two parameters l and v. The function f asymptotically approaches 100. When l increases, the output of function f decreases. When v increases, the output of function f increases. Here are some values and the corresponding output of the function:

The variable l just can have values between 1 and 40.
Whenever v is 0, the output of f is 0.

Here are some more values:

  • f(l;v) = y

  • f(1;0) = 0

  • f(1;4) = 16 2/3

  • f(1;5) = 20

  • f(1;20) = 50

  • f(1;30) = 60

  • f(1;60) = 75

  • f(l, infinite) = 100

  • f(10;1) = 1,25

  • f(10;11) = 12,25

  • f(10;20) = 20,25

  • f(10;11) = 12,25

  • f(10;40) = 33 2/3

  • f(10;78) = 49,75

  • f(10;100) = 55,93

  • f(10;150) = 65,56

  • f(10;200) = 71,74

  • f(10;250) = 76,04

  • f(40;1) = 0,26

  • f(40;10) = 2,51

  • f(40;50) = 11,39

  • f(40;100) = 20,45

  • f(40;200) = 33,95

  • f(40;500) = 56,24

  • f(40;1000) = 71,99

  • f(40;1500) = 79,4

I am looking for a mathematical way to find the function. I think it may be a logarithmical or asymptotical function.

Greetings and thanks for reading
Felix

2

You can start with the list of triples and then find a fit with your favourite formula:

triples = {{1, 0, 0}, {1, 4, 50/3}, {1, 5, 20},
   {1, 20, 50}, {1, 30, 60}, {1, 60, 75},
   {10, 1, 1.25}, {10, 11, 12.25}, {10, 20, 20.25},
   {10, 11, 12.25}, {10, 40, 101/3}, {10, 78, 49.75},
   {10, 100, 55.93}, {10, 150, 65.56}, {10, 200, 71.74},
   {10, 250, 76.04}, {40, 1, 0.26}, {40, 10, 2.51},
   {40, 50, 11.39}, {40, 100, 20.45}, {40, 200, 33.95},
   {40, 500, 56.24}, {40, 1000, 71.99}, {40, 1500, 79.4}};
FindFit[triples, formula, parameters, {l, v}]

I found that the following function gives a good fit:

53.5507 ArcTan[
  0.557648 (1 - 0.00014188 (-10 + l) (-1 + l)) Sqrt[v/(2 + l)]]^2