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I PDE with recursive relation

  1. Mar 28, 2017 #1
    hi, I do not know much about PDEs and programs like wolfram alpha and maple dont give me a solution.
    it is possible to calculate the function through PDE?.
    I would appreciate any help

    $$\frac{\partial }{\partial a}f(a,b,n)+\frac{\partial }{\partial b}f(a,b,n)=-n f(a,b,n+1)$$

    $$f(a,b,0)=\frac{\pi }{2} \: \; \, \,, \, \, \, f(a,b,1)=\frac{\pi }{2\sqrt{a b}}$$


    as we know ##f(a,b,1)## We can calculate ##f(a,b,2)\,,\,f(a,b,3),...##

    But we could calculate closed form expression for ##f(a,b,n)## ?

    thanks
     
  2. jcsd
  3. Mar 28, 2017 #2
    i reduce the problem to solve
    $$\frac{\partial }{\partial c} \, g_{n}(c) = -n \, g_{n+1}(c)$$
    with
    $$g_{1}(c)=\frac{\pi }{2\sqrt{c^{2}+c}}$$
     
  4. Apr 14, 2017 #3
    Do you need the result for all c?
     
  5. Apr 15, 2017 #4
    i found the solution, The function was the result of a definite integral.
    thanks anyway :thumbup::thumbup:
     
  6. Apr 17, 2017 #5
    Well if you found the solution I would like to see it. Please post it :)
     
  7. Apr 17, 2017 #6
    the function is:
    $$\int_{0}^{\frac{\pi }{2}}\left ( a \, cos^2(x) +b \, sin^2(x) \right )^{-n}$$
     
  8. Apr 19, 2017 #7
    Ehh but this is not dependent on c as specified in post number 2.
     
  9. Apr 19, 2017 #8
    evaluate the function as ##f(c,c+1,n)## and check
     
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