I have formulated a problem in which I come across an equation of the form ∂2u/∂x2 + ∂2u/∂y2 - k sinh(u) = 0 . How can I solve this equation analytically? Can anyone help me to solve this anlytically or using any mathematical packages like Mathematica or Maple.
Solution
The problem of solution of this equation can be easily solved with the substitution:
f(x,y)=exp(u(x,y))
Then Maple can be used
u:=(x,y)->ln(f(x,y)):
pdsolve(diff(u(x,y),x,x)+diff(u(x,y),y,y)-k*sinh(u(x,y))=0,f(x,y));
I recieved following solution (I qoute it without any analis)
u(x,y)=ln(tanh(_C1+1/2*(k-4*_C3^2)^(1/2)*x+_C3*y)^2);
In reply to Solution by Vladimir Shneider
Uniqueness of solution
This solution given by Valdimir satisfies the PDE (which is a form of Poisson's equation). This solution is one possible solution but probably not a general solution. You can also try solving the PDE by hand using the appropriate Green's functions if you know the geometry of the problem (and it's simple).
As Roozbeh says, we have to know the geometry of the boundary and the boundary conditions to get the correct (and unique) solution.
A taste for what's involved can be found at
http://eqworld.ipmnet.ru/en/solutions/lpde/lpde302.pdf
I Dont Think So!
That is a Partial Differential Equation and that's solution depends on the boundary in which defined.you must determine that's boundary for obtain solution