Question: Analytical solution to PDE given by Kummer functions

Dear all,

Maple gives me the complete solution to the attached PDE. This solution is expressed in terms of KummerU and KummerM. What I want is to obtain the solution for real numbers only, i.e. I want to get rid of the possible imaginary numbers in the solution. This implies that the separation constant, _c1, must be less or equal to 2r. 

How can I do this?

Furthermore, the four arbitrary constants will need to be determined through boundary conditions. Is the separation constant, _c1, also determined by boundary conditions?

Any ideas are welcome!

pde := (1/2)*sigma[xi]^2*(diff(f(chi, xi), xi, xi))+(1/2)*sigma[chi]^2*(diff(f(chi, xi), chi, chi))+mu[xi]*(diff(f(chi, xi), xi))-kappa*chi*(diff(f(chi, xi), chi))-r*f(chi, xi) = 0

(1/2)*sigma[xi]^2*(diff(diff(f(chi, xi), xi), xi))+(1/2)*sigma[chi]^2*(diff(diff(f(chi, xi), chi), chi))+mu[xi]*(diff(f(chi, xi), xi))-kappa*chi*(diff(f(chi, xi), chi))-r*f(chi, xi) = 0

(1)

pdesolution := pdsolve(pde, build)

f(chi, xi) = _C1*KummerM((1/4)*(2*kappa+_c[1])/kappa, 3/2, kappa*chi^2/sigma[chi]^2)*chi*_C3*exp(xi*(mu[xi]^2-sigma[xi]^2*_c[1]+2*sigma[xi]^2*r)^(1/2)/sigma[xi]^2)/exp(xi*mu[xi]/sigma[xi]^2)+_C1*KummerM((1/4)*(2*kappa+_c[1])/kappa, 3/2, kappa*chi^2/sigma[chi]^2)*chi*_C4/(exp(xi*mu[xi]/sigma[xi]^2)*exp(xi*(mu[xi]^2-sigma[xi]^2*_c[1]+2*sigma[xi]^2*r)^(1/2)/sigma[xi]^2))+_C2*KummerU((1/4)*(2*kappa+_c[1])/kappa, 3/2, kappa*chi^2/sigma[chi]^2)*chi*_C3*exp(xi*(mu[xi]^2-sigma[xi]^2*_c[1]+2*sigma[xi]^2*r)^(1/2)/sigma[xi]^2)/exp(xi*mu[xi]/sigma[xi]^2)+_C2*KummerU((1/4)*(2*kappa+_c[1])/kappa, 3/2, kappa*chi^2/sigma[chi]^2)*chi*_C4/(exp(xi*mu[xi]/sigma[xi]^2)*exp(xi*(mu[xi]^2-sigma[xi]^2*_c[1]+2*sigma[xi]^2*r)^(1/2)/sigma[xi]^2))

(2)

pdetest(pdesolution, pde)

0

(3)

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