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mathy_girl
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Using perturbation theory, I'm trying to solve the following problem
[tex]\frac{\partial P}{\partial \tau} = \frac{1}{2}\varepsilon^2 \alpha^2 \frac{\partial^2 P}{\partial f^2} + \rho \varepsilon^2 \nu \alpha^2 \frac{\partial^2 P}{\partial f \partial \alpha} + \frac{1}{2}\varepsilon^2 \nu^2 \alpha^2 \frac{\partial^2 P}{\partial \alpha^2}, \quad \mbox{for } \tau>0,
[/tex]
with initial condition [tex]P = \alpha^2~\delta(f-K), \quad \mbox{for } \tau=0. [/tex]
Expanding [tex] P_\varepsilon=P_0 + \varepsilon^2 P_1 + \ldots[/tex] the [tex]\mathcal{O}(1)[/tex] equation is given by
[tex]\frac{\partial P_0}{\partial \tau} = 0, \quad \mbox{for } \tau>0,[/tex]
with boundary condition [tex]P_0 = \alpha^2~\delta(f-K) \mbox{ for } \tau=0[/tex].
Obviously, this gives [tex]P_0 = \alpha^2~\delta(f-K).[/tex]
Now I would like to solve the [tex]\mathcal{O}(\varepsilon^2)[/tex] problem
[tex]\frac{\partial P_1}{\partial \tau} = \frac{1}{2} \alpha^2 \frac{\partial^2 P_0}{\partial f^2} + \rho \nu \alpha^2 \frac{\partial^2 P_0}{\partial f \partial \alpha} + \frac{1}{2} \nu^2 \alpha^2 \frac{\partial^2 P_0}{\partial \alpha^2}, \quad \mbox{for } \tau>0[/tex]
with initial condition [tex]P_1 = 0 \mbox{ for } \tau=0[/tex].
Does anyone of you know how to handle the Dirac Delta function in the initial condition and O(1) solution here?
[tex]\frac{\partial P}{\partial \tau} = \frac{1}{2}\varepsilon^2 \alpha^2 \frac{\partial^2 P}{\partial f^2} + \rho \varepsilon^2 \nu \alpha^2 \frac{\partial^2 P}{\partial f \partial \alpha} + \frac{1}{2}\varepsilon^2 \nu^2 \alpha^2 \frac{\partial^2 P}{\partial \alpha^2}, \quad \mbox{for } \tau>0,
[/tex]
with initial condition [tex]P = \alpha^2~\delta(f-K), \quad \mbox{for } \tau=0. [/tex]
Expanding [tex] P_\varepsilon=P_0 + \varepsilon^2 P_1 + \ldots[/tex] the [tex]\mathcal{O}(1)[/tex] equation is given by
[tex]\frac{\partial P_0}{\partial \tau} = 0, \quad \mbox{for } \tau>0,[/tex]
with boundary condition [tex]P_0 = \alpha^2~\delta(f-K) \mbox{ for } \tau=0[/tex].
Obviously, this gives [tex]P_0 = \alpha^2~\delta(f-K).[/tex]
Now I would like to solve the [tex]\mathcal{O}(\varepsilon^2)[/tex] problem
[tex]\frac{\partial P_1}{\partial \tau} = \frac{1}{2} \alpha^2 \frac{\partial^2 P_0}{\partial f^2} + \rho \nu \alpha^2 \frac{\partial^2 P_0}{\partial f \partial \alpha} + \frac{1}{2} \nu^2 \alpha^2 \frac{\partial^2 P_0}{\partial \alpha^2}, \quad \mbox{for } \tau>0[/tex]
with initial condition [tex]P_1 = 0 \mbox{ for } \tau=0[/tex].
Does anyone of you know how to handle the Dirac Delta function in the initial condition and O(1) solution here?
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