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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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