General solution to partial differential equation (PDE)

  • #1
Hi,
I have the following PDE


[itex]-S\frac{\partial\vartheta}{\partial\tau}+\frac{1}{2}\sigma^2\frac{X^2}{S}\frac{\partial^2\vartheta}{\partial\xi^{2}} + [\frac{S}{T} + (r-D)X]\frac{\partial\vartheta}{\partial\xi}[/itex]


I am asked to seek a solution of the form [itex]\vartheta=\alpha_1(\tau)\xi + \alpha_0(\tau)[/itex] and give a general solution for [itex]\alpha_1(\tau)[/itex] and [itex]\alpha_0(\tau)[/itex]

where we have
[itex]\tau=T-t[/itex]
and
[itex]\xi=\frac{t}{T}-\frac{X}{S}[/itex]

I have tried doing the partial differentials of [itex]\vartheta[/itex] with respect to τ and ε, but the answer doesnt allow me to get a general solution for the two unknown functions of τ.
If anyone could help i would be really grateful.
Thanks

NOTE: the word 'partial' in the equation should be a symbol for the partial derivative.
 
Last edited:

Answers and Replies

  • #2
798
34
Hi,
I have the following PDE


[itex]-S\frac{\partial\vartheta}{\partial\tau}+\frac{1}{2}\sigma^2\frac{X^2}{S}\frac{\partial^2\vartheta}{\partial\xi^{2}} + [\frac{S}{T} + (r-D)X]\frac{\partial\vartheta}{\partial\xi}[/itex]


I am asked to seek a solution of the form [itex]\vartheta=\alpha_1(\tau)\xi + \alpha_0(\tau)[/itex] and give a general solution for [itex]\alpha_1(\tau)[/itex] and [itex]\alpha_0(\tau)[/itex]

where we have
[itex]\tau=T-t[/itex]
and
[itex]\xi=\frac{t}{T}-\frac{X}{S}[/itex]

I have tried doing the partial differentials of [itex]\vartheta[/itex] with respect to τ and ε, but the answer doesnt allow me to get a general solution for the two unknown functions of τ.
If anyone could help i would be really grateful.
Thanks

NOTE: the word 'partial' in the equation should be a symbol for the partial derivative.

Hi !

Sorry to say, but the wording of the problem seems very fishy (see attachment)
 

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