Suppose that we have the following time-dependent partial differential equation:

\begin{equation} \frac{\partial V(t, x)}{\partial t} = \frac{1}{2}\frac{\partial^2 V(t, x)}{\partial x^2} - wxV(t, x), \quad t> 0, x\in \mathbb{R} \\ V(0, x ) = f(x) = e^{-ux}, \quad x\in \mathbb{R} \end{equation}

How can I come up with $\tilde{V}(t, x)$ as a solution for the above partial differential equation system? Could you please give me the general instruction on how I should proceed to find such a solution from the above partial differential equation?

\begin{equation} \tilde{V}(t, x) = \exp\Big\{\frac{w^2}{6}t^3 + \frac{wu}{2}t^2 +\frac{u^2}{2}t - [wt+u]x\Big\} \end{equation} where $u$ and $w$ are real constants.



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.