We obtain fundamental solutions for PDEs of the form u t = σ x γ u xx + f (x)u x − μx r u by showing that if the symmetry group of the PDE is nontrivial, it contains a standard integral transform of the fundamental solution. We show that in this case, the problem of finding a fundamental solution can be reduced to inverting a Laplace transform or some other classical transform.
This paper uses Lie symmetry methods to calculate certain expectations for a large class of Itô diffusions. We show that if the problem has sufficient symmetry, then the problem of computing functionals of the form Ex(e −λXt− t 0
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.