Symmetry group methods for fundamental solutions

Abstract

This paper uses Lie symmetry group methods to study PDEs of the form u t =xu xx +f (x)u x . We show that when the drift function f is a solution of a family of Ricatti equations, then symmetry techniques can be used to find a fundamental solution.

Lie symmetry analysis and exact explicit solutions for general Burgers’ equation

“…It is well-known that the Lie group method is a powerful and direct approach to construct exact solutions of nonlinear differential equations. Furthermore, based on the Lie group method, many other types of exact solutions of PDEs can be obtained, such as traveling wave solutions, soliton solutions, power series solutions, fundamental solutions [9,10], and so on.…”

Lie symmetry analysis and exact explicit solutions of three-dimensional Kudryashov–Sinelshchikov equation

“…(2) by means of the combination of Lie symmetry analysis [12][13][14][15][16] and the Riccati equation method [17][18]. The rest of this paper is organized as follows: in Section 2, the Lie symmetry analysis is performed on Eq.…”

Exact scenario simulation for selected multi-dimensional stochastic processes

“…For instance, for squared Bessel processes of integer dimension, see [41], we will explain how to simulate exact solutions. Based on results in [11], exact solutions can be simulated by sampling from the explicitly available transition density of some nonlinear SDEs where the drift function a(·) in (1.1) takes a particular form. These then include also squared Bessel processes of noninteger dimension.…”