#### Year

2012

#### Degree Name

Master of Mathematics - Research

#### Department

School of Mathematics and Applied Statistics

#### Recommended Citation

Al-Nassar, Samar Kadhum, Nonclassical symmetry analysis of second order parabolic partial differential equations, Master of Mathematics - Research thesis, School of Mathematics and Applied Statistics, University of Wollongong, 2012. https://ro.uow.edu.au/theses/3752

#### Abstract

In this thesis we investigate the use of the classical and nonclassical symmetry methods on second-order linear parabolic partial differential equations (PDEs) of the form *u _{t} = u_{xx} + f(x, t)u* . This includes first finding the symmetries and then using them to reduce and solve equations; formulating general initial value problems solvable via the symmetries (including those with initial conditions that are not left invariant under the symmetries); and establishing how symmetries can be used to find functionally separable solutions.

In our nonclassical analysis, we present new strictly nonclassical symmetries to the governing equation and use these to find new solutions to the governing PDEs. In particular, we establish relationships between the infinitesimal *X(x, t, u),* (the coefficient of ϑ /ϑ*x* of the symmetry generator) and the function *f(x; t)* in the governing equation. For given *X(x, t, u)* these relationships can be used to generate function *f(x, t)* for which non-classical symmetries exist.

As well, we investigated the use of symmetries in solving initial value problems (IVPs). It is generally believed that in order to be able to solve IVPs, the given condition of the form *u(x, 0) = F(x)* need be left invariant under the one-parameter Lie group of transformations that leaves the PDE invariant. Under this procedure, and using the symmetries found for our governing PDE, general initial conditions solvable with our PDE were established. However in recent years it has been proven that some initial conditions that are not left invariant by the symmetries that leave invariant the PDE might still be able to solve the IVP. With this in mind we found additional initial conditions for which the corresponding IVPs could be solved by using classical and nonclassical symmetries. Many examples are provided to illustrate the method. Further, we examined a paper by Zhiyong Zhang and Yufu Chen on symmetries and initial conditions and extended the main result of the paper which gave the most general form for a first-order initial condition to be admitted by a given classical or nonclassical symmetry. Their result assumed that the generator needs leave to the initial condition invariant. In this thesis this result is extended to the case where the first-order initial condition need not be left invariant, and thus many more initial value problems could be solved with the symmetries. We then applied the result to our governing PDE with first-order initial conditions and provided examples with classical and nonclassical symmetries.

In addition, we have demonstrated the usefulness of classical and nonclassical symmetries of a PDE with one dependent (u) and two independent *(x, t)* variables to finding functionally separable solutions of different forms such as, *q(u) = G(t) + (z), q(u) = F(x) + (z),* and *q(u) = F(x) + G(t) + (z)*, where *z = z(x, t*). The method is illustrated on our governing PDE and examples in each case are provided.