Conditional symmetry properties for ordinary differential equations

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2015.This work deals with conditional symmetries of ordinary di erential equations (ODEs). We re ne the de nition of conditional symmetries of systems of ODEs in general and provide an algorithmic viewpoint to compute such symmetries subject to root di erential equations. We prove a proposition which gives important and precise criteria as to when the derived higher-order system inherits the symmetries of the root system of ODEs. We rstly study the conditional symmetry properties of linear nth-order (n 3) equations subject to root linear second-order equations. We consider these symmetries for simple scalar higherorder linear equations and then for arbitrary linear systems. We prove criteria when the derived scalar linear ODEs and even order linear system of ODEs inherit the symmetries of the root linear ODEs. There are special symmetries such as the homogeneity and solution symmetries which are inherited symmetries. We mention here the constant coe cient case as well which has translations of the independent variable symmetry inherited. Further we show that if a system of ODEs has exact solutions, then it admits a conditional symmetry subject to the rst-order ODEs related to the invariant curve conditions which arises from the known solution curves. This is even true if the system has no Lie point sy

Thermodynamics of charged and rotating black strings

We study thermodynamics of cylindrically symmetric black holes. Uncharged as well as charged and rotating objects have been discussed. We derive surface gravity and hence the Hawking temperature and entropy for all these cases. We correct some results in the literature and present new ones. It is seen that thermodynamically these black configurations behave differently from spherically symmetric objects

Integrability analysis of the partial differential equation describing the classical bond-pricing model of mathematical finance

The invariant approach is employed to solve the Cauchy problem for the bond-pricing partial differential equation (PDE) of mathematical finance. We first briefly review the invariant criteria for a scalar second-order parabolic PDE in two independent variables and then utilize it to reduce the bond-pricing equation to different Lie canonical forms. We show that the invariant approach aids in transforming the bond-pricing equation to the second Lie canonical form and that with a proper parametric selection, the bond-pricing PDE can be converted to the first Lie canonical form which is the classical heat equation. Different cases are deduced for which the original equation reduces to the first and second Lie canonical forms. For each of the cases, we work out the transformations which map the bond-pricing equation into the heat equation and also to the second Lie canonical form. We construct the fundamental solutions for the bond-pricing model via these transformations by utilizing the fundamental solutions of the classical heat equation as well as solution to the second Lie canonical form. Finally, the closed-form analytical solutions of the Cauchy initial value problems for the bond-pricing model with proper choice of terminal conditions are obtained

A short remark on the integrability of a nonlinear reaction–diffusion equation arising in mathematical biology: Compatibility analysis

AbstractAn analytical approach based on the compatibility concept is employed to solve a nonlinear reaction–diffusion model arising in mathematical biology. The solution process makes it extremely easy to obtain a relatively accurate closed-form solution of the model. The pencil-and-paper solution procedure can be extended to other class of nonlinear problems of similar kind

Group Theoretical Analysis and Invariant Solutions for Unsteady Flow of a Fourth-Grade Fluid over an Infinite Plate Undergoing Impulsive Motion in a Darcy Porous Medium

Abstract In this study, an incompressible time-dependent flow of a fourth-grade fluid in a porous half space is investigated. The flow is generated due to the motion of the flat rigid plate in its own plane with an impulsive velocity. The partial differential equation governing the motion is reduced to ordinary differential equations by means of the Lie group theoretic analysis. A complete group analysis is performed for the governing nonlinear partial differential equation to deduce all possible Lie point symmetries. One-dimensional optimal systems of subalgebras are also obtained, which give all possibilities for classifying meaningful solutions in using the Lie group analysis. The conditional symmetry approach is also utilised to solve the governing model. Various new classes of group-invariant solutions are developed for the model problem. Travelling wave solutions, steady-state solution, and conditional symmetry solutions are obtained as closed-form exponential functions. The influence of pertinent parameters on the fluid motion is graphically underlined and discussed.</jats:p

Noether symmetries and exact solutions of an Euler–Bernoulli beam model

In this paper, a Noether symmetry analysis is carried out for an Euler–Bernoulli beam equation via the standard Lagrangian of its reduced scalar second-order equation which arises from the standard Lagrangian of the fourth-order beam equation via its Noether integrals. The Noether symmetries corresponding to the reduced equation is shown to be the inherited Noether symmetries of the standard Lagrangian of the beam equation. The corresponding Noether integrals of the reduced Euler–Lagrange equations are deduced which remarkably allows for three families of new exact solutions of the static beam equation. These are shown to contain all the previous solutions obtained from the standard Lie analysis and more. </jats:p

Algebraic aspects of evolution partial differential equation arising in the study of constant elasticity of variance model from financial mathematics

The optimal investment-consumption problem under the constant elasticity of variance (CEV) model is investigated from the perspective of Lie group analysis. The Lie symmetry group of the evolution partial differential equation describing the CEV model is derived. The Lie point symmetries are then used to obtain an exact solution of the governing model satisfying a standard terminal condition. Finally, we construct conservation laws of the underlying equation using the general theorem on conservation laws