A differential invariant is a function defined on the jet space of functions that remains the same under a group action. It is an important concept to solve the equivalence problem. This paper presents an effective method to derive a special type of affine differential invariants. Given some functions defined on the plane and an affine group acting on the plane, there are induced actions of the group on the functions and on the derivative functions of the functions. Affine differential invariants of these functions are useful in many applications. However, there has been little systematic study of this problem at present. No clear and simple results are available for application users to use directly. We propose a direct and simple method to construct affine differential invariants in this situation. Some useful explicit formulas of affine differential invariants of 2D functions are presented.

We present a systematic procedure for the determination of a complete set of kth-order ($k\ge 2$) differential invariants corresponding to vector fields in three variables for three-dimensional Lie algebras. In addition, we give a procedure for the construction of a system of two kth-order ODEs admitting three-dimensional Lie algebras from the associated complete set of invariants and show that there are 29 classes for the case of k = 2 and 31 classes for the case of $k\ge 3$. We discuss the singular invariant representations of canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras. Furthermore, we give an integration procedure for canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras which comprises of two approaches, namely, division into four types I, II, III, and IV and that of integrability of the invariant representations. We prove that if a system of two second-order ODEs has a three-dimensional solvable Lie algebra, then, its general solution can be obtained from a partially linear, partially coupled or reduced invariantly represented system of equations. A natural extension of this result is provided for a system of two kth-order ($k\ge 3$) ODEs. We present illustrative examples of familiar integrable physical systems which admit three-dimensional Lie algebras such as the classical Kepler problem and the generalized Ermakov systems that give rise to closed trajectories.

The migration of melt through the mantle of the Earth is governed by a third-order nonlinear partial differential equation for the voidage or volume fraction of melt. The partial differential equation depends on the permeability of the medium which is assumed to be a function of the voidage. It is shown that the partial differential equation admits, as well as translations in time and space, other Lie point symmetries provided the permeability is either a power law or an exponential law of the voidage or is a constant. A rarefactive solitary wave solution of the partial differential equation is derived in the form of a quadrature for the exponential law for the permeability.

We perform the group classification of the generalized Lane-Emden system ${xu}^{\mathrm{\prime }\mathrm{\prime }}+n{u}^{\mathrm{\prime }}+xH\left(v\right)=0,x{v}^{\mathrm{\prime }\mathrm{\prime }}+n{v}^{\mathrm{\prime }}+xg\left(u\right)=0$, which occurs in many applications of physical phenomena such as pattern formation, population evolution, and chemical reactions. We obtain four cases depending on the values of n.

The steady heat transfer through a rectangular longitudinal fin is studied. The thermal conductivity and heat transfer coefficient are assumed to be temperature dependent making the resulting ordinary differential equation (ODE) highly nonlinear. An asymptotic solution is used as a means of understanding the relationship between key parameters. A dynamical analysis is also employed for the same purpose.

A coupled Kadomtsev-Petviashvili equation, which arises in various problems in many scientific applications, is studied. Exact solutions are obtained using the simplest equation method. The solutions obtained are travelling wave solutions. In addition, we also derive the conservation laws for the coupled Kadomtsev-Petviashvili equation.

A third-order ordinary differential equation with application in the flow of a thin liquid film is considered. The boundary conditions come from Tanner's problem for the surface tension driven flow of a thin film. Symmetric and nonsymmetric finite difference schemes are implemented in order to obtain steady state solutions. We show that a central difference approximation to the third derivative in the model equation produces a solution curve with oscillations. A difference scheme based on a combination of forward and backward differences produces a smooth accurate solution curve. The stability of these schemes is analysed through the use of a von Neumann stability analysis.

We provide a supplementation of the results on the canonical forms for scalar fourth-order ordinary differential equations (ODEs) which admit four-dimensional Lie algebras obtained recently. Together with these new canonical forms, a complete list of scalar fourth-order ODEs that admit four-dimensional Lie algebras is available.

We study a two-dimensional integrable generalization of the Kaup-Kupershmidt equation, which arises in various problems in mathematical physics. Exact solutions are obtained using the Lie symmetry method in conjunction with the extended tanh method and the extended Jacobi elliptic function method. In addition to exact solutions we also present conservation laws which are derived using the multiplier approach.