## Publications

- [Show abstract] [Hide abstract]

**ABSTRACT:**By use of our newly developed methodology (Naz et al., 2014 [1]), for solving the dynamical system of first-order ordinary differential equations (ODEs) arising from first-order conditions of optimal control problems, we derive closed-form solutions for the standard Lucas–Uzawa growth model. We begin by showing how our new methodology yields a series of first integrals for the dynamical system associated with this model and two cases arise. In the first case, two first integrals are obtained and we utilize these to derive closed-form solutions and show that our methodology yields the same results as in the previous literature. In the second case, our methodology yields three first integrals under certain restrictions on the parameters. We use these three integrals to obtain new solutions for all the variables which in turn yield new solutions for the growth rates of these variables. Our results are significant as our approach is applicable to an arbitrary system of ODEs which means that it can also be invoked for more complex models. - [Show abstract] [Hide abstract]

**ABSTRACT:**We develop a new approach termed as a discount free or partial Lagrangian method for construction of first integrals for dynamical systems of ordinary differential equations (ODEs). It is shown how one can utilize the Legendre transformation in a more general setting to provide the equivalence between a current value Hamiltonian and a partial or discount free Lagrangian when it exists. As a consequence, we develop a discount factor free Lagrangian framework to deduce reductions and closed-form solutions via first integrals for ODEs arising from economics by proving three important propositions. The approach is algorithmic and applies to many state variables of the Lagrangian. In order to show its effectiveness, we apply the method to models, one linear and two nonlinear, with one state variable. We obtain new exact solutions for the last model. The discount free Lagrangian naturally arises in economic growth theory and many other economic models when the control variables can be eliminated at the outset which is not always possible in optimal control theory applications of economics. We explain our method with the help of few widely used economic growth models. We point out the difference between this approach and the more general partial Hamiltonian method proposed earlier for a current value Hamiltonian (Naz et al. in Commun Nonlinear Sci Numer Simul 19:3600–3610, 2014) which is applicable in a general setting involving time, state, costate and control variables. - [Show abstract] [Hide abstract]

**ABSTRACT:**We provide an algorithmic approach to the construction of point transformations for scalar ordinary differential equations that admit three-dimensional symmetry algebras which lead to their respective canonical forms. - [Show abstract] [Hide abstract]

**ABSTRACT:**We provide an algorithmic approach to the construction of point transformations for scalar ordinary differential equations that admit three-dimensional symmetry algebras which lead to their respective canonical forms. - [Show abstract] [Hide abstract]

**ABSTRACT:**We provide an algorithmic approach to the construction of point transformations for scalar ordinary differential equations that admit three-dimensional symmetry algebras which lead to their respective canonical forms. -

**ABSTRACT:**We provide an algorithmic approach to the construction of point transformations for scalar ordinary differential equations that admit three-dimensional symmetry algebras which lead to their respective canonical forms. - [Show abstract] [Hide abstract]

**ABSTRACT:**Scalar complex partial differential equations which admit variational formulations are studied. Such a complex partial differential equation, via a complex dependent variable, splits into a system of two real partial differential equations. The decomposition of the Lagrangian of the complex partial differential equation in the real domain is shown to yield two real Lagrangians for the split system. The complex Maxwellian distribution, transonic gas flow, Maxwellian tails, dissipative wave and Klein–Gordon equations are considered. The Noether symmetries and gauge terms of the split system that correspond to both the Lagrangians are constructed by the Noether approach. In the case of coupled split systems, the same Noether symmetries are obtained. The Noether symmetries for the uncoupled split systems are different. The conserved vectors of the split system which correspond to both the Lagrangians are compared to the split conserved vectors of the complex partial differential equation for the examples. The split conserved vectors of the complex partial differential equation are the same as the conserved vectors of the split system of real partial differential equations in the case of coupled systems. Moreover a Noether-like theorem for the split system is proved which provides the Noether-like conserved quantities of the split system from knowledge of the Noether-like operators. An interesting result on the split characteristics and the conservation laws is shown as well. The Noether symmetries and gauge terms of the Lagrangian of the split system with the split Noether-like operators and gauge terms of the Lagrangian of the given complex partial differential equation are compared. Folklore suggests that the split Noether-like operators of a Lagrangian of a complex Euler–Lagrange partial differential equation are symmetries of the Lagrangian of the split system of real partial differential equations. This is not the case. They are proved to be the same if the Noether symmetries of the Lagrangian of the complex partial differential equation have either pure real or pure imaginary form. -
##### Technical Report: A point symmetry based method for transforming ODEs with three-dimensional symmetry algebras to their canonical forms

**ABSTRACT:**We provide an algorithmic approach to the construction of point transformations for scalar ordinary differential equations that admit three-dimensional symmetry algebras which lead to their respective canonical forms. - [Show abstract] [Hide abstract]

**ABSTRACT:**The governing nonlinear equation for unidirectional flow of a Sisko fluid in a cylindrical tube due to translation of the tube wall is modelled in cylindrical polar coordinates. The exact steady-state solution for the nonlinear problem is obtained. The reduction of the nonlinear initial value problem is carried out by using a similarity transformation. The partial differential equation is transformed into an ordinary differential equation, which is integrated numerically taking into account the influence of the exponent n and the material parameter b of the Sisko fluid. The initial approximation for the fluid velocity on the axis of the cylinder is obtained by matching inner and outer expansions for the fluid velocity. A comparison of the velocity, vorticity, and shear stress of Newtonian and Sisko fluids is presented. - [Show abstract] [Hide abstract]

**ABSTRACT:**An efficient compatibility criterion is proposed to solve the nonlinear boundary problem arising in the study of the classical problem of viscous fluid flow due to a stretching sheet due to Crane (Crane, 1970). The compatibility and generalized group analysis make it simple to obtain the exact solution of the classical boundary layer problem. - [Show abstract] [Hide abstract]

**ABSTRACT:**Second-order dynamical systems are of paramount importance as they arise in mechanics and many applications. It is essential to have workable explicit criteria in terms of the coefficients of the equations to effect reduction and solutions for such types of equations. One important aspect is linearization by invertible point transformations which enables one to reduce a non-linear system to a linear system. The solution of the linear system allows one to solve the non-linear system by use of the inverse of the point transformation. It was proved that the n-dimensional system of second-order ordinary differential equations obtained by projecting down the system of geodesics of a flat (n+1)-dimensional space can be converted to linear form by a point transformation. This is a generalization of the Lie linearization criteria for a scalar second-order equation. In this case it is of the maximally symmetric class for a system and the linearizing transformation as well as the solution can be directly written down. This was explicitly used for two-dimensional dynamical systems. The criteria were written down in terms of the coefficients and the linearizing transformation allowed for the general solution of the original system. Here the work is extended to a three-dimensional dynamical system and we find explicit criteria, including the linearization test given in terms of the coefficients of the cubic in the first derivatives of the system and the construction of the transformations, that result in linearization. Applications to equations of classical mechanics and relativity are given to illustrate our results. -
##### Article: Symmetry classification and joint invariants for the scalar linear (1 + 1) elliptic equation

[Show abstract] [Hide abstract]

**ABSTRACT:**The equations for the classification of symmetries of the scalar linear (1 + 1) elliptic partial differential equation (PDE) are obtained in terms of Cotton’s invariants. New joint differential invariants of the scalar linear elliptic (1 + 1) PDE in two independent variables are derived in terms of Cotton’s invariants by application of the infinitesimal method. Joint differential invariants of the scalar linear elliptic equation are also deduced from the basis of the joint differential invariants of the scalar linear (1 + 1) hyperbolic equation under the application of the complex linear transformation. We also find a basis of joint differential invariants for such type of equations by utilization of the operators of invariant differentiation. The other invariants are functions of the basis elements and their invariant derivatives. Examples are given to illustrate our results. - [Show abstract] [Hide abstract]

**ABSTRACT:**We study a dynamic fourth-order Euler-Bernoulli partial differential equation having a constant elastic modulus and area moment of inertia, a variable lineal mass density g ( x ) , and the applied load denoted by f ( u ) , a function of transverse displacement u ( t , x ) . The complete Lie group classification is obtained for different forms of the variable lineal mass density g ( x ) and applied load f ( u ) . The equivalence transformations are constructed to simplify the determining equations for the symmetries. The principal algebra is one-dimensional and it extends to two- and three-dimensional algebras for an arbitrary applied load, general power-law, exponential, and log type of applied loads for different forms of g ( x ) . For the linear applied load case, we obtain an infinite-dimensional Lie algebra. We recover the Lie symmetry classification results discussed in the literature when g ( x ) is constant with variable applied load f ( u ) . For the general power-law and exponential case the group invariant solutions are derived. The similarity transformations reduce the fourth-order partial differential equation to a fourth-order ordinary differential equation. For the power-law applied load case a compatible initial-boundary value problem for the clamped and free end beam cases is formulated. We deduce the fourth-order ordinary differential equation with appropriate initial and boundary conditions. - [Show abstract] [Hide abstract]

**ABSTRACT:**We investigate the turbulent planar classical wake and derive the governing equations using the eddy viscosity closure model. The Lie point symmetry associated with the elementary conserved vector is used to generate the invariant solution. We first consider the case where the eddy viscosity depends only on the distance along the wake. We then relax this condition to include the dependence of the eddy viscosity on the perpendicular distance from the axis of the wake. The profiles of the mean velocity show that the role of the eddy viscosity is to increase the effective width of the wake and decrease the magnitude of the maximum mean velocity deficit. - [Show abstract] [Hide abstract]

**ABSTRACT:**New systematic method to find the relative invariant differentiation operators is developed. We incorporate this new approach with Lie's infinitesimal method to study the general class $y'''=f(x,y,y',y'')$ under general point equivalence transformations in the generic case. As a result, all third-order differential invariants, relative and absolute invariant differentiation operators are determined for third-order ODEs $y'''=f(x,y,y',y'')$, which are not quadratic in the second-order derivative. These relative invariant differentiation operators are used to determine the fourth-order differential invariants and absolute invariant differentiation operators in a degenerate case of interest. As an application, invariant descriptions of all the canonical forms in the complex plane with four infinitesimal symmetries for third-order ODEs $y'''=f(x,y,y',y'')$, which are not quadratic in the second-order derivative, are provided. - [Show abstract] [Hide abstract]

**ABSTRACT:**An alternative proof of Lie’s approach for the linearization of scalar second-order ordinary differential equations is derived by using the relationship between λ-symmetries and first integrals. This relation further leads to a new λ-symmetry linearization criterion for second-order ordinary differential equations which provides a new approach for constructing the linearization transformations with lower complexity. The effectiveness of the approach is illustrated by obtaining the local linearization transformations for the linearizable nonlinear ordinary differential equations of the form y''+F1(x,y)y'+F(x,y)=0. Examples of linearizable nonlinear ordinary differential equations which are quadratic or cubic in the first derivative are also presented. - [Show abstract] [Hide abstract]

**ABSTRACT:**An analysis is carried out to study the time-dependent flow of an incompressible electrically conducting fourth-grade fluid over an infinite porous plate. The flow is caused by the motion of the porous plate in its own plane with an impulsive velocity V(t). The governing nonlinear problem is solved by invoking the Lie group theoretic approach and a numerical technique. Travelling wave solutions of the forward and backward type, together with a steady state solution, form the basis of our analytical analysis. Further, the closed-form solutions are also compared against numerical results. The essential features of the embedded parameters are described. In particular, the physical significance of the plate suction/injection and magnetic field is studied. - [Show abstract] [Hide abstract]

**ABSTRACT:**Criteria are established for higher order ordinary differential equations to be compatible with lower order ordinary differential equations. Necessary and sufficient compatibility conditions are derived which can be used to construct exact solutions of higher order ordinary differential equations subject to lower order equations. We provide the connection to generalized groups through conditional symmetries. Using this approach of compatibility and generalized groups, new exact solutions of nonlinear flow problems arising in the study of Newtonian and non-Newtonian fluids are derived. The ansatz approach for obtaining exact solutions for nonlinear flow models of Newtonian and non-Newtonian fluids is unified with the application of the compatibility and generalized group criteria. - [Show abstract] [Hide abstract]

**ABSTRACT:**We refine the definition of conditional symmetries of ordinary differential equations and provide an algorithm to compute such symmetries. A proposition is proved which provides criteria as to when the symmetries of the root system of ODEs are inherited by the derived higher-order system. We provide examples and then investigate the conditional symmetry properties of linear nth-order equations subject to root linear second-order equations. First this is considered for simple linear equations and then for arbitrary linear systems. We prove criteria when the symmetries of the root linear ODEs are inherited by the derived scalar linear ODEs and even order linear system of ODEs. Furthermore, we show that if a system of ODEs has exact solutions, then it admits a conditional symmetry subject to the first-order ODEs related to the invariant curve conditions which arises from the known solution curves. Moreover, we give examples of the conditional symmetries of non-linear third-order equations which are linearizable by second-order Lie linearizable equations. Applications to classical and fluid mechanics are presented. - [Show abstract] [Hide abstract]

**ABSTRACT:**The Lie and Noether point symmetry analyses of a kth-order system of m complex ordinary differential equations (ODEs) with m dependent variables are performed. The decomposition of complex symmetries of the given system of complex ODEs yields Lie- and Noether-like operators. The system of complex ODEs can be split into 2m coupled real partial differential equations (PDEs) and 2m Cauchy–Riemann (CR) equations. The classical approach is invoked to compute the symmetries of the 4m real PDEs and these are compared with the decomposed Lie- and Noether-like operators of the system of complex ODEs. It is shown that, in general, the Lie- and Noether-like operators of the system of complex ODEs and the symmetries of the decomposed system of real PDEs are not the same. A similar analysis is carried out for restricted systems of complex ODEs that split into 2m coupled real ODEs. We summarize our findings on restricted complex ODEs in two propositions.

- Questions & Answers
- Open Reviews
- Research Feedback