## Publications

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**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.Communications in Nonlinear Science and Numerical Simulation 01/2016; 30(1):299-306. DOI:10.1016/j.cnsns.2015.06.033 · 2.87 Impact Factor - [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.Communications in Nonlinear Science and Numerical Simulation 10/2015; 27(1-3). DOI:10.1016/j.cnsns.2015.03.002 · 2.87 Impact Factor -
##### Article: Symmetry classification and joint invariants for the scalar linear (1 + 1) elliptic equation

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**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.Communications in Nonlinear Science and Numerical Simulation 08/2015; 25(1-3). DOI:10.1016/j.cnsns.2014.11.022 · 2.87 Impact Factor - [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.Communications in Nonlinear Science and Numerical Simulation 06/2015; 23(1-3). DOI:10.1016/j.cnsns.2014.10.006 · 2.87 Impact Factor - [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.Mathematical Methods in the Applied Sciences 05/2015; DOI:10.1002/mma.3544 · 0.92 Impact Factor - [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.Communications in Nonlinear Science and Numerical Simulation 02/2015; 26:45-51. DOI:10.1016/j.cnsns.2015.01.017 · 2.87 Impact Factor - [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.Brazilian Journal of Physics 02/2015; 45(1). DOI:10.1007/s13538-014-0292-9 · 0.81 Impact Factor - [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.International Journal of Non-Linear Mechanics 01/2015; DOI:10.1016/j.ijnonlinmec.2015.01.003 · 1.98 Impact Factor - [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.Mathematical Problems in Engineering 01/2015; 2015:1-7. DOI:10.1155/2015/520491 · 0.76 Impact Factor - [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.International Journal of Non-Linear Mechanics 12/2014; 67. DOI:10.1016/j.ijnonlinmec.2014.08.013 · 1.98 Impact Factor - [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.Pramana 07/2014; 83(1):9-20. DOI:10.1007/s12043-014-0762-1 · 0.65 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We develop a partial Hamiltonian framework to obtain reductions and closed-form solutions via first integrals of current value Hamiltonian systems of ordinary differential equations (ODEs). The approach is algorithmic and applies to many state and costate variables of the current value Hamiltonian. However, we apply the method to models with one control, one state and one costate variable to illustrate its effectiveness. The current value Hamiltonian systems arise in economic growth theory and other economic models. We explain our approach with the help of a simple illustrative example and then apply it to two widely used economic growth models: the Ramsey model with a constant relative risk aversion (CRRA) utility function and Cobb Douglas technology and a one-sector AK model of endogenous growth are considered. We show that our newly developed systematic approach can be used to deduce results given in the literature and also to find new solutions.Communications in Nonlinear Science and Numerical Simulation 06/2014; DOI:10.1016/j.cnsns.2014.03.023 · 2.87 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The invariants of a general linear system of two hyperbolic equations have been derived under transformations of the dependent and independent variables by the real infinitesimal method. Here a subclass of the general system of linear hyperbolic partial differential equations (PDEs) is investigated for the associated invariants by complex as well as real methods. The complex procedure relies on the correspondence of the system and associated invariants with the base complex equation and related complex invariants, respectively. A comparison of all the invariant quantities obtained by complex and real methods is presented which shows that the complex procedure provides a few invariants different from those extracted by real symmetry analysis. - [Show abstract] [Hide abstract]

**ABSTRACT:**We present a method for finding a complete set of kth-order (k≥2) differential invariants including bases of invariants corresponding to vector fields in three variables of four-dimensional real Lie algebras. As a consequence, we provide a complete list of second-order differential invariants and canonical forms for vector fields of four-dimensional Lie algebras and their admitted regular systems of two second-order ODEs. Moreover, we classify invariant representations of these canonical forms of ODEs into linear, partial linear, uncoupled, and partial uncoupled cases. In addition, we give an integration procedure for invariant representations of canonical forms for regular systems of two second-order ODEs admitting four-dimensional Lie algebras.Nonlinear Dynamics 12/2013; 74(4). DOI:10.1007/s11071-013-1016-3 · 2.85 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We study the linearization of a class of Liénard type nonlinear second-order ordinary differential equations from the generalized Sundman transformation viewpoint. The linearizing generalized Sundman transformation for the class of equations is constructed. The transformation is used to map the underlying class of equations into a linear second-order ordinary differential equation which is not in the Laguerre form. The general solution of this class of equations is obtained by integrating the linearized equation and applying the generalized Sundman transformation. Moreover, we apply a Riccati transformation to a general linear third-order variable coefficients ordinary differential equation to extend the underlying class of equations and we also derive the conditions of linearizability of this new class of nonlinear second-order ordinary differential equations using the generalized Sundman transformation method and obtain its general solution.Applied Mathematics & Information Sciences 11/2013; 7(6):2355-2359. DOI:10.12785/amis/070627 · 1.23 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**It has recently been shown that the fourth-order static Euler–Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, in the maximal case has three symmetries. This corresponds to the negative fractional power law y −5/3, and the equation has the nonsolvable algebra ${sl(2, \mathbb{R})}$ . We obtain new two- and three-parameter families of exact solutions when the equation has this symmetry algebra. This is studied via the symmetry classification of the three-parameter family of second-order ordinary differential equations that arises from the relationship among the Noether integrals. In addition, we present a complete symmetry classification of the second-order family of equations. Hence the admittance of ${sl(2, \mathbb{R})}$ remarkably allows for a three-parameter family of exact solutions for the static beam equation with load a fractional power law y −5/3.Journal of Engineering Mathematics 10/2013; 82(1). DOI:10.1007/s10665-012-9583-8 · 0.80 Impact Factor -
##### Article: Non-linear time-dependent flow models of third grade fluids: A conditional symmetry approach

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**ABSTRACT:**In this communication some non-linear flow problems dealing with the unsteady flow of a third grade fluid in porous half-space are analyzed. A new class of closed-form conditionally invariant solutions for these flow models are constructed by using the conditional or non-classical symmetry approach. All possible non-classical symmetries of the model equations are obtained and various new classically invariant solutions have been constructed. The solutions are valid for a half-space and also satisfy the physically relevant initial and the boundary conditions.International Journal of Non-Linear Mechanics 09/2013; 54:55-65. DOI:10.1016/j.ijnonlinmec.2013.03.013 · 1.98 Impact Factor -
##### Dataset: Imran CV

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**ABSTRACT:**This study is based upon constructing a new class of closed-form shock wave solutions for some nonlinear problems arising in the study of a third grade fluid model. The Lie symmetry reduction technique has been employed to reduce the governing nonlinear partial differential equations into nonlinear ordinary differential equations. The reduced equations are then solved analytically, and the shock wave solutions are constructed. The conditions on the physical parameters of the flow problems also fall out naturally in the process of the derivation of the solutions.Mathematical Problems in Engineering 06/2013; 2013. DOI:10.1155/2013/602902 · 0.76 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Noether symmetries of some of the well known spherically symmetric static solutions of the Einstein’s field equations are classified. The resulting Noether symmetries in each case are compared with conservation laws given by Killing vectors and collineations of the Ricci and Riemann tensors for corresponding solutions.International Journal of Theoretical Physics 06/2013; DOI:10.1007/s10773-013-1656-6 · 1.18 Impact Factor

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