## Publications

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**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; · 2.77 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.03/2014; - [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 01/2014; · 1.35 Impact Factor -
##### Article: Non-linear time-dependent flow models of third grade fluids: A conditional symmetry approach

International Journal of Non-Linear Mechanics 09/2013; · 1.35 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. · 1.38 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; · 1.09 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**This study focuses on obtaining a new class of closed-form shock wave solution also known as soliton solution for a nonlinear partial differential equation which governs the unsteady magnetohydrodynamics (MHD) flow of an incompressible fourth grade fluid model. The travelling wave symmetry formulation of the model leads to a shock wave solution of the problem. The restriction on the physical parameters of the flow problem also falls out naturally in the course of derivation of the solution.Mathematical Problems in Engineering 05/2013; 2013. · 1.38 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The unsteady unidirectional flow of an incompressible fourth grade fluid over an infinite porous plate is studied. The flow is induced due to the motion of the plate in its own plane with an arbitrary velocity. Symmetry reductions are performed to transform the governing nonlinear partial differential equations into ordinary differential equations. The reduced equations are then solved analytically and numerically. The influence of various physical parameters of interest on the velocity profile are shown and discussed through several graphs. A comparison of the present analysis shows excellent agreement between analytical and numerical solutions.Applied Mathematics and Computation 05/2013; 219(17):9187–9195. · 1.35 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The simplest equation method is employed to construct some new exact closed-form solutions of the general Prandtl's boundary layer equation for two-dimensional flow with vanishing or uniform mainstream velocity. We obtain solutions for the case when the simplest equation is the Bernoulli equation or the Riccati equation. Prandtl's boundary layer equation arises in the study of various physical models of fluid dynamics. Thus finding the exact solutions of this equation is of great importance and interest.Mathematical Problems in Engineering 04/2013; 2013. · 1.38 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We present a systematic procedure for the determination of a complete set of kth-order () 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 . 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 () 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.Journal of Applied Mathematics 04/2013; 2013. · 0.83 Impact Factor -
##### Article: A Note on Four-Dimensional Symmetry Algebras and Fourth-Order Ordinary Differential Equations

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**ABSTRACT:**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.Journal of Applied Mathematics 02/2013; 2013. · 0.83 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 01/2013; · 1.08 Impact Factor - [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 01/2013; 74(4). · 3.01 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We construct approximate conservation laws for non-variational nonlinear perturbed (1+1) heat and wave equations by utilizing the partial Lagrangian approach. These perturbed nonlinear heat and wave equations arise in a number of important applications which are reviewed. Approximate symmetries of these have been obtained in the literature. Approximate partial Noether operators associated with a partial Lagrangian of the underlying perturbed heat and wave equations are derived herein. These approximate partial Noether operators are then used via the approximate version of the partial Noether theorem in the construction of approximate conservation laws of the underlying perturbed heat and wave equations.Nonlinear Analysis Real World Applications 12/2012; 13(6):2823–2829. · 2.20 Impact Factor -
##### Article: Group invariant solutions for the unsteady MHD flow of a third grade fluid in a porous medium

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**ABSTRACT:**This work describes the time-dependent flow of an incompressible non-Newtonian fluid over an infinite rigid plate. The flow is induced due to the arbitrary velocity V(t) of the plate. The fluid occupies the porous half space y>0y>0 and is also electrically conducting in the presence of a constant applied magnetic field in the transverse direction to the flow. Analytical solutions of the governing non-linear partial differential equation for the unidirectional flow of a third grade fluid are established using the symmetry approach. We construct three types of analytical solutions by employing the Lie symmetry method and the better solution from the physical point of view is shown to be the non-travelling wave solution. We also present numerical solutions of the governing PDE and the reduced ODE and compare with the analytical results. Finally, the influence of emerging parameters are studied through several graphs with emphasis on the study of the effects of the magnetic field and non-Newtonian fluid parameters.International Journal of Non-Linear Mechanics 09/2012; 47(7):792–798. · 1.35 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We obtain new semi-invariants for a system of two linear parabolic type partial differential equations (PDEs) in two independent variables under equivalence transformations of the dependent variables only. This is achieved for a class of systems of two linear parabolic type PDEs that correspond to a scalar complex linear (1 + 1) parabolic equation. The complex transformations of the dependent variables which map the complex scalar linear parabolic PDE to itself provide us with real transformations that map the corresponding system of linear parabolic type PDEs to itself with different coefficients in general. The semi-invariants deduced for this class of systems of two linear parabolic type equations correspond to the complex Ibragimov invariants of the complex scalar linear parabolic equation. We also look at particular cases of the system of parabolic type equations when they are uncoupled or coupled in a special manner. Moreover, we address the inverse problem of when systems of linear parabolic type equations arise from analytic continuation of a scalar linear parabolic PDE. Examples are given to illustrate the method implemented.Communications in Nonlinear Science and Numerical Simulation 08/2012; 17(8):3140–3147. · 2.77 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**An expression for the coefficients of a linear iterative equation in terms of the parameters of the source equation is given both for equations in standard form and for equations in reduced normal form. The operator generating an iterative equation of a general order in reduced normal form is also obtained and some other properties of iterative equations are established. In particular, a simple necessary and sufficient condition for an equation to be iterative is given for the general fourth-order linear equation solely in terms of its coefficients.07/2012; - [Show abstract] [Hide abstract]

**ABSTRACT:**We obtain the complete Lie symmetry group classification of the dynamic fourth-order Euler-Bernoulli partial differential equation, where the elastic modulus, the area moment of inertia are constants and the applied load is a nonlinear function. In the Lie analysis, the principal Lie algebra which is two-dimensional extends in three cases, viz., the linear, the exponential, and the general power law. For each of the nontrivial cases, we determine symmetry reductions to ordinary differential equations which are of order four. In only one case related to the power law we are able to have a compatible initial-boundary value problem for a clamped end and a free beam. For these cases we deduce the corresponding fourth-order ordinary differential equations with appropriate boundary conditions. We provide an asymptotic solution for the reduced fourth-order ordinary differential equation corresponding to a clamped or free beam.Journal of Mathematical Physics 04/2012; 53(4). · 1.30 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The approach of Noether symmetries with gauge term in the Gauss–Bonnet dilatonic theory of gravity is used for different values of the state parameter, ω, of the background matter. It is found that for ω = –1 (dark energy Universe), ω = 0 (matter-dominated Universe), and ω = 1 (the case of stiff matter), for the existence of Noether gauge symmetries, the potential V() and the coupling of the Gauss–Bonnet term with gravity Λ() are obtained as exponential functions of the scalar field . For the case of arbitrary ω the coupling parameter Λ() is found as a linear function of the scalar field and the potential is a constant, for the existence of Noether gauge symmetries. Here it is observed that both the potential and coupling parameter depend on the evolutionary history of the Universe.Canadian Journal of Physics 04/2012; 90(5):467-471. · 0.90 Impact Factor

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