R. Naz

A partial Hamiltonian approach for current value Hamiltonian systems

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.

Symmetry Solutions of a Third-Order Ordinary Differential Equation which Arises from Prandtl Boundary Layer Equations

Abstract The similarity solution to Prandtl’s boundary layer equations for two-dimensional and radial flows with vanishing or constant mainstream velocity gives rise to a thirdorder ordinary differential equation which depends on a parameter a. For special values of a the third-order ordinary differential equation admits a three-dimensional symmetry Lie algebra L 3. For solvable L 3 the equation is integrated by quadrature. For non-solvable L 3 the equation reduces to the Chazy equation. The Chazy equation is reduced to a first-order differential equation in terms of differential invariants which is transformed to a Riccati equation. In general the third-order ordinary differential equation admits a two-dimensional symmetry Lie algebra L 2. For L 2 the differential equation can only be reduced to a first-order equation. The invariant solutions of the third-order ordinary differential equation are also derived.

Comparison of Different Approaches to Construct First Integrals for Ordinary Differential Equations

Different approaches to construct first integrals for ordinary differential equations and systems of ordinary differential equations are studied here. These approaches can be grouped into three categories: direct methods, Lagrangian or partial Lagrangian formulations, and characteristic (multipliers) approaches. The direct method and symmetry conditions on the first integrals correspond to first category. The Lagrangian and partial Lagrangian include three approaches: Noether’s theorem, the partial Noether approach, and the Noether approach for the equation and its adjoint as a system. The characteristic method, the multiplier approaches, and the direct construction formula approach require the integrating factors or characteristics or multipliers. The Hamiltonian version of Noether’s theorem is presented to derive first integrals. We apply these different approaches to derive the first integrals of the harmonic oscillator equation. We also study first integrals for some physical models. The first integrals for nonlinear jerk equation and the free oscillations of a two-degree-of-freedom gyroscopic system with quadratic nonlinearities are derived. Moreover, solutions via first integrals are also constructed.

The applications of the partial Hamiltonian approach to mechanics and other areas

Abstract The partial Hamiltonian systems of the form q i = ∂ H ∂ p i , p i = − ∂ H ∂ q i + Γ i ( t , q i , p i ) arise widely in different fields of the applied mathematics. The partial Hamiltonian systems appear for a mechanical system with non-holonomic nonlinear constraints and non-potential generalized forces. In dynamic optimization problems of economic growth theory involving a non-zero discount factor the partial Hamiltonian systems arise and are known as a current value Hamiltonian systems. It is shown that the partial Hamiltonian approach proposed earlier for the current value Hamiltonian systems arising in economic growth theory Naz et al. (2014) [1] is applicable to mechanics and other areas as well. The partial Hamiltonian approach is utilized to construct first integrals and closed form solutions of optimal growth model with environmental asset, equations of motion for a mechanical system with non-potential forces, the force-free Duffing Van der Pol Oscillator and Lotka–Volterra models.

Group-Invariant Solutions for Two-Dimensional Free, Wall, and Liquid Jets Having Finite Fluid Velocity at Orifice

The group-invariant solutions for nonlinear third-order partial differential equation (PDE) governing flow in two-dimensional jets (free, wall, and liquid) having finite fluid velocity at orifice are constructed. The symmetry associated with the conserved vector that was used to derive the conserved quantity for the jets (free, wall, and liquid) generated the group invariant solution for the nonlinear third-order PDE for the stream function. The comparison between results for two-dimensional jet flows having finite and infinite fluid velocity at orifice is presented. The general form of the group invariant solution for two-dimensional jets is given explicitly.

Reductions and New Exact Solutions of ZK, Gardner KP, and Modified KP Equations via Generalized Double Reduction Theorem

We study here the Lie symmetries, conservation laws, reductions, and new exact solutions of () dimensional Zakharov-Kuznetsov (ZK), Gardner Kadomtsev-Petviashvili (GKP), and Modified Kadomtsev-Petviashvili (MKP) equations. The multiplier approach yields three conservation laws for ZK equation. We find the Lie symmetries associated with the conserved vectors, and three different cases arise. The generalized double reduction theorem is then applied to reduce the third-order ZK equation to a second-order ordinary differential equation (ODE) and implicit solutions are established. We use the Sine-Cosine method for the reduced second-order ODE to obtain new explicit solutions of ZK equation. The Lie symmetries, conservation laws, reductions, and exact solutions via generalized double reduction theorem are computed for the GKP and MKP equations. Moreover, for the GKP equation, some new explicit solutions are constructed by applying the first integral method to the reduced equations.

Nonclassical Symmetry Analysis of Boundary Layer Equations

The nonclassical symmetries of boundary layer equations for two-dimensional and radial flows are considered. A number of exact solutions for problems under consideration were found in the literature, and here we find new similarity solution by implementing the SADE package for finding nonclassical symmetries.

Conservation Laws for Some Systems of Nonlinear Partial Differential Equations via Multiplier Approach

The conservation laws for the integrable coupled KDV type system, complexly coupled kdv system, coupled system arising from complex-valued KDV in magnetized plasma, Ito integrable system, and Navier stokes equations of gas dynamics are computed by multipliers approach. First of all, we calculate the multipliers depending on dependent variables, independent variables, and derivatives of dependent variables up to some fixed order. The conservation laws fluxes are computed corresponding to each conserved vector. For all understudying systems, the local conservation laws are established by utilizing the multiplier approach.

Comparison of Closed-Form Solutions for the Lucas-Uzawa Model via the Partial Hamiltonian Approach and the Classical Approach

In this paper we derive the closed-form solutions for the Lucas-Uzawa growth model with the aid of the partial Hamiltonian approach and then compare our results with those derived by the classical approach \cite{chil}. The partial Hamiltonian approach provides two first integrals \cite{naz2016} in the case where there are no parameter restrictions and these two first integrals are utilized to construct three sets of closed form solutions for all the variables in the model. First two first integrals are used to find two closed form solutions, one of which is new to the literature. We then use only one of the first integrals to derive a third solution that is the same as that found in the previous literature. We also show that all three solutions converge to the same long run balance growth path.

Conservation Laws of Some Physical Models via Symbolic Package GeM

We study the conservation laws of evolution equation, lubrication models, sinh-Poisson equation, Kaup-Kupershmidt equation, and modified Sawada-Kotera equation. The symbolic software GeM (Cheviakov (2007) and (2010)) is used to derive the multipliers and conservation law fluxes. Software GeM is Maple-based package, and it computes conservation laws by direct method and first homotopy and second homotopy formulas.