Classification of first integrals and closed form solutions for dynamical systems of differential equations
A specific part of this dissertation is dedicated to the coupled nonlinear first order systems of differential equations. Such systems emerge widely in the field of epidemiology and physical sciences. A two-stream model of tuberculosis and dengue fever is studied using partial Hamiltonian approach. Under certain parametric restrictions, we obtained the first integrals and exact solutions of the governing system of equations. We have further constructed the exact solutions of some disease models which include gonorrhea dynamics model, core group model of sexually transmitted disease and SIS model with standard incidence. We have graphically presented the solution curves to describe the prognosis of the disease as time progresses. The analysis of first order nonlinear systems has been extended to several other dynamical systems from physics and engineering. Likewise, we have presented the closed-form solutions of Duffing-Van der Pol type oscillator, laser photon model, heat convection model and parity-time symmetric oscillator.
The partial Hamiltonian approach can be effectively applied to optimal control problems arising in economics. In this regard, our area of interest is two-sector models of endogenous growth. Motivated by the pioneering contribution of Paul Romer in this field, the closed-form solutions for a modified variant of Romer model have been constructed. Using two first integrals, we have obtained two distinct solutions of the model for control and state variables. Furthermore, the growth rates for all these variables are presented and their long run behavior is predicted. We have also explored the two sector models of optimal population growth and optimal factor tax incidence in Lucas model and computed the exact solutions for both these models. The notion of partial Hamiltonian has been extended to the perturbed system of equations. We have constructed the series of first integrals for galaxy model of astronomy. The combination of stable and unstable first integrals allowed us to evaluate a variety of different approximate solutions for both resonant and non-resonant cases
List of Publications:
1. Haq, B. U., Naeem, I.: First integrals and analytical solutions of some dynamicalsystems. Nonlinear Dyn. 95(3), 1747-1765 (2019). doi:10.1007/s11071-018-4657-4
2. Haq, B. U., Naeem, I.: First integrals and exact solutions of some compartmental disease models. Z. Naturforsch. A. 74(4), 293-304 (2019). doi:10.1515/zna-2018-0450
3. Haq, B. U., Naeem, I.: Closed-from solutions of two sector Romer model of endogenous growth using partial Hamiltonian approach. Math. Meth. Appl. Sci. 2020,1-11. doi:10.1002/mma.6303