Classification of first integrals and closed form solutions for dynamical systems of differential equations
Majority of the real-world natural processes arising in physical, biological and engineering sciences are governed by nonlinear differential equations. To analyze the dynamics and obtaining the closed-form solutions have always been a challenging task. It requires adequate attention and interest of researchers and scientists to explore and quest for various techniques that may be helpful to analyze these systems and provide the readers with detailed overview about the dynamics of the problem under discussion. However, it is not always possible to express the solutions of these nonlinear differential equations explicitly in the form of variables. It is sometimes possible to obtain the functions that are constant on solution curves, i.e., the first integrals. In the literature, the properties of various dynamical systems are studied for only particular cases, or at equilibrium points to obtain conserved quantities. Partial Hamiltonian systems exist for a wide range of problems in interdisciplinary applied mathematics. Most of the natural and physical processes that arise are mathematically modeled as a coupled nonlinear systems of differential equations. These equations usually evade from the existence of exact solutions. In literature, the dynamics of these nonlinear systems are studied using variety of numerical methods. A particular area of literature analyzes these models using stability analysis. The exact solutions are very crucial in order to provide a detailed overview regarding the dynamics and physical properties of the system. These solutions can also serve as a benchmark to examine various numerical schemes.
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