Abstract:

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

Abstract: 

Spectral theory is a generalization of ideas of eigenvalues and eigenvectors of square matrices to broader classes of linear operators defined in a variety of infinite dimensional vector spaces. Spectrum of non-compact quantum star graphs, star shaped metric graph equipped with differential operators and set of vertex conditions, is studied. The selection of vertex or boundary conditions play a vital role in making the operators self-adjoint as well as ensuring that edges are connected or coupled with each other. Generally, a closed and densely defined self-adjoint linear differential operator defined on a separable Hilbert space has real spectrum which consists of a set of negative eigenvalues with finite multiplicity and a continuous part covering the positive semi-axis. The number of negative eigenvalues are finite if the potential function(s) are rapidly decaying at the boundary or a countable infinite set if the potential decays slowly with zero as the only accumulation point. In case of complex-valued potential the corresponding non-selfadjoint graphs has complex eigenvalues. For Schrödinger operators we develop trace formulas for resolvent difference of perturbed and unperturbed operators. Which lead to explicit expressions for perturbation determinant, Levinson's formula and spectral shift function for star shaped quantum graphs. These mathematical objects describe continuous as well as discrete part of the spectrum in terms of quantities which are directly to the operator (potential functions) and boundary parameters. Moreover, among many spectral problems associated with differential operators, one is, to find the bounds on the series of moments of negative eigenvalues in terms of potentials and boundary parameters of the operator. We derive sharp upper bounds for a fourth order differential operator with two real-valued potentials on semi-axis with the "Generic" three parameters self-adjoint boundary conditions at zero. Such bounds are commonly known as Lieb-Thirring inequalities. We also obtain bounds for eigenvalues of star graphs with complex valued potentials having n edges with δ and δ’ vertex conditions. For two edge graphs sharp and explicit bounds on the complex eigenvalues are obtained in terms of potential and boundary parameters. 

Publications:  

• M. D. Zia and M. Usman, Sharp Spectral Inequalities for Fourth Order Differential Operators on Semi-Axis, Math. Phys. Anal. Geom. 22(24), (2019). https://link.springer.com/article/10.1007/s11040-019-9321-9 
• M. D. Zia and M. Usman, A Trace Formula, Perturbation Determinant and Levinson’s Theorem for a class of star graphs. (Submitted). 
• M. D. Zia and M. Usman, Perturbation Determinant and Levinson’s Formula for Schrödinger Operators with Generalized Point Interaction. (Submitted). 
• M. D. Zia and M. Usman, Eigenvalue Bounds for Non-selfadjoint Star Graphs, in preparation. 

Abstract:

Physical models of viscoelastic flow in terms of fractional order derivatives is a fascinating subject, in various fields of engineering, fluid dynamics, quantum computing, etc. The viscoelastic fluid models represent more realistic behavior as compared to the integer order derivatives in fluid dynamics. This is mainly because of the hereditary effects are taken into account by the fractional order models while formulating the flow problems. Mathematical models for such phenomenon usually result in the form of fractional partial differential equations (PDEs). In order to solve the fractional order PDEs, there are several analytical methods such as Laplace transform and Fourier transform methods in literature which had been usually employed to solve linear fractional order PDEs. However, it is difficult to solve non-linear fractional order PDEs using these analytical techniques. Hence, the researchers use numerical techniques to find approximate solutions for such fractional PDEs.

In this thesis, we have highlighted the numerical techniques alongwith the error and stability analysis of the schemes have been presented. Numerical methods, such as finite element and finite difference methods, along with L1 algorithm approximation, have been employed to investigate the approximate solutions. The simulations of flow fields as well as pertinent physical parameters and fractional order parameter effects on the fluid flow of some non-linear viscoelastic fluid problems have been discussed. A code in MATHLAB has been developed for simulations of the underlying models. Finally, the graphical illustration along with the physical interpretations have been presented for each problem. Conclusions and future directions are discussed at the end of thesis.

Publications:

• Effects of exponential variable viscosity on heat transfer flow of MHD fractional Maxwell fluid. https://link.springer.com/content/pdf/10.1007/s40819-020-00870-x.pdf

• Mixed Convection Magnetohydrodynamics Flow of a Nanofluid with Heat Transfer: A Numerical Study. https://www.hindawi.com/journals/mpe/2019/8129564/

• Numerical simulation of fractional Maxwell fluid flow through Forchheimer medium. https://www.sciencedirect.com/science/article/abs/pii/S0735193320304000

Abstract: 

The field of fractional calculus has been exceptionally developing since the inception of the idea in 1695 by Leibniz and L’Hôpital. The mathematical research groups extended their research interests to the fractional derivatives as they can candidly be presumed to generalize the integer-order derivatives. Moreover, very few research studies present logical inferences from the real-world physical phenomena studied through fractional derivatives. The field of fractional calculus has also captured the interest of many researchers from the field of Physics, acoustics, bio engineering, seismology, medical physics, imaging, and stochastics, etc. due to the fact that it can be used to model the non-locality, memory, spatial heterogeneity, and anomalous diffusion inherited in a number of real-world problems. The investigation of non-Newtonian fluid is a significant illustration of fractional calculus when the transport in a viscoelastic non-Newtonian fluid converts into “anisotropic” due to the distribution of fluid density and heterogeneity internal temperature while an external force created by shear flow is applied. Since the non-Newtonian fluids contain intricate multi-term time-fractional dynamical systems having diffusion and time-fractional diffusion terms in addition to a non-linear reaction term, so the conventional research techniques of universal applications and mathematical characterization utilized to investigate the Newtonian fluids are no longer valid. The aforesaid multifarious systems are modeled with the aid of fractional calculus by means of intercalating among the integer orders of differential equations in order to comprehend the non-local association in time and space using a fractional operator. The available techniques in literature cannot determine the exact solutions of coupled viscoelastic non-linear differential equations, and due attention is also not extended to the numerical solutions of atypical non-linear fractional flows. Therefore, in the recent past, several researchers have performed focused studies to discover novel and up-to-date theoretical and numerical techniques to determine solutions of fractional hydro-dynamical systems. Various concerns regarding viscoelastic simulations have been addressed by researchers with the passage of time. These issues may form a basis for restructuring the model in order to develop an efficient numerical scheme. In our thesis, we have taken into consideration only the viscoelastic fluids with different flow formations. Primarily, a structural description of the problem is offered, followed by fundamental laws essential for modeling the presented problem. Further, we have developed an up-to-date model for viscoelastic fluids where various physical effects like a magnetic field, variable thermal heat flux, joule heating phenomenon, etc., are applied over different geometrical surfaces keeping in view their industrial utility. In order to solve the aforesaid fractional viscoelastic fluid model, we have considered computational approaches of Finite difference approximations along with the $L_1$ algorithm to discretize the fractional time derivative and Finite Element discretization for spatial variables to determine the solution of fractional viscoelastic models. The benefits of using this model are manifold and will result in an efficient and precise solution with reference to different rheological models and kinds of nonlinearities that occur in the Non-Newtonian systems. We have formulated a code for our proposed numerical scheme and carried out numerical convergence and error analysis intending to establish the validity of the code. To conclude, we are confident enough that this thesis will be a valuable contribution to the studies of computational and fractional calculus over a viscoelastic fluid in addition to the beneficial count in industrial and engineering applications. 

Publications:

1.       Mumtaz Khan, Amer Rasheed, Scott-Blair model with unequal diffusivities of chemical species through a Forchheimer medium, Journal of Molecular Liquids, Vol. 341(3),pp. 117351 2021.

2.       Mumtaz Khan, Amer Rasheed, The Space-Time Coupled Fractional Cattaneo-Friedrich Maxwell Model with Caputo Derivatives, International Journal of Applied and Computational Mathematics, Vol. 7(3), 2021.

3.       Mumtaz Khan, Amer Rasheed, Taimoor Salahuddin, Shafqat Ali, Chemically reactive flow of hyperbolic tangent fluid flow having thermal radiation and double stratification embedded in porous medium, Ain Shams Engineering Journal, 12(3) 3209-3216, 2021.

4.       Mumtaz Khan, Amer Rasheed, Exothermic impact on Eyring-Powell nanofluid flow through a heated surface embedded in porous medium with thermal radiation: A numerical study, Physica Scripta, Vol. 96, pp. 035213, 2021.

5.       Mumtaz Khan, Amer Rasheed, Slip velocity and temperature jump effects on molybdenum disulfide MoS2 and silicon oxide SiO2 hybrid nano-fuid near irregular 3D surface, Alexandria Engineering Journal, Vol. 60(1), pp. 1689 - 1701, 2020.

Abstract:

The thesis concerns the development of the theory of functional (nonlocal) partial differential equations (PDEs). Such equations arise in various applications including size structured cell growth models, collection of current in an electric train, light absorption in the Milky Way, fragmentation in polymers, droplets, telecommunication systems to describe certain internet protocols, and more recently granular Physics. The equations are supplemented with appropriate initial and boundary conditions. There are no general methods for solving such problems due to the presence of nonlocal terms in the equation.

We first consider an initial boundary value problem that involves a functional partial differential equation with two nonlocal terms and a broad class of coefficients. We establish the existence of an attracting solution, towards which solutions to the problem converge for large time.

We then devise a method of solving an initial boundary value problem that involves a second-order parabolic partial differential equation with two nonlocal terms and constant coefficients. The presence of two nonlocal terms is a consequence of asymmetry in cell division. The solution techniques for solving such problems are rare due to nonlocal terms. We obtain separable solutions, as well as the general solution to the full problem, and show that the solutions to the problem converge to the separable solution for large time. The dispersion term does not affect the rate of convergence to the separable solution.

We then establish the existence of a unique solution to an initial boundary value problem that involves a non-homogeneous functional partial differential equation of the pantograph type with singular time dependent coefficients. The problem is also motivated from a cell division equation.

We also study an initial boundary value problem that involves a functional partial differential equation with two nonlocal terms and a certain class of time dependent coefficients. We establish the existence of a large time attracting solution at least for a certain class of coefficients. The rate of convergence of solutions towards the large time attracting solution is affected by the choice of coefficients and remains unaffected by the number of nonlocal terms. The uniqueness of solutions to the initial boundary value problem is also established.

Publications:

1. STH Shah, AA Zaidi, The existence of solutions to an inhomogeneous pantograph type equation with singular coefficients, Journal of Elliptic and Parabolic Equations, (2020) 6:935-945 DOI 10.1007/s41808-020-00089-3

2. M Efendiev, B van Brunt, AA Zaidi, STH Shah, Asymmetric cell division with stochastic growth rate. Dedicated to the memory of the late Spartak Agamirzayev, Mathematical Methods in the Applied Sciences (2018), 41 (17), 8059-8069

3. AA Zaidi, STH Shah, A cell division equation with time dependent growth and division rates, Under review, 2021.