Analytical solutions to nonlinear differential equations arising in physical problems

Nonlinear partial differential equations are difficult to solve, with many of the approximate solutions in the literature being numerical in nature. In this work, we apply the Homotopy Analysis Method to give approximate analytical solutions to nonlinear ordinary and partial differential equations. The main goal is to apply different linear operators, which can be chosen, to solve nonlinear problems. In the first three chapters, we study ordinary differential equations (ODEs) with one or two linear operators. As we progress, we apply the method to partial differential equations (PDEs) and use several linear operators. The results are all purely analytical, meaning these are approximate solutions that we can evaluate at points and take their derivatives. Another main focus is error analysis, where we test how good our approximations are. The method will always produce approximations, but we use residual errors on the domain of the problem to find a measure of error. In the last two chapters, we apply similarity transforms to PDEs to transform them into ODEs. We then use the Homotopy Analysis Method on one, but are able to find exact solutions to both equations

Mass, momentum, and energy flux conservation for nonlinear wave-wave interaction

A fully nonlinear solution for bi-chromatic progressive waves in water of finite depth in the framework of the homotopy analysis method (HAM) is derived. The bi-chromatic wave field is assumed to be obtained by the nonlinear interaction of two monochromatic wave trains that propagate independently in the same direction before encountering. The equations for the mass, momentum, and energy fluxes based on the accurate high-order homotopy series solutions are obtained using a discrete integration and a Fourier series-based fitting. The conservation equations for the mean rates of the mass, momentum, and energy fluxes before and after the interaction of the two nonlinear monochromatic wave trains are proposed to establish the relationship between the steady-state bi-chromatic wave field and the two nonlinear monochromatic wave trains. The parametric analysis on ε1 and ε2, representing the nonlinearity of the bi-chromatic wave field, is performed to obtain a sufficiently small standard deviation Sd, which is applied to describe the deviation from the conservation state (Sd = 0) in terms of the mean rates of the mass, momentum, and energy fluxes before and after the interaction. It is demonstrated that very small standard deviation from the conservation state can be achieved. After the interaction, the amplitude of the primary wave with a lower circular frequency is found to decrease; while the one with a higher circular frequency is found to increase. Moreover, the highest horizontal velocity of the water particles underneath the largest wave crest, which is obtained by the nonlinear interaction between the two monochromatic waves, is found to be significantly higher than the linear superposition value of the corresponding velocity of the two monochromatic waves. The present study is helpful to enrich and deepen the understanding with insight to steady-state wave-wave interactions

Analytical and Numerical Methods for Differential Equations and Applications

The book is a printed version of the Special issue Analytical and Numerical Methods for Differential Equations and Applications, published in Frontiers in Applied Mathematics and Statistic

Applied Mathematics and Fractional Calculus

In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia

Pattern Formation in Certain Classical and Quantum Systems

My research predominantly focuses on micromagnetic simulations of cobalt nanoparticles. These simulations are carried out by using the Object-Oriented MicroMagnetic Framework: OOMMF) distributed by the National Institute of Standards and Technology: NIST). In performing these simulations, observations of skyrmionic states were observed for large enough nanoparticles. Skyrmions are magnetic vortex states which have a core that is antiparallel with the outermost local magnetic moments. These states are being explored as an exciting branch of research that are applicable in many applications including, but not limited to, magnetic storage devices. Most simulations were carried out using a hemispherical geometry because collaborators at the University of Tennessee, Knoxville use laser-induced dewetting to create arrays of hemispheres on the nanometer scale. In conjunction with these simulations, we have performed analytical and numerical calculations of the demagnetizing factor of a hemispherical particle. The demagnetizing factor of a system is a parameter that characterizes shape anisotropy and is useful in calculating the energy of a uniformly magnetized body. Demagnetizing factors have been calculated for different geometries, but never for a hemisphere. Other projects that my research includes research into rogue waves and a mapping between classical and quantum systems. The rogue wave work deals with analysis of the Nonlinear Schr {o}dinger Equation and developing new forms of rogue wave solutions. This process includes focussing on the time-reversal invariance of the Nonlinear Shr {o}dinger Equation and also generalizes a wave form known as compactons to generate approximate, traveling wave solutions that are like a rogue wave in nature. These can be applied to oceanic, optical, and even economical situations. The mapping that we have generalized here indicates that classical and quantum correlation functions can be calculated from one another simply by allowing time to go to imaginary time, and taking the real part of the expression. This is partially based on previous work in a more specific version of this mapping, but generalizes it to any operator rather than specifically the density operator

Spacecraft/Rover Hybrids for the Exploration of Small Solar System Bodies

This study investigated a mission architecture that allows the systematic and affordable in-situ exploration of small solar system bodies, such as asteroids, comets, and Martian moons (Figure 1). The architecture relies on the novel concept of spacecraft/rover hybrids,which are surface mobility platforms capable of achieving large surface coverage (by attitude controlled hops, akin to spacecraft flight), fine mobility (by tumbling), and coarse instrument pointing (by changing orientation relative to the ground) in the low-gravity environments(micro-g to milli-g) of small bodies. The actuation of the hybrids relies on spinning three internal flywheels. Using a combination of torques, the three flywheel motors can produce a reaction torque in any orientation without additional moving parts. This mobility concept allows all subsystems to be packaged in one sealed enclosure and enables the platforms to be minimalistic. The hybrids would be deployed from a mother spacecraft, which would act as a communication relay to Earth and would aid the in-situ assets with tasks such as localization and navigation (Figure 1). The hybrids are expected to be more capable and affordable than wheeled or legged rovers, due to their multiple modes of mobility (both hopping and tumbling), and have simpler environmental sealing and thermal management (since all components are sealed in one enclosure, assuming non-deployable science instruments). In summary, this NIAC Phase II study has significantly increased the TRL (Technology Readiness Level) of the mobility and autonomy subsystems of spacecraft/rover hybrids, and characterized system engineering aspects in the context of a reference mission to Phobos. Future studies should focus on improving the robustness of the autonomy module and further refine system engineering aspects, in view of opportunities for technology infusion