Partial Differential Equations Commons^™

Isotropic Oscillator Under A Magnetic And Spatially Varying Electric Field, David L. Frost Mr., Frank Hagelberg

Undergraduate Honors Theses

We investigate the energy levels of a particle confined in the isotropic oscillator potential with a magnetic and spatially varying electric field. Here we are able to exactly solve the Schrodinger equation, using matrix methods, for the first excited states. To this end we find that the spatial gradient of the electric field acts as a magnetic field in certain circumstances. Here we present the changes in the energy levels as functions of the electric field, and other parameters.

Options Pricing And Hedging In A Regime-Switching Volatility Model, Melissa A. Mielkie

Electronic Thesis and Dissertation Repository

Both deterministic and stochastic volatility models have been used to price and hedge options. Observation of real market data suggests that volatility, while stochastic, is well modelled as alternating between two states. Under this two-state regime-switching framework, we derive coupled pricing partial differential equations (PDEs) with the inclusion of a state-dependent market price of volatility risk (MPVR) term.

Since there is no closed-form solution for this pricing problem, we apply and compare two approaches to solving the coupled PDEs, assuming constant Poisson intensities. First we solve the problem using numerical solution techniques, through the application of the Crank-Nicolson numerical scheme. …

Reliable Study Of Nonhomogeneous Bbm Equation With Time-Dependent Coefficients By The Modified Sine-Cosine Method, Aminah Qawasmeh, Marwan Alquran

Applications and Applied Mathematics: An International Journal (AAM)

The modified sine-cosine method is an efficient and powerful mathematical tool in finding exact traveling wave solutions to nonlinear partial differential equations (NLPDEs) with time-dependent coefficients. In this paper, the proposed approach is applied to study a nonhomogeneous generalized form of Benjamin-Bona-Mahony (BBM) equation with time-dependent coefficients. Explicit traveling wave solutions of the equation are obtained under certain constraints on the coefficient functions.

Stability Of An Inhomogeneous Damped Vibrating String, Siddhartha Misra, Ganesh C. Gorain

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we consider the vibrations of an inhomogeneous damped string under a distributed disturbing force which is clamped at both ends. The well-possedness of the system is studied. We prove that the amplitude of such vibrations is bounded under some restriction of the disturbing force. Finally, we establish the uniform exponential stabilization of the system when the disturbing force is insignificant. The results are established directly by means of an exponential energy decay estimate.

Numerical Solution For The Systems Of Variable-Coefficient Coupled Burgers’ Equation By Two-Dimensional Legendre Wavelets Method, Hossein Aminikhah, Sakineh Moradian

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, a numerical method for solving the systems of variable-coefficient coupled Burgers’ equation is proposed. The method is based on two-dimensional Legendre wavelets. Two-dimensional operational matrices of integration are introduced and then employed to find a solution to the systems of variable-coefficient coupled Burgers’ equation. Two examples are presented to illustrate the capability of the method. It is shown that the numerical results are in good agreement with the exact solutions for each problem.

Application Of The Extended G'/G-Expansion Method To The Improved Eckhaus Equation, Nasir Taghizadeh, Seyyedeh R. Moosavi Noori, Seyyedeh B. Moosavi Noori

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, the extended (G'/G)-expansion method is used to seek more general exact solutions of the improved Eckhaus equation and the (2+1)-dimensional improved Eckhaus equation. As a result, hyperbolic function solutions, trigonometric function solutions and rational function solutions with free parameters are obtained. When the parameters are taken as special values the solitary wave solutions are also derived from the traveling wave solutions. Moreover, it is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations in mathematical physics.

Analysis Of A Partial Differential Equation Model Of Surface Electromigration, Selahittin Cinar

Masters Theses & Specialist Projects

A Partial Differential Equation (PDE) based model combining surface electromigration and wetting is developed for the analysis of the morphological instability of mono-crystalline metal films in a high temperature environment typical to operational conditions of microelectronic interconnects. The atomic mobility and surface energy of such films are anisotropic, and the model accounts for these material properties. The goal of modeling is to describe and understand the time-evolution of the shape of film surface. I will present the formulation of a nonlinear parabolic PDE problem for the height function h(x,t) of the film in the horizontal …

An Applied Functional And Numerical Analysis Of A 3-D Fluid-Structure Interactive Pde, Thomas J. Clark

Department of Mathematics: Dissertations, Theses, and Student Research

We will present qualitative and numerical results on a partial differential equation (PDE) system which models a certain fluid-structure dynamics. In Chapter \ref{ChWellposedness}, the wellposedness of this PDE model is established by means of constructing for it a nonstandard semigroup generator representation; this representation is essentially accomplished by an appropriate elimination of the pressure. This coupled PDE model involves the Stokes system which evolves on a three dimensional domain $\mathcal{O}$ being coupled to a fourth order plate equation, possibly with rotational inertia parameter $\rho >0$, which evolves on a flat portion $\Omega$ of the boundary of $\mathcal{O}$. The coupling on …

A Radial Basis Function Partition Of Unity Method For Transport On The Sphere, Kevin Aiton

Boise State University Theses and Dissertations

The transport phenomena dominates geophysical fluid motions on all scales making the numerical solution of the transport problem fundamentally important for the overall accuracy of any fluid solver. In this thesis, we describe a new high-order, computationally efficient method for numerically solving the transport equation on the sphere. This method combines radial basis functions (RBFs) and a partition of unity method (PUM). The method is mesh-free, allowing near optimal discretization of the surface of the sphere, and is free of any coordinate singularities. The basic idea of the method is to start with a set of nodes that are quasi-uniformly …

Mathematical Modeling Of The American Lobster Cardiac Muscle Cell: An Investigation Of Calcium Ion Permeability And Force Of Contractions, Lauren A. Skerritt

Honors Projects

In the American lobster (Homarus americanus), neurogenic stimulation of the heart drives fluxes of calcium (Ca²⁺) into the cytoplasm of a muscle cell resulting in heart muscle contraction. The heartbeat is completed by the active transport of calcium out of the cytoplasm into extracellular and intracellular spaces. An increase in the frequency of calcium release is expected to increase amplitude and duration of muscle contraction. This makes sense because an increase in cytoplasmic calcium should increase the activation of the muscle contractile elements (actin and myosin). Since calcium cycling is a reaction-diffusion process, the extent to …

Well-Posedness And Stability Of A Semilinear Mindlin-Timoshenko Plate Model, Pei Pei

Department of Mathematics: Dissertations, Theses, and Student Research

I will discuss well-posedness and long-time behavior of Mindlin-Timoshenko plate equations that describe vibrations of thin plates. This system of partial differential equations was derived by R. Mindlin in 1951 (though E. Reissner also considered an analogous model earlier in 1945). It can be regarded as a generalization of the Timoshenko beam model (1937) to flat plates, and is more accurate than the classical Kirchhoff-Love plate theory (1888) because it accounts for shear deformations.

I will present a semilinear version of the Mindlin-Timoshenko system. The primary feature of this model is the interplay between nonlinear frictional forces (``damping”) and nonlinear …

Study Of Virus Dynamics By Mathematical Models, Xiulan Lai

Electronic Thesis and Dissertation Repository

This thesis studies virus dynamics within host by mathematical models, and topics discussed include viral release strategies, viral spreading mechanism, and interaction of virus with the immune system.

Firstly, we propose a delay differential equation model with distributed delay to investigate the evolutionary competition between budding and lytic viral release strategies. We find that when antibody is not established, the dynamics of competition depends on the respective basic reproduction numbers of the two viruses. If the basic reproductive ratio of budding virus is greater than that of lytic virus and one, budding virus can survive. When antibody is established for …

Euler-Poincar´E Equations For G-Strands, Darryl Holm, Rossen Ivanov

Conference papers

The G-strand equations for a map R×R into a Lie group G are associated to a G-invariant Lagrangian. The Lie group manifold is also the configuration space for the Lagrangian. The G-strand itself is the map g(t,s):R×R→G, where t and s are the independent variables of the G-strand equations. The Euler-Poincar'e reduction of the variational principle leads to a formulation where the dependent variables of the G-strand equations take values in the corresponding Lie algebra g and its co-algebra, g^∗ with respect to the pairing provided by the variational derivatives of the Lagrangian. We review examples of different G-strand …

On A Nonlocal Nonlinear Schrodinger Equation, Tihomir Valchev

Conference papers

We consider a nonlocal nonlinear Schr\"odinger equation recently proposed by Ablowitz and Musslimani as a theoretical model for wave propagation in {\it PT}-symmetric coupled wave-guides and photonic crystals. This new equation is integrable by means of inverse scattering method, i. e. it possesses a Lax pair, infinite number of integrals of motion and exact solutions. We aim to describe here some of the basic properties of the nonlocal Schr\"odinger equation and its scattering operator. In doing this we shall make use of methods alternative to those applied by Ablowitz and Musslimani which seem to be better suited for treating possible …

Examples Of G-Strand Equations, Darryl Holm, Rossen Ivanov

Conference papers

The G-strand equations for a map R×R into a Lie group G are associated to a G-invariant Lagrangian. The Lie group manifold is also the configuration space for the Lagrangian. The G-strand itself is the map g(t,s):R×R→G, where t and s are the independent variables of the G-strand equations. The Euler-Poincare´ reduction of the variational principle leads to a formulation where the dependent variables of the G-strand equations take values in the corresponding Lie algebra g and its co-algebra, g∗ with respect to the pairing provided by the variational derivatives of the Lagrangian. We review examples of G-strand constructions, including …

Zakharov-Shabat System With Constant Boundary Conditions. Reflectionless Potentials And End Point Singularities, Tihomir Valchev, Rossen Ivanov, Vladimir Gerdjikov

Conference papers

We consider scalar defocusing nonlinear Schroedinger equation with constant boundary conditions. We aim here to provide a self contained pedagogical exposition of the most important facts regarding integrability of that classical evolution equation. It comprises the following topics: direct and inverse scattering problem and the dressing method.

A Posteriori Error Estimates For Surface Finite Element Methods, Fernando F. Camacho

Theses and Dissertations--Mathematics

Problems involving the solution of partial differential equations over surfaces appear in many engineering and scientific applications. Some of those applications include crystal growth, fluid mechanics and computer graphics. Many times analytic solutions to such problems are not available. Numerical algorithms, such as Finite Element Methods, are used in practice to find approximate solutions in those cases.

In this work we present L2 and pointwise a posteriori error estimates for Adaptive Surface Finite Elements solving the Laplace-Beltrami equation −△_Γ u = f . The two sources of errors for Surface Finite Elements are a Galerkin error, and a …

Termodynamika Procesowa I Techniczna Lab., Wojciech M. Budzianowski

Wojciech Budzianowski

No abstract provided.

Tematyka Prac Dyplomowych Dla Studentów Wydziału Mechaniczno-Energetycznego Pwr., Wojciech M. Budzianowski

Wojciech Budzianowski

No abstract provided.

Tematyka Prac Dyplomowych Dla Studentów Wydziału Chemicznego Pwr., Wojciech M. Budzianowski

Wojciech Budzianowski

No abstract provided.