Mathematics Commons^™

Differential Equation Of Appell Polynomials Via The Factorization Method, Matthew He, Paolo E. Ricci

Mathematics Faculty Articles

Let {P_n(x)}^∞_n=0 be a sequence of polynomials of degree n. We define two sequences of differential operators Φ_n and Ψ_n satisfying the following properties:

By constructing these two operators for Appell polynomials, we determine their differential equations via the factorization method introduced by Infeld and Hull (Rev. Mod. Phys. 23 (1951) 21). The differential equations for both Bernoulli and Euler polynomials are given as special cases of the Appell polynomials.

Hopf Bifurcation In Models For Pertussis Epidemiology, Herbert W. Hethcote, Yi Li, Zhujun Jing

Mathematics and Statistics Faculty Publications

Pertussis (whooping cough) incidence in the United States has oscillated with a period of about four years since data was first collected in 1922. An infection with pertussis confers immunity for several years, but then the immunity wanes, so that reinfection is possible. A pertussis reinfection is mild after partial loss of immunity, but the reinfection can be severe after complete loss of immunity. Three pertussis transmission models with waning of immunity are examined for periodic solutions. Equilibria and their stability are determined. Hopf bifurcation of periodic solutions around the endemic equilibrium can occur for some parameter values in two …

Geometric Aspects Of Second-Order Scalar Hyperbolic Partial Differential Equations In The Plane, Martin Jurás

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The purpose of this dissertation is to address various geometric aspects of second-order scalar hyperbolic partial differential equations in two independent variables and one dependent variable

F(x, y, u, u_x, u_y, u_xx, u_xy, u_yy) = 0

We find a characterization of hyperbolic Darboux integrable equations at level k (1) in terms of the vanishing of the generalized Laplace invariants and provide an invariant characterization of various cases in the Goursat general classification of hyperbolic Darboux integrable equations (1). In particular we give a contact invariant characterization of equations integrable by …

Conservative Motion Of A Discrete, Nonsymmetric, Hexahedral Gyroscope, Donald Greenspan

Mathematics Technical Papers - Archive

Gyroscopic motion is simulated by applying a molecular dynamics formulation to a rigid hexahedron. The conservative dynamical differential equations are solved numerically in such a fashion that all the system invariants are preserved. The difficult problem of nonsymmetry is emphasized. Examples of the complex motions which result are described and discussed.

A Quadratic Volterra Integral Equation And Its Solution For Various Kernels, Merlynd K. Nestell, Mostafa Ghandehari

Mathematics Technical Papers - Archive

Consider the Volterra integral equation [see pdf for notation], ^ is a parameter Solutions of this equation are discussed for separable and difference kernels using Laplace transform and differential equation techniques. Existence and uniqueness questions are explored for various assumptions on the kernel.

Continuous Deformation Of A Developable Surface, Chili Ping Hsu

Mathematics Technical Papers - Archive

A developable surface can be developed from a piece of planar region, or vice versa. Several methods are known to construct an isometric mapping between the developable and the planar region. A simple and efficient algorithm based on the differential geometry and differential equations is presented to construct such an isometric mapping. This algorithm contains a deformation parameter ^, where ^ varies between 0 and 1, that can trace the development of the surface in a fashion that ^ = 0 corresponds to the planar region and ^ = 1 recovers the developable. An error estimation shows the error, in …

Conservative Motion Of Discrete, Hexahedral Gyroscope, Donald Greenspan

Mathematics Technical Papers - Archive

Gyroscopic motion is simulated by applying a molecular dynamics formulation to a rigid hexahedron. The conservative dynamical differential equations are solved numerically in such a fashion that all the system invariants are preserved. Examples which included precession, nutation, and a combination of looping and cusp formation are described and discussed.

Conservative Motion Of Discrete, Tetrahedral Tops And Gyroscopes, Donald Greenspan

Mathematics Technical Papers - Archive

Tetrahedral tops and gyroscopes are simulated as discrete, rigid bodies in rotation by introducing a molecular mechanics formulation. The conservative, dynamical differential equations are solved numerically in such a fashion that all the system invariants are preserved. Examples which include precession and nutation are described and discussed.

Conservative Motion Of Discrete, Tetrahedral Top On A Smooth Horizontal Plane, Donald Greenspan

Mathematics Technical Papers - Archive

Tetrahedral tops are simulated as discrete, rigid bodies in rotation by introducing a molecular mechanics formulation. The contact point of the top with the XY plane is allowed to move in the plane. The conservative, dynamical differential equations are solved numerically in such a fashion that all the system invariants are preserved. Examples which include precession, nutation, cusp formation, and looping are described and discussed.

An Almost Periodic Function Of Several Variables With No Local Minimum, Gregory S. Spradlin

Publications

No abstract provided.

Projective And Non-Projective Systems Of First Order Nonlinear Differential Equations, Riad A. Rejoub

University of the Pacific Theses and Dissertations

It is well established that many physical and chemical phenomena such as those in chemical reaction kinetics, laser cavities, rotating fluids, and in plasmas and in solid state physics are governed by nonlinear differential equations whose solutions are of variable character and even may lack regularities. Such systems are usually first studied qualitatively by examining their temporal behavior near singular points of their phase portrait.

In this work we will be concerned with systems governed by the time evolution equations [see PDF for mathematical formulas]

The x_i may generally be considered to be concentrations of species in a chemical …

The Fokker-Planck And Related Equations In Theoretical Population Dynamics, George Derise

Mathematics & Statistics Theses & Dissertations

The population growth of a single species is modeled by a differential equation with initial condition(s) so that the number of organisms in the population is derived using some mechanism of growth, i.e. a growth rate function. However, such deterministic models are often highly unrealistic in population dynamics because population growth is basically a random event. There are a large number of chance factors influencing growth that might not be taken into account by deterministic models. The effect of other species (for example, in the chance meeting of a predator), population fluctuations due to weather changes that would alter food …

Multiple Shooting, Monique St-Maurice

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

The purpose of this report was to study the Multiple Shooting method, a numerical method to solve boundary value ordinary differential equation. A FORTRAN program was written to solve the specific problem.

y'' = -y, y(0) = 0, y(π/2) = 1

on a microcomputer, using the microsoft FORTRAN compiler and the 8087 coprocessor.

Sample Solutions Of Stochastic Ordinary Differential Equations, K. Waling

Mathematics Technical Papers - Archive

Motivated by the stochastic differential equation[see pdf for notation] in [see pdf for notation] we prove a measurable dependence on parameters theorem for ODEs in case f is only continuous in [see pdf for notation] is done by means of a known result about measurable selections of multivalued maps. Afterwards we discuss consequences for stochastic ODEs.

A Class Of Elliptic Partial Differential Equations With Exponential Nonlineraities, Pierre A. Vuillermot

Mathematics Technical Papers - Archive

This is the second of a series of papers devoted to systems of ordinary and partial differential equations with exponential nonlinearities [20], in which we prove the existence of at least a countable infinity of one- parameter families of nontrivial, Hölder continuous solutions for a class of second-order, strongly nonlinear Dirichlet boundary-value problems in two and more dimensions. Let Y be a C(2)-Young function, namely a twice continuously differentiable, even convex lR—valued function on IR such that[see pdf for notation] and[see pdf for notation]. By C(2)-function, we shall mean a function which is twice continuously differentiable everywhere except possibly at …

An Algorithm For Finding The Generalized Nucleolus Of A Finite Set And The Multiobjective Discrete Programming Problems, Irinel C. Dragan

Mathematics Technical Papers - Archive

No abstract provided.

General Uniqueness Criteria For Ordinary Differential Equations, V. Lakshmikantham, Mansour Samimi

Mathematics Technical Papers - Archive

This paper generalizes various uniqueness results and offers very general uniqueness criteria which include different types of results considered so far.

Nonuniqueness Criteria For Ordinary Differential Equations, Mansour Samimi

Mathematics Technical Papers - Archive

We consider an initial value problem (1.1) [see pdf for notation] where [see pdf for notation]. Several uniqueness results weaker than a Lipschitz condition are known, see [1,4]. However, results concerning with nonuniqueness criteria are rare. For the case n = 1, a nonuniqueness result was given in [5,6], see also [4]. Very recently the general case was also investigated in [3]. In this paper, we consider nonuniqueness problem from a very general point of view. Our first nonuniqueness result deals with the scalar case which extends the results of [5,6]. It also shows that when the conditions of general …

Monotone Iterative Technique For Differential Equations In A Banach Space, V. Lakshmikantham, Sen-Wo Du

Mathematics Technical Papers - Archive

Let E be a real Banach space with norm [see pdf for notation]. Consider the initial value problem (1.1) [see pdf for notation], where [see pdf for notation]. Generally speaking of approximate solutions of (1.1) consist of three steps, namely, (i) constructing a sequence of approximate solutions of some kinds for (1.1); (ii) showing the convergence of the constructed sequence; (iii) proving that the limit function is a solution. If f is continous, steps (i) and (iii) are standard and straight forward. It is a step (ii) that deserves attention. This in turn leads to three possibilities; namely to show …

The Frobenius Theorem, Hiroshi Nagao

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Many theorems in differential geometry which deal with the existence of certain geometrical structures or properties depend upon various existence and uniqueness theorems for differential equations. Because of its wide range of applications one of the most important of these theorems is the Frobenius Theorem for systems of total differential equations. There are four different forms of the Frobenius Theorem. In applications of the theorem one form is often preferable to the others. In this report we -shall prove the Frobenius Theorem, establish the equivalence of these various forms, and discuss a few applications.

Shallow Water Problem, Shu-Chen Li

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

The equations in shallow water over an obstacle are nonlinear hyperbolic differential equations. We use the method of lines and GEARB package to numerically solve this problem.

Quasi-Solutions And Their Role In The Qualitative Theory Of Differential Equations, V. Lakshmikantham, K. Deimling

Mathematics Technical Papers - Archive

In the study of comparison theorems and extremal solutions for systems of ordinary differential equations [5], one usually imposes a condition on the right hand side known as quasi-monotone nondecreasing property. This property. is also needed in proving comparison theorems for second order, boundary value problems [9] as well as for the initial boundary value problem for parabolic systems [5, 6]. Also, it is well known that in the .method of vector Lyapunov functions which provides an effective tool to investigate the stability of Large Scale Systems [1-4], an unpleasant drawback is the requirement of the quasi-monotone property for the …

Comparison Results For Reaction-Diffusion Equations In A Banach Space, V. Lakshmikantham

Mathematics Technical Papers - Archive

Let T be the temperature and n the concentration of a combustible substance. A simple model governing the combustion of the material is given by [see pdf for notations] (1.1) where the constant Q is the heat of reaction; the constants Kl , K2 are thermal, material diffusion coefficients; the term exp(-E/RT) is the Arrenhius rate factor; E is the activation energy; and R is the univeral gas constant. Equations (1.1) are considered on a bounded domain ^ with the boundary conditions [see pdf for notations] (1.2) together with initial conditions [see pdf for notations] (1.3) under the assumption T0(x) …

Existence And Comparison Theorems For Differential Equations In Banach Spaces, V. Lakshmikantham, K. Deimling

Mathematics Technical Papers - Archive

In our recent paper [3] we have studied the existence of maximal and minimal solutions to the IVP in a Banach space [see pdf for notation]. (1) [see pdf for notation] where [see pdf for notation] maps [see pdf for notation] into [see pdf for notation], with [see pdf for notation] and [see pdf for notation] a cone. The essential hypotheses have been that [see pdf for notation]f is quasimonotone with respect to [see pdf for notation] and that [see pdf for notation] and [see pdf for notation] have some natural properties. If such extremal solutions exist then it is …

Monotone Methods For Nonlinear Boundary Value Problems In Banach Spaces, Stephen R. Bernfeld, V. Lakshmikantham

Mathematics Technical Papers - Archive

Monotone methods have been used to generate multiple solutions of nonlinear boundary value problems for both ordinary and partial differential equations. Keller [10] and Sattinger [13], extending the chord method, considered nonlinear partial differential equations containing no gradient term. The inclusion of the gradient term was first introduced by Chandra and Davis [6] who considered the ordinary boundary value problem (1.1) [see pdf for notation] (1.2) [see pdf for notation] Here [see pdf for notation]. They assumed that f depends linearly on [see pdf for notation]; and this restriction on [see pdf for notation]' was eliminated by Bernfeld and Chandra …

Existence Of Solutions Of Boundary Value Problems For Nonlinear Second Order Systems In A Banach Space, A. Richard Mitchell, V. Chandra

Mathematics Technical Papers - Archive

This paper is concerned with the existence of solutions of boundary value problems (BVP, for short) for nonlinear second order ordinary differential equations of the type (1.1) [see pdf for notation] (1.2) [see pdf for notation] where [see pdf for notation] is a real Banach space. In case [see pdf for notation], existence was proved by first obtaining a priori bounds for [see pdf for notation] of a solution of (1.1) and (1.2) and then employing a theorem of Scorza-Dragoni [3,7,16]. The methods involve assuming inequalities in terms of the second derivative of Lyapunov-like functions relative to H, using comparison …

Group Of Point Transformations Of Time Dependent Harmonic Oscillators, Jose Ricardo Bernal

University of the Pacific Theses and Dissertations

In general, a physical system has invariant quantities which are very often related to its symmetry and to the invariance of the equation that describe it. A detailed study of the invariance property of the differential equation will be helpful in understanding this relation.

The work is concerned with a preliminary investigation of the Lie-group which leaves invariant the Newtonian and Lagrangian equation of motion for a one-dimensional harmonic oscillator. A brief review of Ehrenfest's adiabatic principle and the later treatments on exact and adiabatic invariants will be presented.

On The Zeros Of Monotone Operators Of Retarded Type In A Banach Space, Patrick Sutherland, V. Lakshmikantham

Mathematics Technical Papers - Archive

Recent interest in the Cauchy problem for differential equations in a Banach space [9] has stimulated a similar interest for differential equations of a retarded type in a Banach space. The difficulty in imposing assumptions due to a different range and domain space has been overcome in [4] where existence of solutions is established using a monotoni- city type condition in terms of norm and weaker forms of differential inequalities. The theory of existence of solutions of differential equations has been used in [1,2,5,6,7] to obtain existence of zeros and fixed points for nonlinear operators from E into E. In …

Maximal And Minimal Solutions And Comparison Principle For Differential Equations In Abstract Cones, Roger W. Mitchell, A. Richard Mitchell, V. Lakshmikantham

Mathematics Technical Papers - Archive

Existence of maximal and minimal solutions for differential equations in abstract cones is established without requiring uniform continuity. Utilizing such a result an abstract comparison principle is developed. The results of the paper significantly improve earlier results of the authors and also simplify the proofs of other known results.

On A Measure Of Nonconvexity And Applications, V. Lakshmikantham, Jerome Eisenfeld

Mathematics Technical Papers - Archive

The measure of noncompactness which was introduced by Kuratowski [8] (in 1930) has now become an important tool in nonlinear analysis (although its value in that regard was not appreciated until much later). Following Kuratowski we introduce a measure of nonconvexity which has many properties in common with the measure of noncompactness and therefore we may now have "convex" where previously we had "compact" in the statements of some theorems.