Ordinary Differential Equations and Applied Dynamics Commons^™

Variants Of Meir-Keeler Fixed Point Theorem And Applications Of Soft Set-Valued Maps, Akbar Azam, Mohammed Shehu Shagari

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we prove a Meir-Keeler type common fixed point theorem for two mappings for which the range set of the first one is a family of soft sets, called soft set-valued map and the second is a point-to-point mapping. In addition, it is also shown that under some suitable conditions, a soft set-valued map admits a selection having a unique fixed point. In support of the obtained result, nontrivial examples are provided. The novelty of the presented idea herein is that it extends the Meir-Keeler fixed point theorem and the theory of selections for multivalued mappings from the …

Homotopy Analysis Method For Nonlinear Ordinary Eigenvalue Problems, Subagya Perera

Master's Theses

In this thesis, we solve nonlinear differential equations by the homotopy analysis method (HAM), which is a semi-analytic method first introduced by Shijun Liao in 1992. The modified HAM can be viewed as a more generalized method that encloses many perturbation and non-perturbation methods. It is different from perturbation or other analytical methods in that it allows considerable freedomformanyvariables. Using the modified HAM, especially zero-order and higher-order deformation equations, we solve a nonlinear initial value problem and a nonlinear eigenvalue problem. We adjust the convergence region of a solution by modifying auxiliary parameter values. The results converge in very few …

Teaching And Learning Of Fluid Mechanics, Ashwin Vaidya

Department of Mathematics Faculty Scholarship and Creative Works

Fluid mechanics occupies a privileged position in the sciences; it is taught in various science departments including physics, mathematics, environmental sciences and mechanical, chemical and civil engineering, with each highlighting a different aspect or interpretation of the foundation and applications of fluids. Doll’s fluid analogy [5] for this idea is especially relevant to this issue: “Emergence of creativity from complex flow of knowledge—example of Benard convection pattern as an analogy—dissipation or dispersal of knowledge (complex knowledge) results in emergent structures, i.e., creativity which in the context of education should be thought of as a unique way to arrange information so …

Numerical Solution For Solving Two-Points Boundary Value Problems Using Orthogonal Boubaker Polynomials, Imad Noah Ahmed

Emirates Journal for Engineering Research

In this paper, a new technique for solving boundary value problems (BVPs) is introduced. An orthogonal function for Boubaker polynomial was utilizedand by the aid of Galerkin method the BVP was transformed to a system of linear algebraic equations with unknown coefficients, which can be easily solved to find the approximate result. Some numerical examples were added with illustrations, comparing their results with the exact to show the efficiency and the applicability of the method.

Block And Weddle Methods For Solving Nth Order Linear Retarded Volterra Integro-Differential Equations, Raghad Kadhim Salih

Emirates Journal for Engineering Research

A proposed method is presented to solve n^th order linear retarded Volterra integro-differential equations (RVIDE's) numerically by using fourth-order block and Weddle methods. Comparison between numerical and exact results has been given in numerical examples for conciliated the accuracy of the results of the proposed scheme.

On A Hybrid Technique To Handle Analytical And Approximate Solutions Of Linear And Nonlinear Fractional Order Partial Differential Equations, Kamal Shah, Hammad Khalil, Ahmet Yildirim

Applications and Applied Mathematics: An International Journal (AAM)

This manuscript is devoted to consider Natural transform (NT) coupled with homotopy perturbation method (HPM) for obtaining series solutions to some linear and nonlinear fractional partial differential equations (FPDEs). By means of NT, we obtain the transformed problem which is then solved by using HPM. By means of Stehfest’s numerical algorithm and using the dual relationship of NT and Laplace transform, we calculate inverse NT for approximate solutions. The series solutions we obtain using the proposed method are in close agreement with the exact solutions. We apply the proposed method to some interesting problems to illustrate our main results.

On The Weighted Pseudo Almost Periodic Solutions Of Nicholson’S Blowflies Equation, Ramazan Yazgan, Cemil Tunç

Applications and Applied Mathematics: An International Journal (AAM)

This study is concerned with the existence, uniqueness and global exponential stability of weighted pseudo almost periodic solutions of a generalized Nicholson’s blowflies equation with mixed delays. Using some differential inequalities and a fixed point theorem, sufficient conditions were obtained for the existence, uniqueness of at the least a weighted pseudo almost periodic solutions and global exponential stability of this solution. The results of this study are new and complementary to the previous ones can be found in the literature. At the end of the study an example is given to show the accuracy of our results.

Remarkable Applications Of Measure Of Non-Compactness For Infinite System Of Differential Equations, Merve İlkhan, Emrah E. Kara

Applications and Applied Mathematics: An International Journal (AAM)

The essential goal of our study is to search for a solution of an infinite system of differential equations in two different Banach spaces under certain assumptions by the aid of measure of noncompactness. Also, we establish some interesting examples related to our results.

Generalized Bagley-Torvik Equation And Fractional Oscillators, Mark Naber, Lucas Lymburner

Applications and Applied Mathematics: An International Journal (AAM)

In this paper the Bagley-Torvik Equation is considered with the order of the damping term allowed to range between one and two. The solution is found to be representable as a convolution of trigonometric and exponential functions with the driving force. The properties of the effective decay rate and the oscillation frequency with respect to the order of the fractional damping are also studied. It is found that the effective decay rate and oscillation frequency have a complex dependency on the order of the derivative of the damping term and exhibit properties one might expect of a thermodynamic Equation of …

Study Of Specially And Temporally Dependent Adsorption Coefficient In Heterogeneous Porous Medium, Dilip K. Jaiswal, Gulrana _

Applications and Applied Mathematics: An International Journal (AAM)

One-dimensional advection-dispersion equation (ADE) is studied along unsteady longitudinal flow through a semi-infinite heterogeneous medium. Adsorption coefficient is considered temporally and spatially–dependent function i.e., expressed in degenerate form. The dispersion parameter is considered as inversely proportional to adsorption coefficient. The input source is of pulse type. The Laplace Transformation Technique (LTT) is used to obtain the analytical solution by introducing certain new independent variables through separate transformations. The effects of adsorption, heterogeneity and unsteadiness are investigated and discussed with the help of various graphs.

Characterizing The Permanence And Stationary Distribution For A Family Of Malaria Stochastic Models, Divine Wanduku

Biology and Medicine Through Mathematics Conference

No abstract provided.

On The Complexity Of Computing Galois Groups Of Differential Equations, Mengxiao Sun

Dissertations, Theses, and Capstone Projects

The differential Galois group is an analogue for a linear differential equation of the classical Galois group for a polynomial equation. An important application of the differential Galois group is that a linear differential equation can be solved by integrals, exponentials and algebraic functions if and only if the connected component of its differential Galois group is solvable. Computing the differential Galois groups would help us determine the existence of the solutions expressed in terms of elementary functions (integrals, exponentials and algebraic functions) and understand the algebraic relations among the solutions.

Hrushovski first proposed an algorithm for computing the differential …

Mathematical Models: The Lanchester Equations And The Zombie Apocalypse, Hailey Bauer

Undergraduate Theses and Capstone Projects

This research study used mathematical models to analyze and depicted specific battle situations and the outcomes of the zombie apocalypse. The original models that predicted warfare were the Lanchester models, while the zombie apocalypse models were fictional expansions upon mathematical models used to examine infectious diseases. In this paper, I analyzed and compared different mathematical models by examining each model’s set of assumptions and the impact of the change in variables on the population classes. The purpose of this study was to understand the basics of the discrete dynamical systems and to determine the similarities between imaginary and realistic models. …

Climate Change In A Differential Equations Course: Using Bifurcation Diagrams To Explore Small Changes With Big Effects, Justin Dunmyre, Nicholas Fortune, Tianna Bogart, Chris Rasmussen, Karen Keene

CODEE Journal

The environmental phenomenon of climate change is of critical importance to today's science and global communities. Differential equations give a powerful lens onto this phenomenon, and so we should commit to discussing the mathematics of this environmental issue in differential equations courses. Doing so highlights the power of linking differential equations to environmental and social justice causes, and also brings important science to the forefront in the mathematics classroom. In this paper, we provide an extended problem, appropriate for a first course in differential equations, that uses bifurcation analysis to study climate change. Specifically, through studying hysteresis, this problem highlights …

Sir Models: Differential Equations That Support The Common Good, Lorelei Koss

CODEE Journal

This article surveys how SIR models have been extended beyond investigations of biologically infectious diseases to other topics that contribute to social inequality and environmental concerns. We present models that have been used to study sustainable agriculture, drug and alcohol use, the spread of violent ideologies on the internet, criminal activity, and health issues such as bulimia and obesity.

Simplifying Coefficients In A Family Of Ordinary Differential Equations Related To The Generating Function Of The Laguerre Polynomials, Feng Qi

Applications and Applied Mathematics: An International Journal (AAM)

In the paper, by virtue of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the Lah inversion formula, the author simplifies coefficients in a family of ordinary differential equations related to the generating function of the Laguerre polynomials.

Combinatorial Geometry Of Threshold-Linear Networks, Christopher Langdon

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Introducing The Fractional Differentiation For Clinical Data-Justified Prostate Cancer Modelling Under Iad Therapy, Ozlem Ozturk Mizrak

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Galois Groups Of Differential Equations And Representing Algebraic Sets, Eli Amzallag

Dissertations, Theses, and Capstone Projects

The algebraic framework for capturing properties of solution sets of differential equations was formally introduced by Ritt and Kolchin. As a parallel to the classical Galois groups of polynomial equations, they devised the notion of a differential Galois group for a linear differential equation. Just as solvability of a polynomial equation by radicals is linked to the equation’s Galois group, so too is the ability to express the solution to a linear differential equation in "closed form" linked to the equation’s differential Galois group. It is thus useful even outside of mathematics to be able to compute and represent these …

Masked Instability: Within-Sector Financial Risk In The Presence Of Wealth Inequality, Youngna Choi

Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works

We investigate masked financial instability caused by wealth inequality. When an economic sector is decomposed into two subsectors that possess a severe wealth inequality, the sector in entirety can look financially stable while the two subsectors possess extreme financially instabilities of opposite nature, one from excessive equity, the other from lack thereof. The unstable subsector can result in further financial distress and even trigger a financial crisis. The market instability indicator, an early warning system derived from dynamical systems applied to agent-based models, is used to analyze the subsectoral financial instabilities. Detailed mathematical analysis is provided to explain what financial …

Physical Applications Of The Geometric Minimum Action Method, George L. Poppe Jr.

Dissertations, Theses, and Capstone Projects

This thesis extends the landscape of rare events problems solved on stochastic systems by means of the \textit{geometric minimum action method} (gMAM). These include partial differential equations (PDEs) such as the real Ginzburg-Landau equation (RGLE), the linear Schroedinger equation, along with various forms of the nonlinear Schroedinger equation (NLSE) including an application towards an ultra-short pulse mode-locked laser system (MLL).

Additionally we develop analytical tools that can be used alongside numerics to validate those solutions. This includes the use of instanton methods in deriving state transitions for the linear Schroedinger equation and the cubic diffusive NLSE.

These analytical solutions are …

The Pope's Rhinoceros And Quantum Mechanics, Michael Gulas

Honors Projects

In this project, I unravel various mathematical milestones and relate them to the human experience. I show and explain the solution to the Tautochrone, a popular variation on the Brachistochrone, which details a major battle between Leibniz and Newton for the title of inventor of Calculus. One way to solve the Tautochrone is using Laplace Transforms; in this project I expound on common functions that get transformed and how those can be used to solve the Tautochrone. I then connect the solution of the Tautochrone to clock making. From this understanding of clocks, I examine mankind’s understanding of time and …

Theoretical Analysis Of Nonlinear Differential Equations, Emily Jean Weymier

Electronic Theses and Dissertations

Nonlinear differential equations arise as mathematical models of various phenomena. Here, various methods of solving and approximating linear and nonlinear differential equations are examined. Since analytical solutions to nonlinear differential equations are rare and difficult to determine, approximation methods have been developed. Initial and boundary value problems will be discussed. Several linear and nonlinear techniques to approximate or solve the linear or nonlinear problems are demonstrated. Regular and singular perturbation theory and Magnus expansions are our particular focus. Each section offers several examples to show how each technique is implemented along with the use of visuals to demonstrate the accuracy, …

On The Lp-Spaces Techniques In The Existence And Uniqueness Of The Fuzzy Fractional Korteweg-De Vries Equation’S Solution, F. Farahrooz, A. Ebadian, S. Najafzadeh

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, is proposed the existence and uniqueness of the solution of all fuzzy fractional differential equations, which are equivalent to the fuzzy integral equation. The techniques on L^P-spaces are used, defining the L^p_F F ([0; 1]) for 1≤P≤∞, its properties, and using the functional analysis methods. Also the convergence of the method of successive approximations used to approximate the solution of fuzzy integral equation be proved and an iterative procedure to solve such equations is presented.

Thermoelastic Analysis Of A Nonhomogeneous Hollow Cylinder With Internal Heat Generation, V. R. Manthena, N. K. Lamba, G. D. Kedar

Applications and Applied Mathematics: An International Journal (AAM)

In the present paper, we have determined the heat conduction and thermal stresses of a hollow cylinder with inhomogeneous material properties and internal heat generation. All the material properties except Poisson’s ratio and density are assumed to be given by a simple power law in axial direction. We have obtained the solution of the two dimensional heat conduction equation in the transient state in terms of Bessel’s and trigonometric functions. The influence of inhomogeneity on the thermal and mechanical behavior is examined. Numerical computations are carried out for both homogeneous and nonhomogeneous cylinders and are represented graphically.

Electromagnetic Resonant Scattering In Layered Media With Fabrication Errors, Emily Anne Mchenry

LSU Doctoral Dissertations

In certain layered electromagnetic media, one can construct a waveguide that supports a harmonic electromagnetic field at a frequency that is embedded in the continuous spectrum. When the structure is perturbed, this embedded eigenvalue moves into the complex plane and becomes a “complex resonance” frequency. The real and imaginary parts of this complex frequency have physical meaning. They lie behind anomalous scattering behaviors known collectively as “Fano resonance”, and people are interested in tuning them to specific values in optical devices. The mathematics involves spectral theory and analytic perturbation theory and is well understood [16], at least on a theoretical …

Thermal Stress Analysis In A Functionally Graded Hollow Elliptic-Cylinder Subjected To Uniform Temperature Distribution, V. R. Manthena, N. K. Lamba, G. D. Kedar

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, an analytical method of a thermoelastic problem for a medium with functionally graded material properties is developed in a theoretical manner for the elliptic-cylindrical coordinate system under the assumption that the material properties except for Poisson’s ratio and density are assumed to vary arbitrarily with the exponential law in the radial direction. An attempt has been made to reconsider the fundamental system of equations for functionally graded solids in a two-dimensional state under thermal and mechanical loads. The general solution of displacement formulation is obtained by the introduction of appropriate transformation and carried out the analysis by …

A Novel Approach For Solving Volterra Integral Equations Involving Local Fractional Operator, Hassan K. Jassim

Applications and Applied Mathematics: An International Journal (AAM)

The paper presents an approximation method called local fractional variational iteration method (LFVIM) for solving the linear and nonlinear Volterra integral equations of the second kind with local fractional derivative operators. Some illustrative examples are discussed to demonstrate the efficiency and the accuracy of the proposed method. Furthermore, this method does not require spatial discretization or restrictive assumptions and therefore reduces the numerical computation significantly. The results reveal that the local fractional variational iteration method is very effective and convenient to solve linear and nonlinear integral equations within local fractional derivative operators.

Application Of Kudryashov Method For The Ito Equations, Mozhgan Akbari

Applications and Applied Mathematics: An International Journal (AAM)

In this present work, the Kudryashov method is used to construct exact solutions of the (1+1)- dimensional and the (1+2)-dimensional form of the generalized Ito integro-differential equation. The Kudryashov method is a powerful method for obtaining exact solutions of nonlinear evolution equations. This method can be applied to non-integrable equations as well as integrable ones.

Application Of Symplectic Integration On A Dynamical System, William Frazier

Electronic Theses and Dissertations

Molecular Dynamics (MD) is the numerical simulation of a large system of interacting molecules, and one of the key components of a MD simulation is the numerical estimation of the solutions to a system of nonlinear differential equations. Such systems are very sensitive to discretization and round-off error, and correspondingly, standard techniques such as Runge-Kutta methods can lead to poor results. However, MD systems are conservative, which means that we can use Hamiltonian mechanics and symplectic transformations (also known as canonical transformations) in analyzing and approximating solutions. This is standard in MD applications, leading to numerical techniques known as symplectic …