Applied Mathematics Commons^™

An Efficient Computational Method For Solving A System Of Fdes Via Fractional Finite Difference Method, M. M. Khader, Sunil Kumar

Applications and Applied Mathematics: An International Journal (AAM)

This paper aims to provide a numerical method for solving systems of fractional (Caputo sense) differential equations (FDEs). This method is based on the fractional finite difference method (FDM), where we implemented the Grünwald-Letnikov’s approach. This method is computationally very efficient and gives very accurate solutions. In this study, the stability of the obtained numerical scheme is given. The numerical results show that the proposed approach is easy to be implemented and are accurate when applied to system of FDEs. The method introduces promising tool for solving many systems of FDEs. Two examples are given to demonstrate the applicability and …

Joule's 19th Century Energy Conservation Meta-Law And The 20th Century Physics (Quantum Mechanics And General Relativity): 21st Century Analysis, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

Joule's Energy Conservation Law was the first "meta-law": a general principle that all physical equations must satisfy. It has led to many important and useful physical discoveries. However, a recent analysis seems to indicate that this meta-law is inconsistent with other principles -- such as the existence of free will. We show that this conclusion about inconsistency is based on a seemingly reasonable -- but simplified -- analysis of the situation. We also show that a more detailed mathematical and physical analysis of the situation reveals that not only Joule's principle remains true -- it is actually strengthened: it is …

Dynamics In A Respiratory Control Model With Two Delays, Saroj P. Pradhan, Ferenc Hartung, Janos Turi

Applications and Applied Mathematics: An International Journal (AAM)

In this paper we study ventilation patterns in a set of parameter dependent nonlinear delay equations with two transport delays modeling the human respiratory control system with peripheral and central control loops. We present a convergent numerical scheme suitable to perform simulations when all disturbances and system parameters are known, then we consider the numerical identifiability of various system parameters based on ventilation data. We are especially interested in the identification of the transport delays in the control loops because these parameters are not measurable directly, but they have a strong influence on system stability/instability.

Boundedness And Square Integrability In Neutral Differential Systems Of Fourth Order, Mebrouk Rahmane, Moussadek Remili, Linda D. Oudjedi

Applications and Applied Mathematics: An International Journal (AAM)

The aim of this paper is to study the asymptotic behavior of solutions to a class of fourth-order neutral differential equations. We discuss the stability, boundedness and square integrability of solutions for the considered system. The technique of proofs involves defining an appropriate Lyapunov functional. Our results obtained in this work improve and extend some existing well-known related results in the relevant literature which were obtained for nonlinear differential equations of fourth order with a constant delay. The obtained results here are new even when our equation is specialized to the forms previously studied and include many recent results in …

Understanding The Fundamental Molecular Mechanism Of Osteogenic Differentiation From Mesenchymal Stem Cells, Imelda Trejo, Hristo V. Kojouharov

Applications and Applied Mathematics: An International Journal (AAM)

A mathematical model is presented to study the regulatory effects of growth factors in osteoblastogenesis. The model incorporates the interactions among mesenchymal stem cells, osteoblasts, and growth factors. The resulting system of nonlinear ordinary differential equations is studied analytically and numerically. Mathematical conditions for successful osteogenic differentiation and optimal osteoblasts population are formulated, which can be used in practice to accelerate bone formation. Numerical simulations are also presented to support the theoretical results and to explore different medical interventions to enhance osteoblastogenesis.

Certain Quadruple Hypergeometric Series And Their Integral Representations, Maged Bin-Saad, Jihad Younis

Applications and Applied Mathematics: An International Journal (AAM)

While investigating the Exton's list of twenty one hyper-geometric functions of four variables and the Sharma's and Parihar's list of eighty three hyper-geometric functions of four variables, we noticed existence of new hyper-geometric series of four variables. The principal object of this paper is to introduce new hyper-geometric series of four variables and present a natural further step toward the mathematical integral presentation concerning these new series of four variables. Integral representations of Euler type and Laplace type involving Appell's hyper-geometric functions and the Horn's series of two variables, Exton's and Lauricella's triple functions and Sharma and Parihar hyper-geometric functions …

Numerical Solution Of Fractional Partial Differential Equations With Normalized Bernstein Wavelet Method, Mahsa Entezari, Saeid Abbasbandy, Esmail Babolian

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, normalized Bernstein wavelets are presented. Next, the fractional order integration and Bernstein wavelets operational matrices of integration are derived and finally are used for solving fractional partial differential equations. The operational matrices merged with the collocation method are used in order to convert fractional problems to a number of algebraic equations. In the suggested method the boundary conditions are automatically taken into consideration. An assessment of the error of function approximation based on the normalized Bernstein wavelet is also presented. Some numerical instances are given to manifest the versatility and applicability of the suggested method. Founded numerical …

Analysis Of Two Stage M[X1],M[X2]/G1,G2/1 Retrial G-Queue With Discretionary Priority Services, Working Breakdown, Bernoulli Vacation, Preferred And Impatient Units, G. Ayyappan, B. Somasundaram

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we study M^[X₁] , M^[X2] /G₁ ,G₂ /1 retrial queueing system with discretionary priority services. There are two stages of service for the ordinary units. During the first stage of service of the ordinary unit, arriving priority units can have an option to interrupt the service, but, in the second stage of service it cannot interrupt. When ordinary units enter the system, they may get the service even if the server is busy with the first stage of service of an ordinary unit or may enter into the orbit or leave …

Some Free Boundary Problems For The Nonlinear Degenerate Multidimensional Parabolic Equations Modeling Reaction-Diffusion Processes, Amna Abu Weden

Theses and Dissertations

This dissertation presents a full classification of the short-time behavior of the interfaces or free boundaries for the nonlinear second order degenerate multidimensional parabolic partial differential equation (PDE) ut −∆u m +buβ = 0, x ∈ R N ,0 < t < T (1) with m > 0, β > 0,b ∈ R, arising in various applications in fluid mechanics, filtration of oil or gas in a porous media, plasma physics, reaction-diffusion equations in chemical kinetics, population dynamics in mathematical biology etc. as a mathematical model of nonlinear diffusion phenomena in the presence of the absorption or release of energy. Cauchy problem with compactly supported and nonnegative initial function …

Multi-Point Flux Approximations Via The O-Method, Christen Leggett

Master's Theses

When an oil refining company is drilling for oil, much of the oil gets left behind after the first drilling. Enhanced oil recovery techniques can be used to recover more of that oil, but these methods are quite expensive. When a company is deciding if it is worth their time and money to use enhanced oil recovery methods, simulations can be used to model oil flow, showing the behavior and location of the oil. While methods do exist to model this flow, these methods are often very slow and inaccurate due to a large domain and wide variance in coefficients. …

Blow Up Of Solutions For A Coupled Kirchhoff-Type Equations With Degenerate Damping Terms, Erhan Piskin, Fatma Ekinci

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we investigate a system of coupled Kirchhoff-type equations with degenerate damping terms. We prove a nonexistence of global solutions with positive initial energy. Later, we give some estimates for lower bound of the blow up time.

Local Non-Similar Solution Of Powell-Eyring Fluid Flow Over A Vertical Flat Plate, Hemangini Shukla, Hema C. Surati, M. G. Timol

Applications and Applied Mathematics: An International Journal (AAM)

Our objective is to obtain the non-similarity solution of non-Newtonian fluid for Powell-Eyring model by a local non-similarity method. Here, free stream velocity is considered in power-law form (𝑈=𝑥^m). The governing equations are transformed using non-similar transformations and derived equations are treated as ordinary differential equations. Non-similar solutions are obtained for different values of power-law index 𝑚 and stream-wise location 𝜉. Influence of various parameters on velocity and temperature field are presented graphically using MATLAB bvp4c solver.

Inverse Spectral Problems For Spectral Data And Two Spectra Of N By N Tridiagonal Almost-Symmetric Matrices, Bayram Bala, Manaf D. Manafov, Abdullah Kablan

Applications and Applied Mathematics: An International Journal (AAM)

One way to study the spectral properties of Sturm-Liouville operators is difference equations. The coefficients of the second order difference equation which is equivalent Sturm-Liouville equation can be written as a tridiagonal matrix. One investigation area for tridiagonal matrix is finding eigenvalues, eigenvectors and normalized numbers. To determine these datas, we use the solutions of the second order difference equation and this investigation is called direct spectral problem. Furthermore, reconstruction of matrix according to some arguments is called inverse spectral problem. There are many methods to solve inverse spectral problems according to selecting the datas which are generalized spectral function, …

Numerical Solution Of The Lane-Emden Equations With Moving Least Squares Method, Sasan Asadpour, Hassan Hosseinzadeh, Allahbakhsh Yazdani

Applications and Applied Mathematics: An International Journal (AAM)

No abstract provided.

Function Space Tensor Decomposition And Its Application In Sports Analytics, Justin Reising

Electronic Theses and Dissertations

Recent advancements in sports information and technology systems have ushered in a new age of applications of both supervised and unsupervised analytical techniques in the sports domain. These automated systems capture large volumes of data points about competitors during live competition. As a result, multi-relational analyses are gaining popularity in the field of Sports Analytics. We review two case studies of dimensionality reduction with Principal Component Analysis and latent factor analysis with Non-Negative Matrix Factorization applied in sports. Also, we provide a review of a framework for extending these techniques for higher order data structures. The primary scope of this …

Analysis Of Interfaces For The Nonlinear Degenerate Second Order Parabolic Equations Modeling Diffusion-Convection Processes, Lamees Kadhim Ali Alzaki

Theses and Dissertations

Dissertation pursues analysis of the short-time evolution of interfaces or free boundaries for the non-negative solutions of the nonlinear degenerate second order parabolic partial differential equation (PDE) ut = ( u m ) xx +b ( u γ ) x , x ∈ R,t > 0; m > 1, γ > 0,b ∈ R (1) modeling diffusion-convection processes arising in fluid or gas flow in a porous media, plasma physics, population dynamics in mathematical biology and other applications. Due to the implicit degeneration (m > 1), PDE (1) it possesses a property of the finite speed of propagation and develops interfaces or free boundaries …

Nonlocal Boundary Value Problems For Linear Hyperbolic Systems With Two Independent Variables, Afrah Almutairi

Theses and Dissertations

Nonlocal boundary value problems in a characteristic rectangle for second order linear hyperbolic systems are considered. There are established: (i) Unimprovable sufficient conditions for general boundary value problems to possess the Fredholm property; (ii) Optimal sufficient conditions of unique solvability of general boundary value problems; (iii) Effective sufficient conditions for doubly periodic problems to possess the Fredholm property; (iv) Unimprovable sufficient conditions of unique solvability of doubly periodic problems; (v) Effective sufficient conditions for boundary value problems of periodic type to possess the Fredholm property; (vi) Unimprovable sufficient conditions of unique solvability of boundary value problems of periodic type; (vii) …

Stability Analysis Of Neutral Functional Differential Equations Arising In Partial Element Equivalent Circuit Models, Howard Michael Allison

Theses and Dissertations

Neutral Functional Differential Equations (NFDEs) arise in the study of the Partial Element Equivalent Circuit (PEEC) model with time delays. We present sufficient conditions for asymptotic stability and global stability in the delays of the PEEC NFDE’s, using Lyapunov-Razumikhin function methods.. We develop, for the first time, a standard mixing-type nonlinearity for the PEEC NFDEs. Introducing time invariant and time varying nonlinear perturbation to the PEEC NFDEs, we develop sufficient conditions for stability of the nonlinear perturbed PEEC NFDEs and convergence of the nonlinear system to the original stable linear autonomous system. We also develop sufficient conditions for stability and …

Computational Models For Biological Locomotion In Gels, Hashim Mohammed Alshehri

Theses and Dissertations

We investigated Low Reynold’s Number Locomotion in two-phase biological gels. The gel is composed of two materials: a viscous fluid solvent phase and a viscoelastic polymer network phase. A novel Two-phase Immersed Boundary Method (IBM) is developed to simulate the complicated interactions between an elastic boundary and a mixture of two fluids with very different physical properties. A further extension of the method is developed for the case where fluids satisfy partial-slip and free-slip conditions on the elastic boundary. Our major conclusions are summarized as following: (i) Our numerical scheme is proved to be robust and efficient. It can successfully …

Recover Data In Sparse Expansion Forms Modeled By Special Basis Functions, Abdulmtalb Mohamed Hussen

Dissertations

In data analysis and signal processing, the recovery of structured functions (in terms of frequencies and coefficients) with respect to certain basis functions from the given sampling values is a fundamental problem. The original Prony method is the main tool to solve this problem, which requires the equispaced sampling values.

In this dissertation, we use the equispaced sampling values in the frequency domain after the short time Fourier transform in order to reconstruct some signal expansions, such as the exponential expansions and the cosine expansions. In particular, we consider the case that the phase of the cosine expansion is quadratic. …

Dynamical Modeling In Cell Biology With Ordinary Differential Equations, Renee Marie Dale

LSU Doctoral Dissertations

Dynamical systems have been of interest to biologists and mathematicians alike. Many processes in biology lend themselves to dynamical study. Movement, change, and response to stimuli are dynamical characteristics that define what is 'alive'. A scientific relationship between these two fields is therefore natural. In this thesis, I describe how my PhD research variously related to biological, mathematical, and computational problems in cell biology. In chapter 1 I introduce some of the current problems in the field. In chapter 2, my mathematical model of firefly luciferase in vivo shows the importance of dynamical models to understand systems. Data originally collected …

Optimal Relaxation Weights For Multigrid Reduction In Time (Mgrit), Masumi Sugiyama

Mathematics & Statistics ETDs

Based on current trends in computer architectures, faster compute speeds must come from increased parallelism rather than increased clock speeds, which are stagnate. This situation has created the well-known bottleneck for sequential time-integration, where each individual time-value (i.e., time-step) is computed sequentially. One approach to alleviate this and achieve parallelism in time is with multigrid. In this work, we consider the scheme known as multigrid-reduction-in-time (MGRIT), but note that there exist other parallel-in-time methods such as parareal and the parallel full approximation scheme in space and time (PFASST). MGRIT is a full multi-level method applied to the time dimension and …

Planck's And Callendar's Blackbody Radiation Formulas And Their Fitness To Experimental Data, Max Tran

Publications and Research

In this paper, we compare the blackbody radiation density formula obtained with classical physics by Hugh L Callendar and the formula obtained by Max Planck using quantization of energy. We use R and Maxima to analyze their fitness on coordinating experimental data and indicate some limitations with experiments in this area.

How Can We Explain Different Number Systems?, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

At present, we mostly use decimal (base-10) number system, but in the past, many other systems were used: base-20, base-60 -- which is still reflected in how we divide an hour into minutes and a minute into seconds -- and many others. There is a known explanation for the base-60 system: 60 is the smallest number that can be divided by 2, by 3, by 4, by 5, and by 6. Because of this, e.g., half an hour, one-third of an hour, all the way to one-sixth of an hour all correspond to a whole number of minutes. In this …

Deep Learning (Partly) Demystified, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

Successes of deep learning are partly due to appropriate selection of activation function, pooling functions, etc. Most of these choices have been made based on empirical comparison and heuristic ideas. In this paper, we show that many of these choices -- and the surprising success of deep learning in the first place -- can be explained by reasonably simple and natural mathematics.

The Graphs That Have Antivoltages Using Groups Of Small Order, Vaidy Sivaraman, Dan Slilaty

Mathematics and Statistics Faculty Publications

Given a group Γ of order at most six, we characterize the graphs that have Γ-antivoltages and also determine the list of minor-minimal graphs that have no Γ-antivoltage. Our characterizations yield polynomial-time recognition algorithms for such graphs.

Properly Handling Negative Values In The Calculation Of Binding Constants By Physicochemical Modeling Of Spectroscopic Titration Data, Nathanael P. Kazmierczak, Douglas A. Vander Griend

University Faculty Publications and Creative Works

To implement equilibrium hard-modeling of spectroscopic titration data, the analyst must make a variety of crucial data processing choices that address negative absorbance and molar absorptivity values. The efficacy of three such methodological options is evaluated via high-throughput Monte Carlo simulations, root-mean-square error surface mapping, and two mathematical theorems. Accuracy of the calculated binding constant values constitutes the key figure of merit used to compare different data analysis approaches. First, using singular value decomposition to filter the raw absorbance data prior to modeling often reduces the number of negative values involved but has little effect on the calculated binding constant …

Diabetes And Its Effect On Abdominal Aortic Aneurysm Growth Rate In Hispanic Patients, Monica Betancourt-Garcia, Kristina Vatcheva, Amrit Thakur, Prateek Gupta, Eugene Postevka, Ricardo Martinez, R. Armour Forse

School of Mathematical & Statistical Sciences Faculty Publications

Background

The growth rate of abdominal aortic aneurysms (AAA) can vary depending on age, baseline diameter, blood pressure, race, and history of smoking. Paradoxically, previous studies show evidence of a protective effect of diabetes on the rate of AAA expansion despite its well-established role in the morbidity and mortality of cardiovascular disease. This study aims to investigate the impact diabetes plays on AAA growth within a Hispanic population.

Methods

Data were collected from patients who were predominantly Mexican-American at a single hospital site. Baseline and follow-up measures for AAA diameter were obtained from serial imaging studies. Demographics, medical history, the …

Fluids In Music: The Mathematics Of Pan’S Flutes, Bogdan Nita, Sajan Ramanathan

Department of Mathematics Faculty Scholarship and Creative Works

We discuss the mathematics behind the Pan’s flute. We analyze how the sound is created, the relationship between the notes that the pipes produce, their frequencies and the length of the pipes. We find an equation which models the curve that appears at the bottom of any Pan’s flute due to the different pipe lengths.

Why Does Ramanujan, The Man Who Knew Infinity, Matter?, Ken Ono

Dalrymple Lecture Series

Dr. Ken Ono is the Thomas Jefferson Professor of Mathematics at the University of Virginia, the Asa Griggs Candler Professor of Mathematics at Emory University, and the vice president of the American Mathematical Society.He is an associate producer of the film The Man Who Knew Infinity starring Dev Patel and Jeremy Irons about Srinivasa Ramanujan, a self-trained two-time college dropout who left behind three notebooks filled with equations that mathematicians are still trying to figure out today. Ramanujan claimed that his ideas came to him as visions from an Indian goddess. This lecture is about why Ramanujan matters. The answers …