Applied Mathematics Commons^™

Solution Of Fractional Drinfeld-Sokolov-Wilson Equation Using Homotopy Perturbation Transform Method, P. K. Singh, K. Vishal, T. Som

Applications and Applied Mathematics: An International Journal (AAM)

In this article, the approximate solutions of the non-linear Drinfeld-Sokolov-Wilson equation with fractional time derivative have been obtained. The fractional derivative is described in the Caputo sense. He’s polynomial is used to tackle the nonlinearity which arise in our considered problems. A time fractional nonlinear partial differential equation has been computed numerically. The numerical procedures illustrate the effectiveness and reliability of the method. Effects of fractional order time derivatives on the solutions for different particular cases are presented through graphs.

Application Of Reduced Differential Transform Method For Solving Nonlinear Reaction-Diffusion-Convection Problems, A. Taghavi, A. Babaei, A.` Mohammadpour

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, Reduced differential transform method is presented for solving nonlinear reactiondiffusion- convection initial value problems. The methodology with some known techniques shows that the present approach is simple and effective.To show the efficiency of the present method, four interesting examples is given.

Thermal Stresses In Functionally Graded Hollow Sphere Due To Non-Uniform Internal Heat Generation, S. P. Pawar, K. C. Deshmukh, G. D. Kedar

Applications and Applied Mathematics: An International Journal (AAM)

In this article, the thermal stresses in a hollow thick sphere of functionally graded material subjected to non-uniform internal heat generation are obtained as a function of radius to an exact solution by using the theory of elasticity. Material properties and heat generation are assumed as a function of radius of sphere and Poisson’s ratio as constant. The distribution of thermal stresses for different values of the powers of the module of elasticity and varying power law index of heat generation is studied. The results are illustrated numerically and graphically.

Implicit-Explicit Higher-Order Time Integration Schemes For Computations Of Structural Dynamics With Fluid-Structure Interaction, José C. Pedro, Mapundi K. Banda, Precious Sibanda

Applications and Applied Mathematics: An International Journal (AAM)

In this paper higher order implicit Runge-Kutta schemes are applied to fluid-structure interaction (FSI) simulations. A staggered approach with a structural predictor is applied to an FSI problem. The equations governing the dynamics of the structure are integrated in time by the Explicit Single Diagonal Implicit Runge-Kutta (ESDIRK) schemes and the arbitrary high order finite volume scheme is taken as the fluid solver. The performance of the ESDIRK scheme of order of convergence three to five is tested. Comparative studies with other time integration schemes which have been successfully applied to FSI problems are undertaken. Comparisons to test the performance …

Unsteady Boundary Layer Flow Of Thermophoretic Mhd Nanofluid Past A Stretching Sheet With Space And Time Dependent Internal Heat Source/Sink, N. Sandeep, C. Sulochana, C. S. K. Raju, M. J. Babu, V. Sugunamma

Applications and Applied Mathematics: An International Journal (AAM)

In this study we analyze the boundary layer flow of a thermophoretic magnetohydrodynamic dissipative nanofluid over an unsteady stretching sheet in a porous medium with space and time dependent internal heat source/sink. The governing equations are transformed to ordinary differential equations by using similarity transformation. Numerical solutions of these equations are obtained by using the Shooting Technique. The effects of non-dimensional governing parameters on the velocity, temperature, concentration profiles, friction factor, Nusselt and Sherwood numbers are discussed and presented through graphs and tables. Accuracy of the results compared with the existing ones. Excellent agreement is found with earlier studies.

Application Of A Numerical Method And Optimal Control Theory To A Partial Differential Equation Model For A Bacterial Infection In A Chronic Wound, Stephen Guffey

Masters Theses & Specialist Projects

In this work, we study the application both of optimal control techniques and a numerical method to a system of partial differential equations arising from a problem in wound healing. Optimal control theory is a generalization of calculus of variations, as well as the method of Lagrange Multipliers. Both of these techniques have seen prevalent use in the modern theories of Physics, Economics, as well as in the study of Partial Differential Equations. The numerical method we consider is the method of lines, a prominent method for solving partial differential equations. This method uses finite difference schemes to discretize the …

On Mikhailov's Reduction Group, Tihomir Valchev

Articles

We present a generalization of the notion of reduction group which allows one to study in a uniform way certain classes of nonlocal $S$-integrable equations like Ablowitz-Musslimani's nonlocal Schr\"odinger equation. Another aspect of the proposed generalization is the possibility to derive in a systematic way solutions to S-integrable equations with prescribed symmetries.

Time Integration Methods Of Fundamental Solutions And Approximate Fundamental Solutions For Nonlinear Elliptic Partial Differential Equations, Corey Leon Jones

Dissertations

A time-dependent method is coupled with the Method of Approximate Particular Solutions (MAPS) of Delta-shaped basis functions, the Method of Fundamental Solutions (MFS), and the Method of Approximate Fundamental Solutions (MAFS) to solve a second order nonlinear elliptic partial differential equation (PDE) on regular and irregular shaped domains. The nonlinear PDE boundary value problem is first transformed into a time-dependent quasilinear problem by introducing a fictitious time. Forward Euler integration is then used to ultimately convert the problem into a sequence of time-dependent linear nonhomogeneous modified Helmholtz boundary value problems on which the superposition principle is applied to split the …

A Contribution Toward Better Understanding Of Overbanking Tendency In Fixed-Wing Aircraft, Nihad E. Daidzic

Journal of Aviation Technology and Engineering

The phenomenon of overbanking tendency for a rigid-body, fixed-wing aircraft is investigated. Overbanking tendency is defined as a spontaneous, unbalanced rolling moment that keeps increasing an airplane’s bank angle in steep turns and must be arrested by opposite aileron action. As stated by the Federal Aviation Administration, the overbanking tendency may lead to a loss of control, especially in instrument meteorological conditions. It was found in this study that the speed differential over wing halves in horizontal turns indeed creates a rolling moment that achieves maximum values for bank angles between 45 and 55 degrees. However, this induced rolling moment …

A Rank 7 Pfaffian System On A 15-Dimensional Manifold With F4 Symmetry Algebra, Ian M. Anderson

Tutorials on... in 1 hour or less

Let I be a differential system on a manifold M. The infinitesimal symmetry algebra of I is the set of all vectors fields X on M such that preserve I. In this worksheet we present an example, due to E. Cartan of a rank 7 Pfaffian system on a 15-dimensional manifold whose infinitesimal symmetry algebra is the split real form of the exceptional Lie algebra f4 .

A Look At Vanilla Options And Greeks, Kevin Barbian

Mathematics: Student Scholarship & Creative Works

No abstract provided.

On The Dynamics Of Internal Waves Interacting With The Equatorial Undercurrent, Alan Compelli, Rossen Ivanov

Articles

The interaction of the nonlinear internal waves with a nonuniform current with a specific form, characteristic for the equatorial undercurrent, is studied. The current has no vorticity in the layer, where the internal wave motion takes place. We show that the nonzero vorticity that might be occuring in other layers of the current does not affect the wave motion. The equations of motion are formulated as a Hamiltonian system.

A Hamiltonian Approach To Wave-Current Interactions In Two-Layer Fluids, Adrian Constantin, Rossen Ivanov

Articles

We provide a Hamiltonian formulation for the governing equations describing the two-dimensional nonlinear interaction between coupled surfacewaves, internalwaves, and an underlying current with piecewise constant vorticity, in a two-layered fluid overlying a flat bed. This Hamiltonian structure is a starting point for the derivation of simpler models, which can be obtained systematically by expanding the Hamiltonian in dimensionless parameters. These enable an in-depth study of the coupling between the surface and internal waves, and how both these wave systems interact with the background current.

An Applied Mathematics Approach To Modeling Inflammation: Hematopoietic Bone Marrow Stem Cells, Systemic Estrogen And Wound Healing And Gas Exchange In The Lungs And Body, Racheal L. Cooper

Theses and Dissertations

Mathematical models apply to a multitude physiological processes and are used to make predictions and analyze outcomes of these processes. Specifically, in the medical field, a mathematical model uses a set of initial conditions that represents a physiological state as input and a set of parameter values are used to describe the interaction between variables being modeled. These models are used to analyze possible outcomes, and assist physicians in choosing the most appropriate treatment options for a particular situation. We aim to use mathematical modeling to analyze the dynamics of processes involved in the inflammatory process.

First, we create a …

Pulses And Snakes In Ginzburg-Landau Equation, S.C. Mancas, Roy S. Choudhury

Publications

Using a variational formulation for partial differential equations combined with numerical simulations on ordinary differential equations (ODEs), we find two categories (pulses and snakes) of dissipative solitons, and analyze the dependence of both their shape and stability on the physical parameters of the cubic-quintic Ginzburg–Landau equation (CGLE). In contrast to the regular solitary waves investigated in numerous integrable and non-integrable systems over the last three decades, these dissipative solitons are not stationary in time. Rather, they are spatially confined pulse-type structures whose envelopes exhibit complicated temporal dynamics. Numerical simulations reveal very interesting bifurcations sequences as the parameters of the CGLE …

Rainich-Type Conditions For Perfect Fluid Spacetimes, Dionisios Krongos, Charles G. Torre

Research Vignettes

In this worksheet we describe and illustrate a relatively simple set of new Rainich-type conditions on an n-dimensional spacetime which are necessary and sufficient for it to define a perfect fluid solution of the Einstein field equations. Procedures are provided which implement these Rainich-type conditions and which reconstruct the perfect fluid from the metric. These results provide an example of the idea of geometrization of matter fields in general relativity, which is a purely geometrical characterization of matter fields via the Einstein field equations.

On A Nonlinear Hyperbolic Partial Differential Equation With Irregular Data, Victor D´Evou´E

Applications and Applied Mathematics: An International Journal (AAM)

The main purpose of this paper is to study the existence and properties of solutions of a certain nonlinear non-Lipschitz hyperbolic partial differential equation in two independent variables with irregular data. Using regularization techniques, we give a meaning to this problem by replacing it by a tow parameters family of Lipschitz regular problems. We prove existence and uniqueness of the solution in an appropriate algebra of generalized functions and we precise how it depends on the choices made. We study the relationship with the classical solution.

A Semiparametric Estimation For Regression Functions In The Partially Linear Autoregressive Time Series Model, R. Farnoosh, M. Hajebi, S. J. Mortazavi

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, a semiparametric method is proposed for estimating regression function in the partially linear autoregressive time series model . Here, we consider a combination of parametric forms and nonlinear functions, in which the errors are independent. Semiparametric and nonparametric curve estimation provides a useful tool for exploring and understanding the structure of a nonlinear time series data set to make for a more efficient study in the partially linear autoregressive model. The unknown parameters are estimated using the conditional nonlinear least squares method, and the nonparametric adjustment is also estimated by defining and minimizing the local L2 -fitting …

Scaling Group Analysis On Mhd Free Convective Heat And Mass Transfer Over A Stretching Surface With Suction / Injection, Heat Source/Sink Considering Viscous Dissipation And Chemical Reaction Effects, Hunegnaw Dessie, Naikoti Kishan

Applications and Applied Mathematics: An International Journal (AAM)

This paper concerns with scaling group analysis on MHD free convective heat and mass transfer over stretching surface considering effects of thermal-diffusion and diffusion-thermo with suction /injection, heat source/sink and chemical reaction by taking into account viscous dissipation. Scaling group transformations are used to convert the partial differential equations of governing equations into ordinary differential equation and are solved numerically by Keller Box Method. Numerical results obtained for different parameters are drawn graphically and their effects on velocity, temperature and concentration profiles are discussed and shown graphically. Skin-friction coefficient, Nusselt number and Sherwood number are presented in table. It is …

Integrability And Exact Solutions For A (2+1)-Dimensional Variable-Coefficient Kdv Equation, Zhang Yu, Xu Gui-Qiong

Applications and Applied Mathematics: An International Journal (AAM)

By using the WTC method and symbolic computation, we apply the Painlevé test for a (2+1)-dimensional variable-coefficient Kortweg-de Vries (KdV) equation, and the considered equation is found to possess the Painlevé property without any parametric constraints. The auto-Bǎcklund transformation and several types of exact solutions are obtained by using the Painlevé truncated expansion method. Finally, the Hirota’s bilinear form is presented and multi-soliton solutions are also constructed.

The Investigation Of Surplus Of Energy And Signal Propagation At Time-Domain Waveguide Modes, Özlem Işık, Zeynep F. Koçak, Emre Eroğlu

Applications and Applied Mathematics: An International Journal (AAM)

Classical waveguide theory has been developed bearing on Bernoulli’s product method which results in separation of space and time variables in Maxwell’s equations. The time-harmonic waveguide modes have been stated mathematically for transmitting signals along the waveguides. As a starting point, present studies on transverse-electric (TE) and transverse-magnetic (TM) waveguide modes with previous results are taken and exhibited in an advanced form. They have been obtained within the framework of an evolutionary approach to solve Maxwell’s equations with time derivative. As a result every modal field is obtained in the form of a product of vector functions of transverse coordinates …

Centered-Difference Applications For Schrödinger's Equation, Matthew Thomas Murachver

Physics

This project enumerates methods utilizing discretized centered-difference approximations on the second order differential equation for quantum particles known as Schrodinger’s Equation. An eigenvalue-eigenfunction scheme is developed to sieve for valid solutions to The Time Independent Schrodinger Equation. Additionally the Crank-Nicolson method is applied to the Time Dependent Schrodinger Equation to describe wavefunction (eigenfunction) time evolution. The validity of these methods is discussed with applications to several fundamental pedagogical introductory quantum mechanic systems.

Dressing Method And Quadratic Bundles Related To Symmetric Spaces: Vanishing Boundary Conditions, Tihomir Valchev

Articles

We consider quadratic bundles related to Hermitian symmetric spaces of the type SU(m+n)/S(U(m) x U(n)). The simplest representative of the corresponding integrable hierarchy is given by a multi-component Kaup-Newell derivative nonlinear Schroedinger equation which serves as a motivational example for our general considerations. We extensively discuss how one can apply Zakharov-Shabat's dressing procedure to derive reflectionless potentials obeying zero boundary conditions. Those could be used for one to construct fast decaying solutions to any nonlinear equation belonging to the same hierarchy. One can distinguish between generic soliton type solutions and rational solutions.

On Quasilinear Parabolic Evolution Equations In Weighted Lp-Spaces Ii, Jeremy Lecrone, Mathias Wilke, Jan Prüss

Department of Math & Statistics Faculty Publications

Our study of abstract quasi-linear parabolic problems in time-weighted L_p-spaces, begun in 2010, is extended in this paper to include singular lower order terms, while keeping low initial regularity. The results are applied to reaction-diffusion problems, including Maxwell-Stefan diffusion, and to geometric evolution equations like the surface-diffusion flow or the Willmore flow. The method presented here will be applicable to other parabolic systems, including free boundary problems.

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.

One-Dimensional Weakly Nonlinear Model Equations For Rossby Waves, David Henry, Rossen Ivanov

Articles

In this study we explore several possibilities for modelling weakly nonlinear Rossby waves in fluid of constant depth, which propagate predominantly in one direction. The model equations obtained include the BBM equation, as well as the integrable KdV and Degasperis-Procesi equations.

A Soft Condensed Matter Approach Towards Mathematical Modelling Of Mass Transport And Swelling In Food Grains, Michael Chapwanya, N. Misra

Articles

Soft condensed matter (SCM) physics has recently gained importance for a large class of engineering materials. The treatment of food materials from a soft matter perspective, however, is only at the surface and is gaining importance for understanding the complex phenomena and structure of foods. In this work, we present a theoretical treatment of navy beans from a SCM perspective to describe the hydration kinetics. We solve the transport equations within a porous matrix and employ the Flory–Huggin’s equation for polymer–solvent mixture to balance the osmotic pressure. The swelling of the legume seed is modelled as a moving boundary with …

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.

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.

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.