Partial Differential Equations Commons^™

Exact Implicit Solution Of Nonlinear Heat Transfer In Rectangular Straight Fin Using Symmetry Reduction Methods, M. S. Abdel Latif, A. H. Abdel Kader, H. M. Nour

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, the exact implicit solution of the second order nonlinear ordinary differential equation which governing heat transfer in rectangular fin is obtained using symmetry reduction methods. General relationship among the temperature at the fin tip, the temperature gradient at the fin base, the mode of heat transfer, 𝑛 and the fin parameters 𝑁 and ℰ is obtained. Some numerical examples are discussed and it is shown that the temperature of fin increases when approaching from the heat source. The relationship between the fin efficiency and the temperature of fin tip is obtained for any value of the mode …

Boundary-Layer Flow Of Nanofluids Over A Moving Surface In The Presence Of Thermal Radiation, Viscous Dissipation And Chemical Reaction, Eshetu Haile, B. Shankar

Applications and Applied Mathematics: An International Journal (AAM)

The flow problem presented in the paper is boundary-layer flow of nanofluids over a moving surface in the presence of thermal radiation, viscous dissipation and chemical reaction. The plate is assumed to move in the same or opposite direction to the free stream which depends on the sign of the velocity parameter. The partial differential equations appearing in the governing equations are transformed into a couple of nonlinear ordinary differential equations using similarity transformations. The transformed equations in turn are solved numerically by the shooting method along with the fourth order Runge-Kutta integration technique. Influences of the pertinent parameters in …

Numerical Solutions Of Generalized Burgers' Equations For Some Incompressible Non-Newtonian Fluids, Yupeng Shu

LSU New Orleans Theses and Dissertations

The author presents some generalized Burgers' equations for incompressible and isothermal flow of viscous non-Newtonian fluids based on the Cross model, the Carreau model, and the Power-Law model and some simple assumptions on the flows. The author numerically solves the traveling wave equations for the Cross model, the Carreau model, the Power-Law model by using industrial data. The author proves existence and uniqueness of solutions to the traveling wave equations of each of the three models. The author also provides numerical estimates of the shock thickness as well as maximum strain $\varepsilon_{11}$ for each of the fluids.

Kink, Singular Soliton And Periodic Solutions To Class Of Nonlinear Equations, Marwan Alquran, Safwan Al-Shara, Sabreen Al-Nimrat

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we extend the ordinary differential Duffing equation into a partial differential equation. We study the traveling wave solutions to this model by using the G'/G expansion method. Then, based on the obtained results given for the Duffing equation, we generate kink, singular soliton and periodic solutions for a coupled integrable dispersionless nonlinear system. All the solutions given in this work are verified.

New Exact Solutions Of The Perturbed Nonlinear Fractional Schr¨Odinger Equation Using Two Reliable Methods, Nasir Taghizadeh, Mona N. Foumani, Vahid S. Mohammadi

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, the fractional derivatives in the sense of the modified Riemann-Liouville derivative and the first integral method and the Bernoulli sub-ODE method are employed for constructing the exact complex solutions of the perturbed nonlinear fractional Schr ¨odinger equation and comparing the solutions.

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.

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 …

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.

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 …

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.

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 …

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 …

Integrability, Recursion Operators And Soliton Interactions, Boyka Aneva, Georgi Grahovski, Rossen Ivanov, Dimitar Mladenov

Book chapter/book

This volume contains selected papers based on the talks,presentedat the Conference Integrability, Recursion Operators and Soliton Interactions, held in Sofia, Bulgaria (29-31 August 2012) at the Institute for Nuclear Research and Nuclear Energy of the Bulgarian Academy of Sciences. Included are also invited papers presenting new research developments in the thematic area. The Conference was dedicated to the 65-th birthday of our esteemed colleague and friend Vladimir Gerdjikov. The event brought together more than 30 scientists, from 6 European countries to celebrate Vladimir's scientific achievements. All participants enjoyed a variety of excellent talks in a friendly and stimulating atmosphere. …

Dispersion Of A Solute In Hartmann Two-Fluid Flow Between Two Parallel Plates, J. P. Kumar, J. C. Umavathi

Applications and Applied Mathematics: An International Journal (AAM)

The paper presents an analytical solution for the dispersion of a solute in a conducting immiscible fluid flowing between two parallel plates in the presence of a transverse magnetic field. The fluids in both the regions are incompressible, electrically conducting and the transport properties are assumed to be constant. The channel walls are assumed to be electrically insulating. Separate solutions for each fluid are obtained and these solutions are matched at the interface using suitable matching conditions. The results are tabulated for various values of viscosity ratio, pressure gradient and Hartman number on the effective Taylor dispersion coefficient and volumetric …

Hydro-Thermal Convective Solutions For An Aquifer System Heated From Below, Dambaru Bhatta

Applications and Applied Mathematics: An International Journal (AAM)

We investigate the effect of hydro-thermal convection in an aquifer system. It is assumed that the aquifer is bounded below and above by impermeable boundaries and it is heated from below. The solution of the governing system is expressed in terms of the basic steady state solution and perturbed solution. We obtain the critical Rayleigh number and critical wavenumber using Runge-Kutta method in combination of shooting method and present the marginal stability curve. The amplitude equation is derived by introducing the adjoint system. After amplitude is obtained, we compute the linear solutions for super-critical and sub-critical cases. Numerical results for …

How To Find Killing Vectors, Charles G. Torre

How to... in 10 minutes or less

We show how to compute the Lie algebra of Killing vector fields of a metric in Maple using the commands KillingVectors and LieAlgebraData. A Maple worksheet and a PDF version can be found below.

On The Geometrıc Interpretatıons Of The Kleın-Gordon Equatıon And Solution Of The Equation By Homotopy Perturbation Method, Hasan Bulut, H. M. Başkonuş

Applications and Applied Mathematics: An International Journal (AAM)

This paper is organized in the following ways: In the first part, we obtained the Klein Gordon Equation (KGE) in the Galilean space. In the second part, we applied Homotopy Perturbation Method (HPM) to this differential equation. In the third part, we gave two examples for the Klein Gordon equation. Finally, We compared the numerical results of this differential equation with their exact results. We also showed that approach used is easy and highly accurate.

Boundary Stabilization Of Torsional Vibrations Of A Solar Panel, Prasanta K. Nandi, Ganesh C. Gorain, Samarjit Kar

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we study a boundary stabilization of the torsional vibrations of a solar panel. The panel is held by a rigid hub at one end and is totally free at the other. The dynamics of the overall system leads to hybrid system of equations. It is set to a certain initial vibrations with a control torque as a stabilizer at the hub end only. Taking a non-linear damping as boundary stabilizer, a uniform exponential energy decay rate is obtained directly. Thus an explicit form of uniform stabilization of the system is achieved by means of the exponential energy …

Two Numerical Algorithms For Solving A Partial Integro-Differential Equation With A Weakly Singular Kernel, Jeong-Mi Yoon, Shishen Xie, Volodymyr Hrynkiv

Applications and Applied Mathematics: An International Journal (AAM)

Two numerical algorithms based on variational iteration and decomposition methods are developed to solve a linear partial integro-differential equation with a weakly singular kernel arising from viscoelasticity. In addition, analytic solution is re-derived by using the variational iteration method and decomposition method.

New Explicit Solutions For Homogeneous Kdv Equations Of Third Order By Trigonometric And Hyperbolic Function Methods, Marwan Alquran, Roba Al-Omary, Qutaibeh Katatbeh

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we study two-component evolutionary systems of the homogeneous KdV equation of the third order types (I) and (II). Trigonometric and hyperbolic function methods such as the sine-cosine method, the rational sine-cosine method, the rational sinh-cosh method, sech-csch method and rational tanh-coth method are used for analytical treatment of these systems. These methods, have the advantage of reducing the nonlinear problem to a system of algebraic equations that can be solved by computerized packages.

Modelling Two-Dimensional Photopolymer Patterns Produced With Multiple-Beam Holography, Dana Mackey, Tsvetanka Babeva, Izabela Naydenova, Vincent Toal

Conference papers

Periodic structures referred to as photonic crystals attract considerable interest due to their potential applications in areas such as nanotechnology, photonics, plasmonics, etc. Among various techniques used for their fabrication, multiple-beam holography is a promising method enabling defect-free structures to be produced in a single step over large areas.

In this paper we use a mathematical model describing photopolymerisation to simulate two-dimensional structures produced by the interference pattern of three noncoplanar beams. The holographic recording of different lattices is studied by variation of certain parameters such as beam wave vectors, time and intensity of illumination.

Stiefel And Grassmann Manifolds In Quantum Chemistry, Eduardo Chiumiento, Michael Melgaard

Articles

We establish geometric properties of Stiefel and Grassmann manifolds which arise in relation to Slatertype variational spaces in many-particle Hartree-Fock theory and beyond. In particular, we prove thatthey are analytic homogeneous spaces and submanifolds of the space of bounded operators on the single-particle Hilbert space. As a by-product we obtain that they are complete Finsler manifolds. These geometric properties underpin state-of-the-art results on existence of solutions to Hartree-Fock type equations.

Heterogeneous Multiscale Modeling Of Advection-Diffusion Problems, David J. Gardner, Daniel R. Reynolds

Mathematics Research

No abstract provided.

Controlling Nanoparticles Formation In Molten Metallic Bilayers By Pulsed-Laser Interference Heating, Mikhail Khenner, Sagar Yadavali, Ramki Kalyanaraman

Mathematics Faculty Publications

The impacts of the two-beam interference heating on the number of core-shell and embedded nanoparticles and on nanostructure coarsening are studied numerically based on the non-linear dynamical model for dewetting of the pulsed-laser irradiated, thin (< 20 nm) metallic bilayers. The model incorporates thermocapillary forces and disjoining pressures, and assumes dewetting from the optically transparent substrate atop of the reflective support layer, which results in the complicated dependence of light reflectivity and absorption on the thicknesses of the layers. Stabilizing thermocapillary effect is due to the local thickness-dependent, steady- state temperature profile in the liquid, which is derived based on the mean substrate temperature estimated from the elaborate thermal model of transient heating and melting/freezing. Linear stability analysis of the model equations set for Ag/Co bilayer predicts the dewetting length scales in the qualitative agreement with experiment.

Formation Of Organized Nanostructures From Unstable Bilayers Of Thin Metallic Liquids, Mikhail Khenner, Sagar Yadavali, Ramki Kalyanaraman

Mathematics Faculty Publications

Dewetting of pulsed-laser irradiated, thin (< 20 nm), optically reflective metallic bilayers on an optically transparent substrate with a reflective support layer is studied within the lubrication equations model. A steady-state bilayer film thickness (h) dependent temperature profile is derived based on the mean substrate temperature estimated from the elaborate thermal model of transient heating and melting/freezing. Large thermocapillary forces are observed along the plane of the liquid-liquid and liquid-gas interfaces due to this h-dependent temperature, which, in turn, is strongly influenced by the h-dependent laser light reflection and absorption. Consequently the dewetting is a result of the competition between thermocapillary and intermolecular forces. A linear analysis of the dewetting length scales established that the non-isothermal calculations better predict the experimental results as compared to the isothermal case within the bounding Hamaker coefficients. Subsequently, a computational non-linear dynamics study of the dewetting pathway was performed for Ag/Co and Co/Ag bilayer systems to predict the morphology evolution. We found that the systems evolve towards formation of different morphologies, including core-shell, embedded, or stacked nanostructure morphologies.

The Singular Perturbation In The Analysis Of Mode I Fracture Based Upon A New Multiscale Theory, Kai-Bin Fu

Applications and Applied Mathematics: An International Journal (AAM)

A theory of fracture is presented that is based upon an extension of continuum mechanics to the nanoscale fracture through the incorporation of long-range intermolecular forces which correct bulk material descriptions near interfaces. To be consistent with the literature, constant surface energies are assigned to interfaces. In the analysis of mode I fracture for quasi-brittle material, it is observed that the incorporation of these long-range intermolecular forces predicts sharp fracture tip rather than blunt and avoids stress singularity. The details of the singular perturbation employed in this analysis are also presented.