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Articles 1 - 30 of 42
Full-Text Articles in Physical Sciences and Mathematics
Generalised Diffusive Delay Logistic Equations: Semi-Analytical Solutions, H Y. Alfifi, Timothy R. Marchant, M I. Nelson
Generalised Diffusive Delay Logistic Equations: Semi-Analytical Solutions, H Y. Alfifi, Timothy R. Marchant, M I. Nelson
Tim Marchant
This paper considers semi-analytical solutions for a class of generalised logis- tic partial dierential equations with both point and distributed delays. Both one and two-dimensional geometries are considered. The Galerkin method is used to approximate the governing equations by a system of ordinary dierential delay equations. This method involves assuming a spatial structure for the solution and averaging to obtain the ordinary dierential delay equation models. Semi-analytical results for the stability of the system are derived with the critical parameter value, at which a Hopf bifurcation occurs, found. The results show that diusion acts to stabilise the system, compared to …
Numerical Simulation Of Contaminant Flow In A Wool Scour Bowl, James Caunce, S. Barry, Geoff Mercer, Timothy Marchant
Numerical Simulation Of Contaminant Flow In A Wool Scour Bowl, James Caunce, S. Barry, Geoff Mercer, Timothy Marchant
Tim Marchant
No abstract provided.
Asymptotic Solitons On A Non-Zero Mean Level, Timothy Marchant
Asymptotic Solitons On A Non-Zero Mean Level, Timothy Marchant
Tim Marchant
No abstract provided.
Development Of Empirical Relationships For Metallurgical Design Of Hot-Rolled Steel Products, Timothy R. Marchant, Alysha Nickerson, David Scott, Steve Taylor
Development Of Empirical Relationships For Metallurgical Design Of Hot-Rolled Steel Products, Timothy R. Marchant, Alysha Nickerson, David Scott, Steve Taylor
Tim Marchant
An empirical model is developed to predict the mechanical properties that steel inherits from the hot-rolling process and its chemical composition. In this process, slabs of steel are heated and rolled into thinner sheets which are either coiled or cut into plates. The mechanical properties of the coils and plates are sampled and must conform to national and international standards for steel products. The aim here is to use the statistical technique of multiple linear regression to develop relationships between the mechanical properties and the various processing temperatures and concentrations of chemical elements present in the steel. This analytical tool …
Approximate Techniques For Dispersive Shock Waves In Nonlinear Media, Timothy R. Marchant, Noel F. Smyth
Approximate Techniques For Dispersive Shock Waves In Nonlinear Media, Timothy R. Marchant, Noel F. Smyth
Tim Marchant
Many optical and other nonlinear media are governed by dispersive, or diffractive, wave equations, for which initial jump discontinuities are resolved into a dispersive shock wave. The dispersive shock wave smooths the initial discontinuity and is a modulated wavetrain consisting of solitary waves at its leading edge and linear waves at its trailing edge. For integrable equations the dispersive shock wave solution can be found using Whitham modulation theory. For nonlinear wave equations which are hyperbolic outside the dispersive shock region, the amplitudes of the solitary waves at the leading edge and the linear waves at the trailing edge of …
Solitary Waves And Their Stability In Colloidal Media: Semi-Analytical Solutions, Timothy R. Marchant, N F. Smyth
Solitary Waves And Their Stability In Colloidal Media: Semi-Analytical Solutions, Timothy R. Marchant, N F. Smyth
Tim Marchant
Spatial solitary waves in colloidal suspensions of spherical dielectric nanopartides are considered. The interaction of the nanoparticles is modelled as a hard-sphere gas, with the Carnahan-Starling formula used for the gas compressibility. Semi-analytical solutions, for both one and two spatial dimensions, arc derived using an averaged Lagrangian and suitable trial functions for the solitary waves. Power versus propagation constant curves and neutral stability curves are obtained for both cases, which illustrate that multiple solution branches occur for both the one and two dimensional geometries. For the one-dimensional case it is found that three solution branches (with a bistable regime) occur. …
Semi-Analytical Solutions For Dispersive Shock Waves In Colloidal Media, Timothy R. Marchant, N. F. Smyth
Semi-Analytical Solutions For Dispersive Shock Waves In Colloidal Media, Timothy R. Marchant, N. F. Smyth
Tim Marchant
The diffractive resolution of a discontinuity at the edge of an optical beam in a colloidal suspension of spherical dielectric nanoparticles by a collisionless shock, or undular bore, is studied. The interaction of the nanoparticles is modelled as a hard-sphere gas with the Carnahan-Starling formula used for the gas compressibility. The governing equation is a focusing nonlinear Schr¨odinger-type equation with an implicit nonlinearity. It is found that the discontinuity is resolved through the formation of a dispersive shock wave which forms before the eventual onset of modulational instability. A semi-analytical solution is developed in the 1 + 1 dimensional case …
Asymptotic Solitons Of The Extended Korteweg–De Vries Equation, Timothy R. Marchant
Asymptotic Solitons Of The Extended Korteweg–De Vries Equation, Timothy R. Marchant
Tim Marchant
The interaction of two higher-order solitary waves, governed by the extended Korteweg–de Vries (KdV) equation, is examined. A nonlocal transformation is used on the extended KdV equation to asymptotically transform it to the KdV equation. The transformation is used to derive the higher-order two-soliton collision and it is found that the interaction is asymptotically elastic. Moreover, the higher-order corrections to the phase shifts suffered by the solitary waves during the collision are found. Comparison is made with a previous result, which indicated that, except for a special case, the interaction of higher-order KdV solitary waves is inelastic, with a coupling, …
Reorientational Versus Kerr Dark And Gray Solitary Waves Using Modulation Theory, Prof. Tim Marchant
Reorientational Versus Kerr Dark And Gray Solitary Waves Using Modulation Theory, Prof. Tim Marchant
Tim Marchant
We develop a modulation theory model based on a Lagrangian formulation to investigate the evolution of dark and gray optical spatial solitary waves for both the defocusing nonlinear Schrodinger (NLS) equation and the nematicon equations describing nonlinear beams, nematicons, in self-defocusing nematic liquid crystals. Since it has an exact soliton solution, the defocusing NLS equation is used as a test bed for the modulation theory applied to the nematicon equations, which have no exact solitary wave solution. We find that the evolution of dark and gray NLS solitons, as well as nematicons, is entirely driven by the emission of diffractive …
The Analytical Evolution Of Nls Solitons Due To The Numerical Discretization Error, Prof. Tim Marchant
The Analytical Evolution Of Nls Solitons Due To The Numerical Discretization Error, Prof. Tim Marchant
Tim Marchant
Soliton perturbation theory is used to obtain analytical solutions describing solitary wave tails or shelves, due to numerical discretization error, for soliton solutions of the nonlinear Schrodinger equation. Two important implicit numerical schemes for the nonlinear Schrodinger equation, with second-order temporal and spatial discretization errors, are considered. These are the Crank-Nicolson scheme and a scheme, due to Taha [1], based on the inverse scattering transform. The first-order correction for the solitary wave tail, or shelf, is in integral form and an explicit expression is found for large time. The shelf decays slowly, at a rate of t(-1/2), which is characteristic …
Evolution Of Solitary Waves For A Perturbed Nonlinear Schrodinger Equation, Tim Marchant
Evolution Of Solitary Waves For A Perturbed Nonlinear Schrodinger Equation, Tim Marchant
Tim Marchant
Soliton perturbation theory is used to determine the evolution of a solitary wave described by a perturbed nonlinear Schrödinger equation. Perturbation terms, which model wide classes of physically relevant perturbations, are considered. An analytical solution is found for the first-order correction of the evolving solitary wave. This solution for the solitary wave tail is in integral form and an explicit expression is found, for large time. Singularity theory, usually used for combustion problems, is applied to the large time expression for the solitary wave tail. Analytical results are obtained, such as the parameter regions in which qualitatively different types of …
Soliton Perturbation Theory For A Higher-Order Hirota Equation, Tim Marchant
Soliton Perturbation Theory For A Higher-Order Hirota Equation, Tim Marchant
Tim Marchant
Solitary wave evolution for a higher order Hirota equation is examined. For the higher order Hirota equation resonance between the solitary waves and linear radiation causes radiation loss. Soliton perturbation theory is used to determine the details of the evolving wave and its tail. An analytical expression for the solitary wave tail is derived and compared to numerical solutions. An excellent comparison between numerical and theoretical solutions is obtained for both right- and left-moving waves. Also, a two-parameter family of higher order asymptotic embedded solitons is identified.
Modulation Analysis Of Boundary Induced Motion Of Optical Solitary Waves In A Nematic Liquid Crystal, Tim Marchant
Modulation Analysis Of Boundary Induced Motion Of Optical Solitary Waves In A Nematic Liquid Crystal, Tim Marchant
Tim Marchant
We consider the motion of a solitary wave, a nematicon, in a finite cell filled with a nematic liquid crystal. A modulation theory is developed to describe the boundary-induced bouncing of a nematicon in a one-dimensional cell and it is found to give predictions in very good agreement with numerical solutions. The boundary-induced motion is then considered numerically for a two-dimensional cell and a simple extension of the modulation theory from one to two space dimensions is then made, with good agreement being found with numerical solutions for the nematicon trajectory. The role of nematicon shape and relative position to …
A Perturbation Drbem Model For Weakly Nonlinear Wave Run-Ups Around Islands, Tim Marchant
A Perturbation Drbem Model For Weakly Nonlinear Wave Run-Ups Around Islands, Tim Marchant
Tim Marchant
In this paper, the dual reciprocity boundary element method (DRBEM) based on the perturbation method is presented for calculating run-ups of weakly nonlinear long waves scattered by islands. Under the assumption that the incident waves are harmonic, the time-dependent nonlinear Boussinesq equations are transformed into three time-independent linear equations by using the perturbation method, where, besides nonlinearity ε, the dispersion μ2 is also included in the perturbed expansion. The first-order solution η0 is found by using the linear long-wave equations. Then η0 is used in two second-order governing equations related to the dispersion and nonlinearity, respectively. Since no any omission …
Evolution Of Higher-Order Gray Hirota Solitary Waves, Tim Marchant
Evolution Of Higher-Order Gray Hirota Solitary Waves, Tim Marchant
Tim Marchant
The defocusing Hirota equation has dark and gray soliton solutions which are stable on a background of periodic waves of constant amplitude. In this paper, gray solitary wave evolution for a higher-order defocusing Hirota equation is examined. A direct analysis is used to identify families of higher-order gray Hirota solitary waves, which are embedded for certain parameter values. Soliton perturbation theory is used to detmine the detailed behavior of an evolving higher-order gray Hirota solitary wave. An integral expression for the first-order correction to the wave is found and analytical expressions for the steady-state and transient components of the solitary …
Mathematical Modelling Of Nematicons And Their Interactions, Prof. Tim Marchant
Mathematical Modelling Of Nematicons And Their Interactions, Prof. Tim Marchant
Tim Marchant
The mathematical modelling of guided wave (nematicon) propagation in liquid crystals is considered. Model equations are derived based on suitable trial functions in an averaged Lagrangian. These equations are used to model nematicon interactions.
Undular Bores And The Initial-Boundary Value Problem For The Modified Korteweg-De Vries Equation, Tim Marchant
Undular Bores And The Initial-Boundary Value Problem For The Modified Korteweg-De Vries Equation, Tim Marchant
Tim Marchant
Two types of analytical undular bore solutions, of the initial value problem for the modified Korteweg-de Vries (mKdV), are found. The first, an undular bore composed of cnoidal waves, is qualitatively similar to the bore found for the KdV equation, with solitons occurring at the leading edge and small amplitude linear waves occurring at the trailing edge. The second, a newly identified type of undular bore, consists of finite amplitude sinusiodal waves, which have a rational form. At the leading edge is the mKdV algebraic soliton, while, again, small amplitude linear waves occur at the trailing edge. The initial-boundary value …
Collisionless Shock Resolution In Nematic Liquid Crystals, Tim Marchant
Collisionless Shock Resolution In Nematic Liquid Crystals, Tim Marchant
Tim Marchant
The diffractive resolution on a collisionless shock formed along the spatial profile of a beam in a nematic liquid crystal is considered, this material being an example of a self-focusing, nonlocal medium. It is found that the shock is resolved through the formation of an undular bore structure which persists for experimentally relevant propagation distances due to nonlocality delaying the onset of modulational instability. Both 1+1 and 2+1 dimensional bores with circular symmetry are considered (termed line and circular bores, respectively). A semianalytical solution is developed for the line undular bore, approximating it as a train of uniform solitary waves. …
Semi-Analytical Solutions For A Gray-Scott Reaction-Diffusion Cell With An Applied Electric Field, Tim Marchant
Semi-Analytical Solutions For A Gray-Scott Reaction-Diffusion Cell With An Applied Electric Field, Tim Marchant
Tim Marchant
An ionic version of the Gray–Scott chemical reaction scheme is considered in a reaction–diffusion cell, with an applied electric field, which causes migration of the reactant and autocatalyst in a preferred direction. The Galerkin method is used to reduce the governing partial differential equations to an approximate model consisting of ordinary differential equations. This is accomplished by approximating the spatial structure of the reactant and autocatalyst concentrations. Bifurcation analysis of the semi-analytical model is performed by using singularity theory to analyse the static multiplicity and a stability analysis to determine the dynamic multiplicity. The application of the electric field causes …
Nonlocal Validity Of An Asymptotic One-Dimensional Nematicon Solution, Tim Marchant
Nonlocal Validity Of An Asymptotic One-Dimensional Nematicon Solution, Tim Marchant
Tim Marchant
The propagation of coherent, polarized light in a nematic liquid crystal, governed by the nematicon equations, is considered. It is found that in the special case of 1 + 1 dimensions and the highly nonlocal limit, the nematicon equations have an asymptotic bulk solitary wave solution, termed a nematicon, which is given in terms of Bessel functions. This asymptotic solution gives both the ground state and the symmetric and antisymmetric excited states, which have multiple peaks. Numerical simulations of nematicon evolution, for parameters corresponding to experimental scenarios, are presented. It is found, for experimentally reasonable parameter choices, that the validity …
Dipole Soliton Formation In A Nematic Liquid Crystal In The Non-Local Limit, Tim Marchant
Dipole Soliton Formation In A Nematic Liquid Crystal In The Non-Local Limit, Tim Marchant
Tim Marchant
The interaction of two symmetric solitary waves, termed nematicons, in a liquid crystal is considered in the limit of nonlocal response of the liquid crystal. This nonlocal limit is the applicable limit for most experimentally available liquid crystals. In this nonlocal limit, two separate cases for the initial separation of the nematicons are considered, these being large and small separation. Both spinning and nonspinning nematicons are considered. It is found that in the case of large initial separation, the nematicons can form a spinning or nonspinning bound state with a finite steady separation, this being called a nematicon dipole, when …
Self-Heating In Compost Piles Due To Biological Effects, Tim Marchant
Self-Heating In Compost Piles Due To Biological Effects, Tim Marchant
Tim Marchant
The increase in temperature in compost piles/landfill sites due to micro-organisms undergoing exothermic reactions is modelled. A simplified model is considered in which only biological self-heating is present. The heat release rate due to biological activity is modelled by a function which is a monotonic increasing function of temperature over the range 0⩽T⩽a, whilst for T⩾a it is a monotone decreasing function of temperature. This functional dependence represents the fact that micro-organisms die or become dormant at high temperatures. The bifurcation behaviour is investigated for 1-d slab and 2-d rectangular slab geometries. In both cases there are two generic steady-state …
Solitary Wave Interaction For A Higher-Order Nonlinear Schrodinger Equation, Tim Marchant
Solitary Wave Interaction For A Higher-Order Nonlinear Schrodinger Equation, Tim Marchant
Tim Marchant
Solitary wave interaction for a higher-order version of the nonlinear Schrödinger (NLS) equation is examined. An asymptotic transformation is used to transform a higher-order NLS equation to a higher-order member of the NLS integrable hierarchy, if an algebraic relationship between the higher-order coefficients is satisfied. The transformation is used to derive the higher-order one- and two-soliton solutions; in general, the N-soliton solution can be derived. It is shown that the higher-order collision is asymptotically elastic and analytical expressions are found for the higher-order phase and coordinate shifts. Numerical simulations of the interaction of two higher-order solitary waves are also performed. …
Asymptotic Solitons On A Non-Zero Mean Level., Tim Marchant
Asymptotic Solitons On A Non-Zero Mean Level., Tim Marchant
Tim Marchant
The collision of solitary waves for a higher-order modified Korteweg-de Vries (mKdV) equation is examined. In particular, the collision between solitary waves with sech-type and algebraic (which only exist on a non-zero mean level) profiles is considered. An asymptotic transformation, valid if the higher-order coefficients satisfy a certain algebraic relationship, is used to transform the higher-order mKdV equation to an integrable member of the mKdV hierarchy. The transformation is used to show that the higher-order collision is asymptotically elastic and to derive the higher-order phase shifts. Numerical simulations of both elastic and inelastic collisions are performed. For the example covered …
Numerical Simulation Of Contaminant Flow In A Wool Scour Bowl., Tim Marchant
Numerical Simulation Of Contaminant Flow In A Wool Scour Bowl., Tim Marchant
Tim Marchant
Wool scouring is the process of washing dirty wool after shearing. Our model numerically simulates contaminant movement in a wool scour bowl using the advection–dispersion equation. This is the first wool scour model to give time-dependent results and to model the transport of contaminants within a single scour bowl. Our aim is to gain a better understanding of the operating parameters that will produce efficient scouring. Investigating the effects of varying the parameters reveals simple, interesting relationships that give insight into the dynamics of a scour bowl.
An Undular Bore Solution For The Higher-Order Korteweg-De Vries Equation, Tim Marchant
An Undular Bore Solution For The Higher-Order Korteweg-De Vries Equation, Tim Marchant
Tim Marchant
Undular bores describe the evolution and smoothing out of an initial step in mean height and are frequently observed in both oceanographic and meteorological applications. The undular bore solution for the higher-order Korteweg-de Vries (KdV) equation is derived, using an asymptotic transformation which relates the KdV equation and its higher-order counterpart. The higher-order KdV equation considered includes all possible third-order correction terms (where the KdV equation retains second-order terms). The asymptotic transformation is then applied to the KdV undular bore solution to obtain the higher-order undular bore. Examples of higher-order undular bores, describing both surface and internal waves, are presented. …
Solitary Wave Interaction And Evolution For A Higher-Order Hirota Equation, Tim Marchant
Solitary Wave Interaction And Evolution For A Higher-Order Hirota Equation, Tim Marchant
Tim Marchant
Solitary wave interaction and evolution for a higher-order Hirota equation is examined. The higher-order Hirota equation is asymptotically transformed to a higher-order member of the NLS hierarchy of integrable equations, if the higher-order coefficients satisfy a certain algebraic relationship. The transformation is used to derive higher-order one- and two-soliton solutions. It is shown that the interaction is asymptotically elastic and the higher-order corrections to the coordinate and phase shifts are derived. For the higher-order Hirota equation resonance occurs between the solitary waves and linear radiation, so soliton perturbation theory is used to determine the details of the evolving wave and …
Modelling A Wool Scour Bowl, Tim Marchant
Modelling A Wool Scour Bowl, Tim Marchant
Tim Marchant
Wool scouring is the process of washing dirty wool after shearing. Our model simulates, using the advection-diffusion equation, the movement of contaminants within a scour bowl. The effects of varying the important parameters are investigated. Interesting, but simple, relationships are found which give insight into the dynamics of a scour bowl.
Approximate Solutions For Magmon Propagation From A Reservoir, Tim Marchant
Approximate Solutions For Magmon Propagation From A Reservoir, Tim Marchant
Tim Marchant
A 1D partial differential equation (pde) describing the flow of magma in the Earth's mantle is considered, this equation allowing for compaction and distension of the surrounding matrix due to the magma. The equation has periodic travelling wave solutions, one limit of which is a solitary wave, called a magmon. Modulation equations for the magma equation are derived and are found to be either hyperbolic or of mixed hyperbolic/elliptic type, depending on the specific values of the wave number, mean height and amplitude of the underlying modulated wave. The periodic wave train is stable in the hyperbolic case and unstable …
Microwave Thawing Of Cylinders., Tim Marchant
Microwave Thawing Of Cylinders., Tim Marchant
Tim Marchant
Microwave thawing of a cylinder is examined. The electromagnetic field is governed by Maxwell's equations, where the electrical conductivity and the thermal absorptivity are both assumed to depend on temperature. The forced heat equation governs the absorption and diffusion of heat where convective heating occurs at the surface of the cylinder, while the Stefan condition governs the position of the moving phase boundary. A semi-analytical model, which consists of ordinary differential equations, is developed using the Galerkin method. Semi-analytical solutions are found for the temperature, the electric-field amplitude in the cylinder and the position of the moving boundary. Two examples, …