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The higher order corrections to the equations that describe nonlinear wave motion in shallow water are derived from the water wave equations. In particular, the extended cylindrical Korteweg-de Vries and Kadomtsev-Petviashvili equations –which include higher order nonlinear, dispersive and nonlocal terms– are derived from the Euler system in (2+1) dimensions, using asymptotic expansions. It is thus found that the nonlocal terms are inherent only to the higher dimensional problem, both in cylindrical and Cartesian geometry. Asymptotic theory is used to study the resonant radiation generated by solitary waves governed by the extended equations, with an excellent comparison obtained between the theoretical predictions, for the resonant radiation amplitude, and the numerical solutions. In addition, resonant dispersive shock waves (undular bores) governed by the extended equations are studied. It is shown that the asymptotic theory, applicable for solitary waves, also provides an accurate estimate of the resonant radiation amplitude, of the undular bore.

International audience; Epoxidation of oleic acid by peracetic acid (PAA) was studied in a recycled reactor system under conventional heating and microwave irradiation. This reaction system consists of several steps. Thus, a kinetic modelling strategy to diminish the number of parameters to be estimated was developed by investigating each reaction step: PAA synthesis and decomposition, epoxidation and ring-opening. The energy balance for microwave heating was correlated with the concentration of the microwave-absorbent species and the total input power of the microwaves. The epoxidation reaction was conducted in the vigorously stirred tank reactor under isothermal conditions within a temperature range of 50–70 °C. The organic phase content was 32–45% v/v. The interfacial mass transfer was supposed to be faster than the intrinsic reaction kinetics suppressing the use of mass transfer correlations. Nonlinear regression was used to estimate the kinetic parameters. Two models were developed for microwave and conventional heating respectively. The perhydrolysis showed to be the slowest reaction, followed by the epoxidation and the ring-opening. The use of microwave irradiation resulted in considerable process intensification for the epoxidation process. By employing microwave heating, the perhydrolysis step in the aqueous phase was enhanced, and the reaction time was reduced by 50% in best cases, which implies that the reactor size can be diminished by microwave technology.

Undular bores, also termed dispersive shock waves, generated by an initial discontinuity in height as governed by two forms of the Boussinesq system of weakly nonlinear shallow water wave theory, the standard formulation and a Hamiltonian formulation, two related Whitham-Boussinesq equations and the full water wave equations for gravity surface waves are studied and compared. It is found that the Whitham-Boussinesq systems give solutions in excellent agreement with numerical solutions of the full water wave equationsfor the positions of the leading and trailing edges of the bore up until the onset on modulational instability. TheWhitham-Boussinesq systems, which are far simpler than the full water wave equations, can then be used to accurately model surface water wave undular bores. Finally, comparisons with numerical solutions of the full water wave equations show that the Whitham-Boussinesq systems give a slightly lower threshold for the onset of modulational instability in terms of the height of the initial step generating the undular bore.

Optical solitary waves and their stability in focusing thermal optical media, such as lead glasses, are studied numerically and theoretically in (2 + 1) dimensions. The optical medium is a square cell and mixed boundary conditions of Newton cooling and fixed temperature on different sides of the cell are used. Nonlinear thermal optical media have a refractive index which depends on temperature, so that heating from the optical beam and heat flow across the boundaries can change the refractive index of the medium. Solitary wave solutions are found numerically using the Newton conjugate gradient method, while their stability isstudied using a linearised stability analysis and also via numerical simulations.It is found that the position of the solitary wave is dependent on the boundary conditions, with the centre of the beam moving toward the warmer boundaries, as the parameters are varied. The stability of the solitary waves depends on the symmetry of the boundary conditions and the amplitude of the solitary waves.

© 2016 Elsevier Ltd Most constitutive models can only simulate cumulative deformation after a limited number of cycles. However, railroad ballast usually experiences a large number of train passages that cause history-dependent long-term deformation. Fractional calculus is an efficient tool for modelling this phenomenon and therefore is incorporated into a constitutive model for predicting the cumulative deformation. The proposed model is further validated by comparing the model predictions with a series of corresponding experimental results. It is observed that the proposed model can realistically simulate the cumulative deformation of ballast from the onset of loading up to a large number of load cycles.

The Belousov–Zhabotinskii reaction is considered in one and two-dimensional reaction–diffusion cells. Feedback control is examined where the feedback mechanism involves varying the concentrations in the boundary reservoir, in response to the concentrations in the centre of the cell. Semi-analytical solutions are developed, via the Galerkin method, which assumes a spatial structure for the solution, and is used to approximate the governing delay partial differential equations by a system of delay ordinary differential equations. The form of feedback control considered, whilst physically realistic, is non-smooth as it has discontinuous derivatives. A stability analysis of the sets of smooth delay ordinary differential equations, which make up the full non-smooth system, allows a band of Hopf bifurcation parameter space to be obtained. It is found that Hopf bifurcations for the full non-smooth system fall within this band of parameter space. In the case of feedback with no delay a precise semi-analytical estimate for the stability of the full non-smooth system can be obtained, which corresponds well with numerical estimates. Examples of limit cycles and the transient evolution of solutions are also considered in detail.

Dispersive shock waves (DSWs), also termed undular bores in fluid mechanics, governed by the nonlocal Whitham equation are studied in order to investigate short wavelength effects that lead to peaked and cusped waves within the DSW. This is done by combining the weak nonlinearity of the Korteweg-de Vries equation with full linear dispersion relations. The dispersion relations considered are those for surface gravity waves, the intermediate long wave equation and a model dispersion relation introduced by Whitham to investigate the 120o peaked Stokes wave of highest amplitude. A dispersive shock fitting method is used to find the leading (solitary wave) and trailing (linear wave) edges of the DSW. This method is found to produce results in excellent agreement with numerical solutions up until the lead solitary wave of the DSW reaches its highest amplitude. Numerical solutions show that the DSWs for the water wave and Whitham peaking kernels become modulationally unstable and evolve into multi-phase wavetrains after a critical amplitude which is just below the DSW of maximum amplitude.

The flow of a fluid over isolated topography in the long wavelength, weakly nonlinear limit is considered. The upstream flow velocity is assumed to be close to a linear long wave velocity of the unforced flow so that the flow is near resonant. Higher order nonlinear, dispersive and nonlinear-dispersive terms beyond the Korteweg-de Vries approximation are included so that the flow is governed by a forced extended Korteweg-de Vries equation. Modulation theory solutions for the undular bores generated upstream and downstream of the forcing are found and used to study the influence of the higher-order terms on the resonant flow, which increases for steeper waves. These modulation theory solutions are compared with numerical solutions of the forced extended Korteweg-de Vries equation for the case of surface water waves. Good comparison is obtained between theoretical and numerical solutions, for properties such as the upstream and downstream solitary wave amplitudes and the widths of the bores.

The propagation of an optical dispersive shock wave, generated from a jump discontinuity in lightintensity, in a defocusing colloidal medium is analysed. The equations governing nonlinear lightpropagation in a colloidal medium consist of a nonlinear Schrödinger equation for the beam and analgebraic equation for the medium response. In the limit of low light intensity, these equations reduce to aperturbed higher order nonlinear Schrödinger equation. Solutions for the leading and trailing edges of thecolloidal dispersive shock wave are found using modulation theory. This is done for both the perturbednonlinear Schrödinger equation and the full colloid equations for arbitrary light intensity. These resultsare compared with numerical solutions of the colloid equations.

Semi-analytical solutions for autocatalytic reactions with mixed quadratic and cubic terms are considered. The kinetic model is combined with diffusion and considered in a one-dimensional reactor. The spatial structure of the reactant and autocatalyst concentrations are approximated by trial functions and averaging is used to obtain a lower-order ordinary differential equation model, as an approximation to the governing partial differential equations. This allows semi-analytical results to be obtained for the reaction–diffusion cell, using theoretical methods developed for ordinary differential equations. Singularity theory is used to investigate the static multiplicity of the system and obtain a parameter map, in which the different types of steady-state bifurcation diagrams occur. Hopf bifurcations are also found by a local stability analysis of the semi-analytical model. The transitions in the number and types of bifurcation diagrams and the changes to the parameter regions, in which Hopf bifurcations occur, as the relative importance of the cubic and quadratic terms vary, is explored in great detail. A key outcome of the study is that the static and dynamic stability of the mixed system exhibits more complexity than either the cubic or quadratic autocatalytic systems alone. In addition it is found that varying the diffusivity ratio, of the reactant and autocatalyst, causes dramatic changes to the dynamic stability. The semi-analytical results are show to be highly accurate, in comparison to numerical solutions of the governing partial differential equations.

Semi-analytical solutions for the reversible Selkov, or glycolytic oscillations model, are considered. The model is coupled with feedback at the boundary and considered in one-dimensional reaction–diffusion cell. This experimentally feasible scenario is analogous to feedback scenarios in a continuously stirred tank reactor. The Galerkin method is applied, which approximates the spatial structure of both the reactant and autocatalyst concentrations. This approach is used to obtain a lower-order, ordinary differential equation model for the system of governing equations. Steady-state solutions, bifurcation diagrams and the region of parameter space, in which Hopf bifurcations occur, are all found. The effect of feedback strength and delay response on the parameter region in which oscillatory solutions occur, is examined. It is found that varying the strength of the feedback response can stabilize or destabilize the system while increasing the delay response usually destabilizes the reaction–diffusion cell. The semi-analytical model is shown to be highly accurate, in comparison with numerical solutions of the governing equations.

We study soliton solutions of a two-dimensional nonlocal NLS equation of Hartree-type with a Bessel potential kernel. The equation models laser propagation in nematic liquid crystals. Motivated by the experimental observation of spatially localized beams, see Conti et al. (2003), we show existence, stability, regularity, and radial symmetry of energy minimizing soliton solutions in R 2 . We also give theoretical lower bounds for the L 2 -norm (power) of these solitons, and show that small L 2 -norm initial conditions lead to decaying solutions. We also present numerical computations of radial soliton solutions. These solutions exhibit the properties expected by the infinite plane theory, although we also see some finite (computational) domain effects, especially solutions with arbitrarily small power.

Semi-analytical solutions are developed for the diffusive Nicholson's blowflies equation. Both one and two-dimensional geometries are considered. The Galerkin method, which assumes a spatial structure for the solution, is used to approximate the governing delay partial differential equation by a system of ordinary differential delay equations. Both steady-state and transient solutions are presented. Semi-analytical results for the stability of the system are derived and the critical parameter value, at which a Hopf bifurcation occurs, is found. Semi-analytical bifurcation diagrams and phase-plane maps are drawn, which show the initial Hopf bifurcation together with a classical period doubling route to chaos. A comparison of the semi-analytical and numerical solutions shows the accuracy and usefulness of the semi-analytical solutions. Also, an asymptotic analysis for the periodic solution near the Hopf bifurcation point is developed, for the one-dimensional geometry. © 2012 The authors 2012. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

In this article, the method of separation of variables (SOV), as illustrated by Subramanian and White (J Power Sources 96:385, 2001), is applied to determine the concentration variations at any point within a three region simplified lithium-ion cell sandwich, undergoing constant current discharge. The primary objective is to obtain an analytical solution that accounts for transient diffusion inside the cell sandwich. The present work involves the application of the SOV method to each region (cathode, separator, and anode) of the lithium-ion cell. This approach can be used as the basis for developing analytical solutions for battery models of greater complexity. This is illustrated here for a case in which non-linear diffusion is considered, but will be extended to full-order nonlinear pseudo-2D models in later work. The analytical expressions are derived in terms of the relevant system parameters. The system considered for this study has LiCoO2–LiC6 battery chemistry.

Soliton perturbation theory is used to determine the evolution of a solitary wave described by a perturbed nonlinear Schrodinger 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 solitary wave tails occur, the location of zeros and the location and amplitude of peaks, in the solitary wave tail. Two examples, the near-continuum limit of a discrete NLS equation and an explicit numerical scheme for the NLS equation, are considered in detail. For the discrete NLS equation it is found that three qualitatively different types of solitary wave tail can occur, while for the explicit finite-difference scheme, only one type of solitary wave tail occurs. An excellent comparison between the perturbation solution and numerical simulations, for the solitary wave tail, is found for both examples.

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.

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 e , the dispersion μ 2 is also included in the perturbed expansion. The first-order solution η 0 is found by using the linear long-wave equations. Then η 0 is used in two second-order governing equations related to the dispersion and nonlinearity, respectively. Since no any omission and approximation for the seabed slope ∇ h and its derivatives is made, there are the third- and fourth-order partial derivatives of η 0 appeared in the right-hand sides of two governing equations of the second-order. By employing a transformation, those third- and fourth-order partial derivatives are removed therefore large errors in approximating these derivatives are eliminated. To validate the new model, wave diffractions around a large vertical cylinder for 13 cases are first considered. It is found that the nonlinear contributions to the new model are significant for weakly nonlinear waves with a much better comparison with experimental results obtained than for the linear diffraction theory. It is also found that the dispersive effects play an important role in improving the accuracy of the new model as numerical results obtained from the Boussinesq equations (with dispersion terms) are more accurate than those from the Airy's equations (without dispersion term). Then the combined wave diffraction and refraction by a conical island is also modelled and discussed. Our model is not only accurate as the dispersive effects have been included but also computationally efficient since the domain integrals are merely evaluated by distributing collocation points over that surface.

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 determine 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 wave tail are derived. A subtle and complex picture of the development of solitary wave tails emerges. It is found that solitary wave tails develop for two reasons, one is decay of the solitary wave caused by resonance, the second is corrections at first-order to the background wave. Strong agreement is found between the theoretical predictions of the perturbation theory and numerical solutions of the governing equations.

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 (IBV) problem for the mKdV equation is also examined. The solutions of the initial value problem are used to construct approximate analytical solutions of the IBV problem. An alternative analytical solution for the IBV problem, based on the assumption of an uniform train of solitons, is also developed. The parameter regimes, in which the different types of solution occur, for both the initial value and IBV problem are identified and excellent comparisons are obtained between the numerical and approximate solutions, for both problems.

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 variation in the parameter regions, in which multiple steady-state and oscillatory solutions occur. Moreover, as the reactor is not symmetric, reversal of the direction of the electric field can cause bifurcation in the reactor between high and low conversion states. Comparisons with numerical solutions of governing partial differential equations confirms the accuracy and usefulness of the semi-analytical model.

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 diagrams including one in which the temperature–response curve is the standard S-shaped curve familiar from combustion problems. Thus biological self-heating can cause elevated temperature raises due to jumps in the steady temperature. This problem is used to test a recently developed semi-analytical technique. For the 2-d problem a four-term expansion is found to give highly accurate results—the error of the semi-analytical solution is much smaller than the error due to uncertainty in parameter values. We conclude that the semi-analytical technique is a very promising method for the investigation of bifurcations in spatially distributed systems.

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 by the asymptotic theory the numerical results confirm that the collision is elastic and the theoretical predictions for the higher-order phase shifts. For the example not covered by the asymptotic theory the numerical results show that the collision is inelastic; an oscillatory wavetrain is produced by the colliding solitary waves. The higher-order phase shift for the faster (sech-type) solitary wave is found. For the slower (algebraic) wave, however, the shed radiation never completely separates from the solitary wave. Interaction with the shed radiation causes the phase shift to evolve long after collision is over, with no final higher-order phase shift able to be determined. An asymptotic mass-conservation law is to test the accuracy of the finite-difference scheme for the numerical solutions. It is shown that, in general, mass is not conserved by the higher-order mKdV equation, but varies during the interaction of the solitary waves.

Solitary wave interaction for a higher-order version of the nonlinear Schrodinger (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. Two examples are considered, one satisfies the algebraic relationship derived from asymptotic theory, and the other does not. For the example which satisfies the algebraic relationship, the numerical results confirm that the collision is elastic. The numerical and theoretical predictions for the higher-order phase and coordinate shifts are also in strong agreement. For the example which does not satisfy the algebraic relationship, the numerical results show that the collision is inelastic and radiation is shed by the solitary wave collision. As the bed of radiation shed by the waves decays very slowly (like t-1/2), it is computationally infeasible to calculate the final phase and coordinate shifts for the inelastic example. An asymptotic conservation law is derived and used to test the finite-difference scheme for the numerical solutions. © 2007 Oxford University Press.

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 its tail. An analytical expression for the solitary wave tail is derived and it is found that the tail vanishes when the algebraic relationship from the asymptotic theory is satisfied. Hence a two-parameter family of higher-order asymptotic embedded solitons exists. A comparison between the theoretical predictions and numerical solutions shows strong agreement for both solitary wave interaction, where the higher-order coordinate and phase shifts are compared, and solitary wave evolution, with comparisons made of the solitary wave tail.

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. Key properties, such as the amplitude and speed of the lead soliton and the width of the bore, are found. An excellent comparison is obtained between the analytical and numerical solutions. Also, it is illustrated how an asymptotic transformation and numerical solutions can be combined to generate hybrid asymptotic-numerical solutions, thus avoiding the severe instabilities associated with numerical schemes for the higher-order KdV equation.

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 in the mixed case. Solutions of the modulation equations are found for an initial-boundary value problem on the semi-infinite line, these solutions representing the influx of magma from a large reservoir. The modulation solutions are found to consist of a full or partial undular bore. Excellent agreement with numerical solutions of the governing pde is obtained, except in the limit where the wave train becomes a train of magmons. An alternative approximation based on the assumption that the wave train is a series of uniform magmons is also derived and is found to be superior to modulation theory in this limit.

Solitary wave interaction for a higher-order version of the Korteweg–de Vries (KdV) equation is considered. The equation is obtained by retaining third-order terms in the perturbation expansion, where for the KdV equation only first-order terms are retained. The third-order KdV equation can be asymptotically transformed to the KdV equation, if the third-order coefficients satisfy a certain algebraic relationship. The third-order two-soliton solution is derived using the transformation. The third-order phase shift corrections are found and it is shown that the collision is asymptotically elastic. The interaction of two third-order solitary waves is also considered numerically. Examples of an elastic and an inelastic collision are both considered. For the elastic collision (which satisfies the algebraic relationship) the numerical results confirm the theoretical predictions, in particular there is good agreement found when comparing the third-order phase shift corrections. For the inelastic collision (which does not satisfy the algebraic relationship) an oscillatory wavetrain is produced by the interacting solitary waves. Also, the third-order phase shift corrections are found numerically for a range of solitary wave amplitudes. An asymptotic mass-conservation law is used to test the finite-difference scheme for the numerical solutions. In general, mass is not conserved by the third-order KdV equation, but varies during the interaction of the solitary waves.

The problem of heat and mass transfer within a porous catalytic pellet in which an irreversible first–order exothermic reaction occurs is a much–studied problem in chemical–reactor engineering. The system is described by two coupled reaction–diffusion equations for the temperature and the degree of reactant conversion. The Galerkin method is used to obtain a semi–analytical model for the pellet problem with both one– and two–dimensional slab geometries. This involves approximating the spatial structure of the temperature and reactant–conversion profiles in the pellet using trial functions. The semi–analytical model is obtained by averaging the governing partial differential equations. As the Arrhenius law cannot be integrated explicitly, the semi–analytical model is given by a system of integrodifferential equations. The semi–analytical model allows both steady–state temperature and conversion profiles and steady–state diagrams to be obtained as the solution to sets of transcendental equations (the integrals are evaluated using quadrature rules). Both the static and dynamic multiplicity of the semi–analytical model is investigated using singularity theory and a local stability analysis. An example of a stable limit cycle is also considered in detail. Comparison with numerical solutions of the governing reaction–diffusion equations and with other results in the literature shows that the semi–analytical solutions are extremely accurate.

Cubic-autocatalysis with Michaelis–Menten decay is considered in a one-dimensional reaction–diffusion cell. The boundaries of the reactor allow diffusion into the cell from external reservoirs, which contain fixed concentrations of the reactant and catalyst. The Galerkin method is used to obtain a semi-analytical model consisting of ordinary differential equations. This involves using trial functions to approximate the spatial structure of the reactant and autocatalyst concentrations in the reactor. The semi-analytical model is then obtained from the governing partial differential equations by averaging. The semi-analytical model allows steady-state concentration profiles and bifurcation diagrams to be obtained as the solution to sets of transcendental equations. Singularity theory is then used to determine the regions of parameter space in which the four main types of bifurcation diagram occur. The region of parameter space, in which Hopf bifurcations can occur, is found by a local stability analysis of the semi-analytical model. An example of a stable limit-cycle is also considered. Comparison with numerical solutions of the governing partial differential equations shows that the semi-analytical solutions are very accurate.

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, consisting of the no heat-loss (insulated) and large heat-loss (fixed temperature) limits, are considered, and a good comparison is obtained with the numerical solution of the governing equations. The semi-analytical model is coupled with a feedback control process in order to minimise thawing times. A strategy is developed which greatly shortens the thawing time whilst avoiding thermal runaway, hence improving the efficiency of the thawing process.

Semi-analytical solutions for the diffusive Lotka-Volterra predator-prey system with delay are considered in one and two-dimensional domains. The Galerkin method is applied, which approximates the spatial structure of both the predator and prey populations. This approach is used to obtain a lower-order, ordinary differential delay equation model for the system of governing delay partial differential equations. Steady-state and transient solutions and the region of parameter space, in which Hopf bifurcations occur, are all found. In some cases simple linear expressions are found as approximations, to describe steady-state solutions and the Hopf parameter regions. An asymptotic analysis for the periodic solution near the Hopf bifurcation point is performed for the one-dimensional domain. An excellent agreement is shown in comparisons between semi-analytical and numerical solutions of the governing equations.

Solitary wave interaction is examined using an extended Benjamin-Bona-Mahony (eBBM) equation. This equation includes higher-order nonlinear and dispersive effects and is is asymptotically equivalent to the extended Korteweg-de Vries (eKdV) equation. The eBBM formulation is preferable to the eKdV equation for the numerical modelling of solitary wave collisions, due to the stability of its finite-difference scheme. In particular, it allows the interaction of steep waves to be modelled, which due to numerical instability, is not possible using the eKdV equation.Numerical simulations of a number of solitary wave collisions of varying nonlinearity are performed for two special cases corresponding to surface water waves. The mass and energy of the dispersive wavetrain generated by the inelastic collision is tabulated and used to show that the change in solitary wave amplitude after interaction is of O(α4), verifying previously obtained theoretical predictions.

The interaction of solitary waves on shallow water is examined to fourth order. At first order the interaction is governed by the Korteweg-de Vries (KdV) equation, and it is shown that the unidirectional assumption, of right-moving waves only, is incompatible with mass conservation at third order. To resolve this, a mass conserving system of KdV equations, involving both right- and left-moving waves, is derived to third order. A fourth-order interaction term, in which the right- and left-moving waves are coupled, is also derived as this term is crucial in determining the fourth-order change in solitary wave amplitude. The form of the unidirectional KdV equation is also discussed with nonlocal terms derived at fourth order. The solitary wave interaction is examined using the inverse scattering method for perturbed KdV equations. Central to the analysis at fourth order is the left-moving wave, for which the solution, in integral form, is derived. A symmetry property for the left-moving wave is found, which is used to show that no change in solitary wave amplitude occurs to fourth order. Hence it is concluded that, for surface waves on shallow water, the change in solitary wave amplitude is of fifth order.

The microwave heating of one- and two-dimensional slabs, subject to linear feedback control, is examined. A semianalytical model of the microwave heating is developed using the Galerkin method. A local stability analysis of the model indicates that Hopf bifurcations occur; the regions of parameter space in which limit-cycles exist are identified. An efficient numerical scheme for the solution of the governing equations, which consist of the forced heat equation and a Helmholtz equation describing the electric-field amplitude, is also developed. An excellent comparison between numerical solutions of the semianalytical model and the governing equations is obtained for the temporal evolution of the temperature in a number of different heating scenarios. It is also found that the region of parameter space, in which limit-cycles can occur, and their amplitude and period are all accurately all predicted by the semianalytical model.

The GrayScott model of cubicautocatalysis with linear decay is coupled with diffusion and considered in a onedimensional reactor (a reactiondiffusion cell). The boundaries of the reactor are permea...

The initial boundary–value problem for the Korteweg–de Vries (KdV) equation on the negative quarter–plane, x 0, is considered. The formulation of this problem is different to the usual initial boundary–value problem on the positive quarter–plane, for which x > 0 and t > 0. Two boundary conditions are required at x = 0 for the negative quarter–plane problem, in contrast to the one boundary condition needed at x = 0 for the positive quarter–plane problem. Solutions of the KdV equation on the infinite line, such as the soliton, cnoidal wave, mean height variation and undular bore solution, are used to find approximate solutions to the negative quarter–plane problem. Five qualitatively different types of solution are found, depending on the relation between the initial and boundary values. Excellent comparisons are obtained between these solutions and full numerical solutions of the KdV equation.

The microwave heating of three-dimensional blocks, by the transverse magnetic waveguide mode TM 11 , is considered in a long rectangular waveguide. The governing equations are the forced heat equation and a steady-state version of Maxwell's equations, while the boundary conditions take into account both convective and radiative heat loss. Semi-analytical solutions, valid for small thermal absorptivity, are found using the Galerkin method. The electrical conductivity and the thermal absorptivity are assumed to be temperature dependent, while both the electrical permittivity and magnetic permeability are taken to be constant. Both a quadratic relation and an Arrhenius-type law are used for the temperature dependency. As the Arrhenius-type law is not amenable analytically, it is approximated by a rational-cubic function. A multivalued steady-state temperature versus power relationship is found to be possible for both types of temperature dependency. At the critical power level thermal runaway occurs when the temperature jumps from the lower (cool) temperature branch to the upper (hot) temperature branch of the solution. The semi-analytical solutions are compared with numerical solutions of the governing equations for various special cases such as the limits of small and large heat loss at the edges of the block. An excellent comparison is obtained between the semi-analytical and numerical solutions, on both temperature branches for the Arrhenius-type law. For the quadratic temperature dependency the comparison is excellent on the low branch but the semi-analytical theory significantly underpredicts the temperature on the upper solution branch.

A prototype chemical reaction is examined in both one and two-dimensional continuous-flow microwave reactors, which are unstirred so the effects of diffusion are important. The reaction rate obeys the Arrhenius law and the thermal absorptivity of the reactor contents is assumed to be both temperature- and concentration-dependent. The governing equations consist of coupled reaction-diffusion equations for the temperature and reactant concentration, plus a Helmholtz equation describing the electric-field amplitude in the reactor. The Galerkin method is used to develop a semi-analytical microwave reactor model, which consists of ordinary differential equations. A stability analysis is performed on the semi-analytical model. This allows the stability of the system to be determined for particular parameter choices and also allows any regions of parameter space in which Hopf bifurcations (and hence periodic solutions called limit-cycles) occur to be obtained. An excellent comparison is obtained between the semi-analytical and numerical solutions, both for the steady-state solution and for time-varying solutions, such as the limit-cycle.

The steady-state heating of a two-dimensional slab by the TE10 mode in a microwave cavity is considered. The cavity contains an iris with a variable aperture and is closed by a short. Resonance can occur in the cavity, which is dependent on the short position, the aperture width and the temperature of the heated slab.The governing equations for the slab are steady-state versions of the forced heat equation and Maxwell's equations while fixed-temperature boundary conditions are used. An Arrhenius temperature dependency is assumed for both the electrical conductivity and the thermal absorptivity. Semi-analytical solutions, valid for small thermal absorptivity, are found for the steady-state temperature and the electric-field amplitude in the slab using the Galerkin method.With no-iris (a semi-infinite waveguide) the usual S-shaped power versus temperature curve occurs. As the aperture width is varied however, the critical power level at which thermal runaway occurs and the temperature response on the upper branch of the S-shaped curve are both changed. This is due to the interaction between the radiation, the cavity and the heated slab. An example is presented to illustrate these aperture effects. Also, it is shown that an optimal aperture setting and short position exists which minimises the input power needed to obtain a given temperature.

Solitary wave interaction is examined, for the case of surface waves on shallow water, by using an extended BenjaminBonaMahony (eBBM) equation. This equation includes higherorder nonlinear and dispersive effects, and hence is asymptotically equivalent to the extended Kortewegde Vries (eKdV) equation. However, it has certain numerical advantages as it allows the modelling of steeper waves, which, due to numerical instability, is not possible using the eKdV equation. Numerical simulations of a number of collisions of varying nonlinearity are performed. The numerical results show evidence of inelastic behaviour at high order. For waves of small amplitude the evidence of inelastic behaviour is indirect; after collision a dispersive wavetrain of extremely small amplitude is found behind the smaller solitary wave. For steeper waves, however, direct evidence of inelastic behaviour is found; the larger wave is increased and the smaller wave is decreased in amplitude after the collision. Conservation laws for the ...

The near-resonant flow of a fluid over a localized topography is examined. The flow is considered in the weakly nonlinear long-wave limit and is governed by the forced Korteweg–de Vries (fKdV) equation at first order. It is shown that the unidirectional assumption, of right-moving waves only, is incompatible with mass conservation at second order. To resolve this incompatibility, a forced coupled KdV system, which allows left-moving waves, is derived to third order (two orders beyond the fKdV approximation). The second-order fKdV equation is reformulated as an asymptotically equivalent forced Benjamin–Bona–Mahony (fBBM) equation, as its numerical scheme has superior stability. First- and second-order predictions for the resonant flow of surface water waves are compared and the mass flux between the right- and left-moving waves is found. An analytical estimate for the mass flux between the right- and left-moving waves is also derived and good agreement with the numerical solution is obtained.

Microwave thawing of a semi-infinite one-dimensional slab is examined. The system is governed by the forced heat equation and Maxwell's equations. Both the electrical conductivity and the thermal absorptivity are assumed to depend on temperature. Convective and radiative heating occurs at the leading edge of the slab, while the Stefan condition governs the position of the moving phase boundary. An approximate analytical model is developed using the Galerkin method. Approximate analytical solutions are found for the temperature and the electric-field amplitude in the slab, which when combined with the Stefan condition allows the position of the moving front to be found. It is shown that the model produces accurate results in the limits of no heat-loss (insulated) and large heat-loss (fixed temperature) at the leading edge of the slab when compared with the full numerical solution for a number of different parameter choices. The approximate model is coupled with a feedback control process in order to examine and minimise slab melting times. A thawing strategy is developed which greatly shortens the thawing time whilst avoiding thermal runaway, hence improving the efficiency of the thawing process.

The microwave heating of two-dimensional slabs in a long rectangular waveguide propagating the TE 10 mode is examined. The temperature dependency of the electrical conductivity and the thermal absorptivity is assumed to be governed by the Arrhenius law, while both the electrical permittivity and the magnetic permeability are assumed constant. The governing equations are the forced heat equation and the steady-state version of Maxwell's equations while the boundary conditions take into account both convective and radiative heat loss. Approximate analytical solutions, valid for small thermal absorptivity, are found for the temperature and the electric-field amplitude using the Galerkin method. As the Arrhenius law is not amenable analytically, it is approximated by a rational-cubic function. At the steady state the temperature versus power relationship is found to be multivalued; at the critical power level thermal runaway occurs when the temperature jumps from the lower (cool) temperature branch to the upper (hot) temperature branch of the solution. In the steady-state limit the approximate analytical solutions are compared with the numerical solutions of the governing equations for various special cases. These are the limits of small and large heat loss and an intermediate case involving radiative heat loss. Results are also presented for a case where differential cooling occurs on the different sides on the slab. An alternative heating senario, where one end of the waveguide is blocked by a short, is also considered. The approximate solutions are found for this geometry and compared in the small Biot-number limit to Kriegsmann (1997). Also, a control process is presented, which allows thermal runaway to be avoided and the desired final steady state to be reached. Various special cases of the feedback parameters associated with the control process are examined.

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, or interaction, term occuring after collision. It is shown that the two theories are asymptotically equivalent, with the coupling term representing the higher-order phase shift corrections. Finally, it is concluded, with the support of existing numerical evidence, that the interpretation of the coupling term as a higher-order phase shift is physically appropriate; hence, the interaction of higher-order solitary waves is asymptotically elastic.

Marangoni-Benard convection is the process by which oscillatory waves are generated on an interface due to a change in surface tension. This process, which can be mass or temperature driven, is described by a perturbed Korteweg-de Vries (KdV) equation. For a certain parameter range, this perturbed KdV equation has a solitary wave solution with an unique steady-state amplitude for which the excitation and friction terms in the perturbed KdV equation are in balance. The evolution of an initial sech^2 pulse to the steady-state solitary wave governed by the perturbed KdV equation of Marangoni-Benard convection is examined. Approximate equations, derived from mass conservation, and momentum evolution for the perturbed KdV equation, are used to describe the evolution of the initial pulse into steady-state solitary wave(s) plus dispersive radiation. Initial conditions which result in one or two solitary waves are considered. A phase plane analysis shows that the pulse evolves on two timescales, initially to a solution of the KdV equation, before evolving to the unique steady solitary wave of the perturbed KdV equation. The steady-state solitary wave is shown to be stable. A parameter regime for which the steady-state solitary wave is never reached, with the pulse amplitude increasing without bound, is also examined. The results obtained from the approximate conservation equations are found to be in good agreement with full numerical solutions of the perturbed KdV equation governing Marangoni-Benard convection.

Marangoni-Benard convection is the process by which oscillatory waves are generated on an interface due to a change in surface tension. This process, which can be mass or temperature driven is described by a perturbed Korteweg-de Vries (KdV) equation. The evolution and interaction of solitary waves generated by Marangoni-Benard convection is examined. The solitary wave with steady-state amplitude, which occurs when the excitation and friction terms of the perturbed KdV equation are in balance is found to second-order in the perturbation parameter. This solitary wave has a fixed amplitude, which depends on the coefficients of the perturbation terms in the governing equation. The evolution of a solitary wave of arbitrary amplitude to the steady-state amplitude is also found, to first-order in the perturbation parameter. In addition, by using a perturbation method based on inverse scattering, it is shown that the interaction of two solitary waves is not elastic with the change in wave amplitude determined. Numerical solutions of the perturbed KdV equation are presented and compared to the asymptotic solutions.

Although microwave radiation is best known for heating food in the kitchen, in recent years it has found new applications in many industrial processes, such as those involving melting, smelting, sintering, drying, and joining. Heating by microwave radiation constitutes a highly coupled nonlinear problem giving rise to new and unexpected physical behavior, the best known of which is the appearance of “hot spots.” That is, in many industrial applications of microwave heating it has been observed that heating does not take place uniformly but rather regions of very high temperature tend to form. In order to predict the occurrence of such phenomena it is necessary to develop simplified mathematical models from which insight might be gleaned into an inherently complex physical process. The purpose of this paper is to review some of the recent developments in the mathematical modelling of microwave heating, including models that consider in isolation the heat equation with a nonlinear source term, in which case the electric-field amplitude is assumed constant, models involving the coupling between the electric-field amplitude and temperature including both steady-state solutions and the initial heating, and also models that control the process of thermal runaway. Numerical modelling of the microwave heating process is also briefly reviewed.

Soliton interactions for the extended Korteweg-de Vries (KdV) equation are examined. It is shown that the extended KdV equation can be transformed (to its order of approximation) to a higher-order member of the KdV hierarchy of integrable equations. This transformation is used to derive the higher-order, two-soliton solution for the extended KdV equation. Hence it follows that the higher-order solitary-wave collisions are elastic, to the order of approximation of the extended KdV equation. In addition, the higher-order corrections to the phase shifts are found. To examine the exact nature of higher-order, solitary-wave collisions, numerical results for various special cases (including surface waves on shallow water) of the extended KdV equation are presented. The numerical results show evidence of inelastic behaviour well beyond the order of approximation of the extended KdV equation; after collision, a dispersive wavetrain of extremely small amplitude is found behind the smaller, higher-order solitary wave.