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The purpose of this paper is to both determine the effects of the nonlinearity on the wave dynamics and assess the temporal and spatial accuracy of five finite difference methods for the solution of the inviscid generalized regularized long-wave (GRLW) equation subject to initial Gaussian conditions.

Two implicit second- and fourth-order accurate finite difference methods and three Runge-Kutta procedures are introduced. The methods employ a new dependent variable which contains the wave amplitude and its second-order spatial derivative. Numerical experiments are reported for several temporal and spatial step sizes in order to assess their accuracy and the preservation of the first two invariants of the inviscid GRLW equation as functions of the spatial and temporal orders of accuracy, and thus determine the conditions under which grid-independent results are obtained.

It has been found that the steepening of the wave increase as the nonlinearity exponent is increased and that the accuracy of the fourth-order Runge-Kutta method is comparable to that of a second-order implicit procedure for time steps smaller than 100th, and that only the fourth-order compact method is almost grid-independent if the time step is on the order of 1,000th and more than 5,000 grid points are used, because of the initial steepening of the initial profile, wave breakup and solitary wave propagation.

This is the first study where an accuracy assessment of wave breakup of the inviscid GRLW equation subject to initial Gaussian conditions is reported.

The purpose of this paper is to develop a new finite-volume method of lines for one-dimensional reaction-diffusion equations that provides piece-wise analytical solutions in space and is conservative, compare it with other finite-difference discretizations and assess the effects of the nonlinear diffusion coefficient on wave propagation.

A conservative, finite-volume method of lines based on piecewise integration of the diffusion operator that provides a globally continuous approximate solution and is second-order accurate is presented. Numerical experiments that assess the accuracy of the method and the time required to achieve steady state, and the effects of the nonlinear diffusion coefficients on wave propagation and boundary values are reported.

The finite-volume method of lines presented here involves the nodal values and their first-order time derivatives at three adjacent grid points, is linearly stable for a first-order accurate Euler’s backward discretization of the time derivative and has a smaller amplification factor than a second-order accurate three-point centered discretization of the second-order spatial derivative. For a system of two nonlinearly-coupled, one-dimensional reaction-diffusion equations, the amplitude, speed and separation of wave fronts are found to be strong functions of the dependence of the nonlinear diffusion coefficients on the concentration and temperature.

A new finite-volume method of lines for one-dimensional reaction-diffusion equations based on piecewise analytical integration of the diffusion operator and the continuity of the dependent variables and their fluxes at the cell boundaries is presented. The method may be used to study heat and mass transfer in layered media.

The purpose of this study is to develop a new method of lines for one-dimensional (1D) advection-reaction-diffusion (ADR) equations that is conservative and provides piecewise analytical solutions in space, compare it with other finite-difference discretizations and assess the effects of advection and reaction on both 1D and two-dimensional (2D) problems.

A conservative method of lines based on the piecewise analytical integration of the two-point boundary value problems that result from the local solution of the advection-diffusion operator subject to the continuity of the dependent variables and their fluxes at the control volume boundaries is presented. The method results in nonlinear first-order, ordinary differential equations in time for the nodal values of the dependent variables at three adjacent grid points and triangular mass and source matrices, reduces to the well-known exponentially fitted techniques for constant coefficients and equally spaced grids and provides continuous solutions in space.

The conservative method of lines presented here results in three-point finite difference equations for the nodal values, implicitly treats the advection and diffusion terms and is unconditionally stable if the reaction terms are implicitly treated. The method is shown to be more accurate than other three-point, exponentially fitted methods for nonlinear problems with interior and/or boundary layers and/or source/reaction terms. The effects of linear advection in 1D reacting flow problems indicates that the wave front steepens as it approaches the downstream boundary, whereas its back corresponds to a translation of the initial conditions; for nonlinear advection, the wave front exhibits steepening but the wave back shows a linear dependence on space. For a system of two nonlinearly coupled, 2D ADR equations, it is shown that a counter-clockwise rotating vortical field stretches the spiral whose tip drifts about the center of the domain, whereas a clock-wise rotating one compresses the wave and thickens its arms.

A new, conservative method of lines that implicitly treats the advection and diffusion terms and provides piecewise-exponential solutions in space is presented and applied to some 1D and 2D advection reactions.

THE study of fatigue from a physiological aspect is a field which motion study technicians have made little or no attempt to explore. Shame on their heads. The physiological simplification of motions aimed at reducing fatigue could have completely offset the notion that motion study is aimed at converting the operator into an automaton. It may well be that an elaborate motion pattern set‐up designed to simplify the work merely succeeds in setting up stresses in the worker. The superimposing of a time‐studied standard for the job may not have taken into account the adaptation of the speed of motions to the physiological limitations of the operator working at a high level performance. Very few practitioners have attempted to study motions in the factory with a view to reducing fatigue and stress as a prerequisite to studying the set‐up for increased production. Still fewer have attempted to evaluate these factors. It is about time they did.

This paper aims to study qualitative properties and approximate solutions of a thermostat dynamics system with three-point boundary value conditions involving a nonsingular kernel operator which is called Atangana-Baleanu-Caputo (ABC) derivative for the first time. The results of the existence and uniqueness of the solution for such a system are investigated with minimum hypotheses by employing Banach and Schauder's fixed point theorems. Furthermore, Ulam-Hyers (UH) stability, Ulam-Hyers-Rassias UHR stability and their generalizations are discussed by using some topics concerning the nonlinear functional analysis. An efficiency of Adomian decomposition method (ADM) is established in order to estimate approximate solutions of our problem and convergence theorem is proved. Finally, four examples are exhibited to illustrate the validity of the theoretical and numerical results.

This paper considered theoretical and numerical methodologies.

This paper contains the following findings: (1) Thermostat fractional dynamics system is studied under ABC operator. (2) Qualitative properties such as existence, uniqueness and Ulam–Hyers–Rassias stability are established by fixed point theorems and nonlinear analysis topics. (3) Approximate solution of the problem is investigated by Adomain decomposition method. (4) Convergence analysis of ADM is proved. (5) Examples are provided to illustrate theoretical and numerical results. (6) Numerical results are compared with exact solution in tables and figures.

The novelty and contributions of this paper is to use a nonsingular kernel operator for the first time in order to study the qualitative properties and approximate solution of a thermostat dynamics system.

The purpose of this paper is to determine both analytically and numerically the existence of smooth, cusped and sharp shock wave solutions to a one-dimensional model of microfluidic droplet ensembles, water flow in unsaturated flows, infiltration, etc., as functions of the powers of the convection and diffusion fluxes and upstream boundary condition; to study numerically the evolution of the wave for two different initial conditions; and to assess the accuracy of several finite difference methods for the solution of the degenerate, nonlinear, advection--diffusion equation that governs the model.

The theory of ordinary differential equations and several explicit, finite difference methods that use first- and second-order, accurate upwind, central and compact discretizations for the convection terms are used to determine the analytical solution for steadily propagating waves and the evolution of the wave fronts from hyperbolic tangent and piecewise linear initial conditions to steadily propagating waves, respectively. The amplitude and phase errors of the semi-discrete schemes are determined analytically and the accuracy of the discrete methods is assessed.

For non-zero upstream boundary conditions, it has been found both analytically and numerically that the shock wave is smooth and its steepness increases as the power of the diffusion term is increased and as the upstream boundary value is decreased. For zero upstream boundary conditions, smooth, cusped and sharp shock waves may be encountered depending on the powers of the convection and diffusion terms. For a linear diffusion flux, the shock wave is smooth, whereas, for a quadratic diffusion flux, the wave exhibits a cusped front whose left spatial derivative decreases as the power of the convection term is increased. For higher nonlinear diffusion fluxes, a sharp shock wave is observed. The wave speed decreases as the powers of both the convection and the diffusion terms are increased. The evolution of the solution from hyperbolic tangent and piecewise linear initial conditions shows that the wave back adapts rapidly to its final steady value, whereas the wave front takes much longer, especially for piecewise linear initial conditions, but the steady wave profile and speed are independent of the initial conditions. It is also shown that discretization of the nonlinear diffusion flux plays a more important role in the accuracy of first- and second-order upwind discretizations of the convection term than either a conservative or a non-conservative discretization of the latter. Second-order upwind and compact discretizations of the convection terms are shown to exhibit oscillations at the foot of the wave’s front where the solution is nil but its left spatial derivative is largest. The results obtained with a conservative, centered second--order accurate finite difference method are found to be in good agreement with those of the second-order accurate, central-upwind Kurganov--Tadmor method which is a non-oscillatory high-resolution shock-capturing procedure, but differ greatly from those obtained with a non-conservative, centered, second-order accurate scheme, where the gradients are largest.

A new, one-dimensional model for microfluidic droplet transport, water flow in unsaturated flows, infiltration, etc., that includes high-order convection fluxes and degenerate diffusion, is proposed and studied both analytically and numerically. Its smooth, cusped and sharp shock wave solutions have been determined analytically as functions of the powers of the nonlinear convection and diffusion fluxes and the boundary conditions. These solutions are used to assess the accuracy of several finite difference methods that use different orders of accuracy in space, and different discretizations of the convection and diffusion fluxes, and can be used to assess the accuracy of other numerical procedures for one-dimensional, degenerate, convection--diffusion equations.

The main advantage of the proposed method is that the computations can be performed on a Cartesian grid with complex immersed boundaries (IBs). The purpose of this paper is to device a numerical scheme based on an embedding finite element method for the solution of two-dimensional (2D) Navier-Stokes equations.

Geometries featuring the stationary solid obstacles in the flow are embedded in the Cartesian grid with special discretizations near the embedded boundary to ensure the accuracy of the solution in the cut cells. To comprehend the complexities of the viscous flows with IBs, the paper adopts a compact interpolation scheme near the IBs that allows to satisfy the second-order accuracy and the conservation property of the solver. The interpolation scheme is designed by virtue of the shape function in the finite element scheme.

Three numerical examples are selected to demonstrate the accuracy and flexibility of the proposed methodology. Simulation of flow past a circular cylinder for a range of Re=20-200 shows excellent agreements with other results using different numerical schemes. Flows around a pair of tandem cylinders and several bodies are particularly investigated. The paper simulates the time-based variation of the flow phenomena for uniform flow past a pair of cylinders with various streamwise gaps between two cylinders. The results in terms of drag coefficient and Strouhal number show excellent agreements with the results available in the literature.

Details of the flow characteristics, such as velocity distribution, pressure and vorticity fields are presented. It is concluded the combined embedding boundary method and FE discretizations are robust and accurate for solving 2D fluid flows with complex IBs.

Presents an implementation of the algebraic multigrid method. It can work in two ways: as pure multigrid method and as a pre‐conditioner for the conjugate gradient method. Shows applications of the iterative solvers for problems in linear and non‐linear elasticity. Shows the range of possible applications with different examples with regular and non‐regular meshes and three‐dimensional problems.

SCIENTISTS and sociologists have for some time been gravely disquieted about the impact which modern technology is making upon society; a disquiet which has recently been percolating through wider sections of all communities. Man has always recognized, since the first machine usurped the place of the human hand as the tool of production, that progress does good but brings harm in its wake, although it is not as quickly appreciated.

Briefly reviews previous literature by the author before presenting an original 12 step system integration protocol designed to ensure the success of companies or countries in their efforts to develop and market new products. Looks at the issues from different strategic levels such as corporate, international, military and economic. Presents 31 case studies, including the success of Japan in microchips to the failure of Xerox to sell its invention of the Alto personal computer 3 years before Apple: from the success in DNA and Superconductor research to the success of Sunbeam in inventing and marketing food processors: and from the daring invention and production of atomic energy for survival to the successes of sewing machine inventor Howe in co‐operating on patents to compete in markets. Includes 306 questions and answers in order to qualify concepts introduced.