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The purpose of this paper is to develop a new transversal method of lines for one-dimensional reaction–diffusion equations that is conservative and provides piecewise–analytical solutions in space, analyze its truncation errors and linear stability, compare it with other finite-difference discretizations and assess the effects of the nonlinear diffusion coefficients, reaction rate terms and initial conditions on wave propagation and merging.

A conservative, transversal method of lines based on the discretization of time and piecewise analytical integration of the resulting two-point boundary-value problems subject to the continuity of the dependent variables and their fluxes at the control-volume boundaries, is presented. The method provides three-point finite difference expressions for the nodal values and continuous solutions in space, and its accuracy has been determined first analytically and then assessed in numerical experiments of reaction-diffusion problems, which exhibit interior and/or boundary layers.

The transversal method of lines presented here results in three-point finite difference equations for the nodal values, treats the diffusion terms implicitly and is unconditionally stable if the reaction terms are treated implicitly. The method is very accurate for problems with the interior and/or boundary layers. For a system of two nonlinearly-coupled, one-dimensional reaction–diffusion equations, the formation, propagation and merging of reactive fronts have been found to be strong function of the diffusion coefficients and reaction rates. For asymmetric ignition, it has been found that, after front merging, the temperature and concentration profiles are almost independent of the ignition conditions.

A new, conservative, transversal method of lines that treats the diffusion terms implicitly and provides piecewise exponential solutions in space without the need for interpolation is presented and applied to someone.

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 the current flow configurations is to bring to attention the thermophysical aspects of magnetohydrodynamics (MHD) Williamson nanofluid flow under the effects of Joule heating, nonlinear thermal radiation, variable thermal coefficient and activation energy past a rotating stretchable surface.

A mathematical model is examined to study the heat and mass transport analysis of steady MHD Williamson fluid flow past a rotating stretchable surface. Impact of activation energy with newly introduced variable diffusion coefficient at the mass equation is considered. The transport phenomenon is modeled by using highly nonlinear PDEs which are then reduced into dimensionless form by using similarity transformation. The resulting equations are then solved with the aid of fifth-order Fehlberg method.

The rotating fluid, heat and mass transport effects are analyzed for different values of parameters on velocity, energy and diffusion distributions. Parameters like the rotation parameter, Hartmann number and Weissenberg number control the flow field. In addition, the solar radiation, Joule heating, Prandtl number, thermal conductivity, concentration diffusion coefficient and activation energy control the temperature and concentration profiles inside the stretching surface. It can be analyzed that for higher values of thermal conductivity, Eckret number and solar radiation parameter the temperature profile increases, whereas opposite behavior is noticed for Prandtl number. Moreover, for increasing values of temperature difference parameter and thermal diffusion coefficient, the concentration profile shows reducing behavior.

This paper is useful for researchers working in mathematical and theoretical physics. Moreover, numerical results are very useful in industry and daily-use processes.

This study aims to analyze the Prandtl fluid flow in the presence of better mass diffusion and heat conduction models. By taking into account a linearly bidirectional stretchable sheet, flow is produced. Heat generation effect, thermal radiation, variable thermal conductivity, variable diffusion coefficient and Cattaneo–Christov double diffusion models are used to evaluate thermal and concentration diffusions.

The governing partial differential equations (PDEs) have been made simpler using a boundary layer method. Strong nonlinear ordinary differential equations (ODEs) relate to appropriate non-dimensional similarity variables. The optimal homotopy analysis technique is used to develop solution.

Graphs analyze the impact of many relevant factors on temperature and concentration. The physical parameters, such as mass and heat transfer rates at the wall and surface drag coefficients, are also displayed and explained.

The reported work discusses the contribution of generalized flux models to note their impact on heat and mass transport.

The purpose of this study is to use the method of lines to solve the two-dimensional nonlinear advection–diffusion–reaction equation with variable coefficients.

The strictly positive definite radial basis functions collocation method together with the decomposition of the interpolation matrix is used to turn the problem into a system of nonlinear first-order differential equations. Then a numerical solution of this system is computed by changing in the classical fourth-order Runge–Kutta method as well.

Several test problems are provided to confirm the validity and efficiently of the proposed method.

For the first time, some famous examples are solved by using the proposed high-order technique.

This paper aims to examine state adoption of climate action plans (CAPs) and investigates the factors driving the adoption of these climate policies in the states.

The framework that is formulated to explain the state climate actions involves four dimensions: climate risks, climate politics, climate economic and climate policy diffusions. These hypotheses are tested with event history analysis on a panel data set on 48 US continental states from 1994 to 2008.

This paper found empirical evidence to support climate politics, economics and policy diffusion explanations. It also found that climate risks are not taken into account in states’ climate actions. A comparison is conducted to compare the differences in state and local climate policymaking.

The paper investigates the motivations of state governments in adopting CAPs, and makes comparisons with local climate strategies. It contributes to academic understanding of the multilevel governance of climate protection in the USA.

It has been recently shown that diffusion of dopant during doping of inhomogeneous structure could be accelerated or decelerated in comparison with diffusion of dopant in structure with averaged diffusion coefficient. As a continuation of previous work, the purpose of this paper is to introduce an approach of estimating the limited value of acceleration of the dopant diffusion by choosing the dependence of the dopant diffusion coefficient on the coordinates.

The authors analyzed relaxation of concentration of dopant during diffusion in inhomogeneous material. The authors determine conditions for maximal acceleration and deceleration of diffusion of dopant. The authors introduced analytical approach for analysis of dopant diffusion in inhomogeneous material.

The authors determine conditions for maximal acceleration and deceleration of diffusion of dopant.

It has been shown that dopant diffusion could be decelerated essentially to a greater extent, rather than accelerated.

This paper aims to study flow‐induced corrosion mechanisms for carbon steel in high‐velocity flowing seawater and to explain corrosive phenomena.

An overall mathematical model for flow‐induced corrosion of carbon steel in high‐velocity flow seawater was established in a rotating disk apparatus using both numerical simulation and test methods. By studying the impact of turbulent flow using the kinetic energy of a turbulent approach and the effects of the computational near‐wall hydrodynamic parameters on corrosion rates, corrosion behavior and mechanism are discussed here. It is applicable in order to understand in depth the synergistic effect mechanism of flow‐induced corrosion.

It was found that it is scientific and reasonable to investigate carbon steel corrosion through correlation of the near‐wall hydrodynamic parameters, which can accurately describe the influence of fluid flow on corrosion. The computational corrosion rates obtained by this model are in good agreement with measured corrosion data. It is shown that serious flow‐induced corrosion is caused by the synergistic effect between the corrosion electrochemical factor and the hydrodynamic factor, while the corrosion electrochemical factor plays a dominant role in flow‐induced corrosion.

The corrosion kinetics and mechanism of metals in a high‐velocity flowing medium is discussed here. These results will help those interested in flow‐induced corrosion to understand in depth the type of issue.

The purpose of this paper is to formulate and analyse a convergent numerical scheme and apply it to investigate the coupled problem of fluid flow with heat and mass transfer in a porous channel with variable transport properties.

This paper derives the model by assuming a fully developed Brinkman flow with temperature-dependent viscosity and incorporating viscous dissipation, variable transport properties and nonlinear heat and mass sources. For the numerical formulation, the nonlinear sources are treated in semi-implicit manner, whereas the non-constant transport properties are treated by lagging in time leading to decoupled diagonally dominant systems. The consistency, stability and convergence results are derived. The method of manufactured solutions is adopted to numerically verify the theoretical results. The scheme is then applied to investigate the impact of relevant parameters, such as the viscosity parameter, on the flow.

Based on the numerical findings, the proposed scheme was found to be unconditionally stable and convergent with first- and second-order accuracy in time and space, respectively. Physical results showed that the flow parameters have influence on the flow fields, particularly, the flow is enhanced by increasing porosity and viscosity parameters and the concentration decreases with increasing diffusivity, whereas both the temperature and Nusselt number decrease with increasing thermal conductivity.

Numerically, the proposed numerical scheme can be applied without concerns on time steps size restrictions. Non-physical solutions cannot be computed. Physically, the flow can be increased by increasing the viscosity parameters. Pollutants with higher diffusivity will have their concentration decreased faster than those of lower diffusivity. The fluid temperature would decrease faster if its thermal conductivity is higher.

A fully coupled fluid flow with heat and mass transfer problem having nonlinear properties and nonlinear fractional sources and sink terms, presumably, has not been investigated in a general form as done in this study. The detailed numerical analysis of this particular scheme for the identified general model has also not been considered in the past, to the best of the author’s knowledge.

Based on the normalized variable diagram, the weakness of the Gaskell and Lau's convective boundedness criterion (GL‐CBC) is revealed by numerical example. By careful consideration of the smoothness of the normalized variable variation pattern, more rigorous constraints on the interface value interpolation are found. A new CBC is thus proposed, whose feasibility and correctness are demonstrated by the inspection of ten existing bounded schemes and a numerical example.