Search Results (28)

Four formulations based on the Kirchhoff transformation and time linearization for the numerical study of one-dimensional reaction–diffusion equations, whose heat capacity, thermal inertia and reaction rate are only functions of the temperature, are presented. The formulations result in linear, two-point boundary-value problems for the temperature, energy or heat potential, and may be solved by either discretizing the second-order spatial derivative or piecewise analytical integration. In both cases, linear systems of algebraic equations are obtained. The formulation for the temperature is extended to two-dimensional, nonlinear reaction–diffusion equations where the resulting linear two-dimensional operator is factorized into a sequence of one-dimensional ones that may be solved by means of any of the four formulations developed for one-dimensional problems. The multidimensional formulation is applied to a two-dimensional, two-equation system of nonlinearly coupled advection–reaction–diffusion equations, and the effects of the velocity and the parameters that characterize the nonlinear heat capacities and thermal conductivity are studied. It is shown that clockwise-rotating velocity fields result in wave stretching for small vortex radii, and wave deceleration and thickening for counter-clockwise-rotating velocity fields. It is also shown that large-core, clockwise-rotating velocity fields may result in large transient periods, followed by time intervals of apparent little activity which, in turn, are followed by the propagation of long-period waves. Full article

The effects of relaxation, convection, and anisotropy on a two-dimensional, two-equation system of nonlinearly coupled, second-order hyperbolic, advection–reaction–diffusion equations are studied numerically by means of a three-time-level linearized finite difference method. The formulation utilizes a frame-indifferent constitutive equation for the heat and mass diffusion fluxes, taking into account the tensorial character of the thermal diffusivity of heat and mass diffusion. This approach results in a large system of linear algebraic equations at each time level. It is shown that the effects of relaxation are small although they may be noticeable initially if the relaxation times are smaller than the characteristic residence, diffusion, and reaction times. It is also shown that the anisotropy associated with one of the dependent variables does not have an important role in the reaction wave dynamics, whereas the anisotropy of the other dependent variable results in transitions from spiral waves to either large or small curvature reaction fronts. Convection is found to play an important role in the reaction front dynamics depending on the vortex circulation and radius and the anisotropy of the two dependent variables. For clockwise-rotating vortices of large diameter, patterns similar to those observed in planar mixing layers have been found for anisotropic diffusion tensors. Full article

This study introduces the higher-order unconditionally positive finite difference (HUPFD) methods to solve the linear, nonlinear, and system of advection–diffusion–reaction (ADR) equations. The stability and consistency of the developed methods are analyzed, which are necessary and sufficient for the numerical approach to converge to the exact solution. The problem under consideration is of the Cauchy type, and hence, Von Neumann stability analysis is used to analyze the stability of the proposed schemes. The HUPFD’s efficacy and efficiency are investigated by calculating the error, convergence rate, and computing time. For validation purposes, the higher-order unconditionally positive finite difference solutions are compared to analytical calculations. The numerical results demonstrate that the proposed methods produce accurate solutions to solve the advection diffusion reaction equations. The results also show that increasing the order of the unconditionally positive finite difference leads an implicit scheme that is conditionally stable and has a higher order of accuracy with respect to time and space. Full article

In this paper, the enhanced higher-order unconditionally positive finite difference method is developed to solve the linear, non-linear and system advection diffusion reaction equations. Investigation into the effectiveness and efficiency of the proposed method is carried out by calculating the convergence rate, error and computational time. A comparison of the solutions obtained by the enhanced higher-order unconditionally positive finite difference and exact solution is conducted for validation purposes. The numerical results show that the developed method reduced the time taken to solve the linear and non-linear advection diffusion reaction equations as compared to the results obtained by the higher-order unconditionally positive finite difference method. Full article

The concept of an ideal free distribution (IFD) is extended to a predator–prey system in a heterogeneous environment. We consider reaction–diffusion–advection equations which describe the evolution of spatial distributions of predators and prey under directed migration. Modification of local interaction terms is introduced, if some coefficients depend on resource. Depending on coefficients of local interaction, the different scenarios of predator distribution are possible. We pick out three cases: proportionality to prey (and respectively to resource), indifferent distribution and inversely proportional to the prey. These scenarios apply in the case of nonzero diffusion and taxis under additional conditions on diffusion and migration rates. We examine migration functions for which there are explicit stationary solutions with nonzero densities of both species. To analyze solutions with violation of the IFD conditions, we apply asymptotic expansions and a numerical approach with staggered grids. The results for a two-dimensional domain with no-flux boundary conditions are presented. Full article

In regions with sanitary landfills, unsuitable liner designs can result in significant soil and groundwater contamination, leading to substantial environmental remediation costs. Addressing this challenge, we propose a semi-analytical model for solute transport that uses the advection–dispersion–reaction equation in a multi-layered liner system. A distinctive feature of our model is its ability to account for infiltration velocity, arbitrary numbers of layers, thin layers such as geomembranes, and mass flow. We validated our model against existing published models and applied it to a case study of a real sanitary landfill in the capital of Brazil. Through parametric analyses, we simulated contaminant transport across various layers, including the geomembrane (GM), geosynthetic clay liner (GCL), soil liner (SL), and compacted clay liner (CCL). The analyses showed the importance of choosing the most appropriate construction system based on the location and availability of materials. Considering toluene contamination, a GM molecular diffusion coefficient (D_GM) greater than 10⁻¹³ m² s⁻¹ exhibited similar efficiency when compared with CCL (60 cm thick). In addition, the results showed that the liner system may have the same efficiency in changing SL (60 cm thick) for a GCL (1 cm thick). Full article

Complex systems of classical physics in certain situations are characterized by intensive fluctuations of the quantities governing their dynamics. These include important phenomena such as (continuous) second-order phase transitions, fully developed turbulence, magnetic hydrodynamics, advective–diffusive processes, the kinetics of chemical reactions, percolation, and processes in financial markets. The theoretical goal of the study of such systems is to determine and predict the temporal and spatial dependence of statistical correlations of fluctuating variables. Modern methods of quantum field theory, originally developed in high-energy physics to describe the properties of elementary particles, allow for quantitative analysis of such correlations. Peculiarities of quantum field theory in solving two paradigmatic statistical problems related to percolation are reviewed, and new results on calculating representative universal parameters such as critical exponents that describe asymptotic behavior are presented. Full article

This work aims at providing a concise review of various agri-food models that employ fractional differential operators. In this context, various mathematical models based on fractional differential equations have been used to describe a wide range of problems in agri-food. As a result of this review, we found out that this new area of research is finding increased acceptance in recent years and that some reports have employed fractional operators successfully in order to model real-world data. Our results also show that the most commonly used differential operators in these problems are the Caputo, the Caputo–Fabrizio, the Atangana–Baleanu, and the Riemann–Liouville derivatives. Most of the authors in this field are predominantly from China and India. Full article

We present a modeling tool capable of computing carbon dioxide (CO₂) fluxes over a non-uniform boreal peatland. The three-dimensional (3D) hydrodynamic model is based on the “one-and-a-half” closure scheme of the system of the Reynolds-Averaged Navier–Stokes and continuity equations. Despite simplifications used in the turbulence description, the model allowed obtaining the spatial steady-state distribution of the averaged wind velocities and coefficients of turbulent exchange within the atmospheric surface layer, taking into account the surface heterogeneity. The spatial pattern of CO₂ fluxes within and above a plant canopy is derived using the “diffusion–reaction–advection” equation. The model was applied to estimate the spatial heterogeneity of CO₂ fluxes over a non-uniform boreal ombrotrophic peatland, Staroselsky Moch, in the Tver region of European Russia. The modeling results showed a significant effect of vegetation heterogeneity on the spatial pattern of vertical and horizontal wind components and on vertical and horizontal CO₂ flux distributions. Maximal airflow disturbances were detected in the near-surface layer at the windward and leeward forest edges. The forest edges were also characterized by maximum rates of horizontal CO₂ fluxes. Modeled turbulent CO₂ fluxes were compared with the mid-day eddy covariance flux measurements in the southern part of the peatland. A very good agreement of modeled and measured fluxes (R² = 0.86, p < 0.05) was found. Comparisons of the vertical profiles of CO₂ fluxes over the entire peatland area and at the flux tower location showed significant differences between these fluxes, depending on the prevailing wind direction and the height above the ground. Full article

This work presents an original 3D code in FreeFem++ to recreate the behavior of anaerobic microorganisms in non-stirred anaerobic reactors with an intermittent feed. The physical and biochemical phenomena have been considered using a mathematical model based on a set of partial differential equations: Stokes, advection–diffusion, and diffusion–reaction. The description of the anaerobic metabolism was carried out by implementing the structured AMD1 model. The Galerkin finite element method has been used to solve the partial differential equations defined in the model. Finally, the methodology and procedures are presented by means of a concrete example. Thanks to the inclusion of this e-learning tool for use in virtual laboratories, it is possible to improve the understanding of engineering students on the functioning of the metabolism that takes place inside non-stirred anaerobic reactors that are fed discontinuously. This proposal reinforces to students, in a transversal way, both environmental sensitivity and awareness of the circular economy focused on the implementation of natural wastewater treatment systems in rural areas. Full article

We construct a stabilized finite element method for linear and nonlinear unsteady advection–diffusion–reaction equations using the method of lines. We propose a residual minimization strategy that uses an ad-hoc modified discrete system that couples a time-marching schema and a semi-discrete discontinuous Galerkin formulation in space. This combination delivers a stable continuous solution and an on-the-fly error estimate that robustly guides adaptivity at every discrete time. We show the performance of advection-dominated problems to demonstrate stability in the solution and efficiency in the adaptivity strategy. We also present the method’s robustness in the nonlinear Bratu equation in two dimensions. Full article

This paper is devoted to apply the Lie methods to a class of reaction diffusion advection systems of two interacting species u and v with two arbitrary constitutive functions f and g. The reaction term appearing in the equation for the species v is a logistic function of Lotka-Volterra type. Once obtained the Lie algebra for any form of f and g a Lie classification is carried out. Interesting reduced systems are derived admitting wide classes of exact solutions. Full article

In this article, a nonlinear autocatalytic chemical reaction glycolysis model with the appearance of advection and diffusion is proposed. The occurrence and unicity of the solutions in Banach spaces are investigated. The solutions to these types of models are obtained by the optimization of the closed and convex subsets of the function space. Explicit estimates of the solutions for the admissible auxiliary data are formulated. An elegant numerical scheme is designed for an autocatalytic chemical reaction model, that is, the glycolysis model. The fundamental traits of the prescribed numerical method, for instance, the positivity, consistency, stability, etc., are also verified. The authenticity of the proposed scheme is ensured by comparing it with two extensively used numerical techniques. A numerical example is presented to observe the graphical behavior of the continuous system by constructing the numerical algorithm. The comparison depicts that the projected numerical design is more productive as compared to the other two schemes, as it holds all the important properties of the continuous model. Full article

Osteosarcoma is the most common malignant bone tumor in children and adolescents with a poor prognosis. To describe the progression of osteosarcoma, we expanded a system of data-driven ODE from a previous study into a system of Reaction-Diffusion-Advection (RDA) equations and coupled it with Biot equations of poroelasticity to form a bio-mechanical model. The RDA system includes the spatio-temporal information of the key components of the tumor microenvironment. The Biot equations are comprised of an equation for the solid phase, which governs the movement of the solid tumor, and an equation for the fluid phase, which relates to the motion of cells. The model predicts the total number of cells and cytokines of the tumor microenvironment and simulates the tumor’s size growth. We simulated different scenarios using this model to investigate the impact of several biomedical settings on tumors’ growth. The results indicate the importance of macrophages in tumors’ growth. Particularly, we have observed a high co-localization of macrophages and cancer cells, and the concentration of tumor cells increases as the number of macrophages increases. Full article

In this paper we propose a new mathematical model for describing the complex interplay between skin cell populations with fibroblast growth factor and bone morphogenetic protein, occurring within deformable porous media describing feather primordia patterning. Tissue growth, in turn, modifies the transport of morphogens (described by reaction-diffusion equations) through diverse mechanisms such as advection from the solid velocity generated by mechanical stress, and mass supply. By performing an asymptotic linear stability analysis on the coupled poromechanical-chemotaxis system (assuming rheological properties of the skin cell aggregates that reside in the regime of infinitesimal strains and where the porous structure is fully saturated with interstitial fluid and encoding the coupling mechanisms through active stress) we obtain the conditions on the parameters—especially those encoding coupling mechanisms—under which the system will give rise to spatially heterogeneous solutions. We also extend the mechanical model to the case of incompressible poro-hyperelasticity and include the mechanisms of anisotropic solid growth and feedback by means of standard Lee decompositions of the tensor gradient of deformation. Because the model in question involves the coupling of several nonlinear PDEs, we cannot straightforwardly obtain closed-form solutions. We therefore design a suitable numerical method that employs backward Euler time discretisation, linearisation of the semidiscrete problem through Newton–Raphson’s method, a seven-field finite element formulation for the spatial discretisation, and we also advocate the construction and efficient implementation of tailored robust solvers. We present a few illustrative computational examples in 2D and 3D, briefly discussing different spatio-temporal patterns of growth factors as well as the associated solid response scenario depending on the specific poromechanical regime. Our findings confirm the theoretically predicted behaviour of spatio-temporal patterns, and the produced results reveal a qualitative agreement with respect to the expected experimental behaviour. We stress that the present study provides insight on several biomechanical properties of primordia patterning. Full article