Solving Reaction-diffusion Equations Research Articles

Multiscale topology optimization of structures by using isogeometrical level set approach

This study aims to optimize topology of structures at macro and micro scales, simultaneously, by using a level set method in an isogeometric analysis (IGA) framework. To achieve this, equilibrium and homogenization equations in the model are solved by IGA method. The level set functions are defined over a grid in parameter space of associating b-splines of the IGA model. Therefore, control net of the model and level set grid are separated and there is no need to refine the control net for having smooth boundaries. Sensitivity analyses for both scales are performed to calculate the velocity of boundary points and the level set functions are updated by solving reaction-diffusion equations. Finally, several 2D and 3D examples with different geometry and boundary conditions are provided to show performance and efficiency of the method. Obtained results show good agreement with examples in literature in terms of both topology and final value of objective function. Also, by using IGA level set method, smooth boundaries are achieved in the final topology of micro and macro structures.

The auxiliary equation approach for solving reaction-diffusion equations

The paper concerns the auxiliary equation method and proposes an approach for finding the most general nonlinear term that can generalize a nonlinear differential equation, so that it keeps solutions expressed in terms of the same auxiliary equation. More precisely we will consider the second order reaction-diffusion equations and we will find the most general nolinear term of this type of equations, for which the solutions can be expressed in terms of the Riccati equation. The procedure is exemplified on Fitzhugh-Nagumo, Dodd-Bullough-Mikhailov, and Klein-Gordon models, seen as reaction-diffusion equations.

Modeling and experimental study of hollow cylindrical catalysts for prediction of transient behaviors of suspension photoreaction systems by Langmuir-Hinshelwood kinetics

Effectiveness factor was derived for hollow cylindrical pellets with finite length by assuming Langmuir-Hinshelwood (LH) kinetics. Transient intra-particle concentration was predicted by solving reaction-diffusion equations using finite element method for various values of reaction parameters in LH kinetics and thickness of hollow core. At stationary state, numerical values of intra-particle concentration was integrated to calculate effectiveness factor (η) under various shape factors of hollow cylindrical pellet including infinite length as well as finite height. The results were compared with those of infinitely long cylinder and spherical pellets to determine optimum morphologies of catalytic pellets. η could be enhanced by increasing hollow-core size and decreasing aspect ratio of the pellet. For the same value of Thiele modulus, which was calculated from volume-equivalent radius, hollow cylindrical catalyst exhibited largest value of η, compared to solid cylindrical or spherical pellets. For simulation of catalytic reactors such as batch reactor, CSTR, and fixed bed, pseudo-steady state was adopted in material balance equation using computed values of η. The modeling results were compared with experimental data from batch photocatalytic reactor containing hollow catalytic fibers synthesized by electrospinning to determine reaction parameters in LH kinetics by minimizing errors of transient concentration between calculated and measured values using optimization method.

Modeling of Core-shell Catalytic Pellets of Inert Shell with Various Morphologies for Series Reactions

Modeling of core-shell catalytic pellets with inert shell was performed assuming three different particle shapes such as sphere, cylinder, and slab. For isothermal irreversible first order reaction, A→B, analytical solutions of intra-particle concentration of reactant were derived by solving reaction-diffusion equations, considering thickness of catalytically active core (r2), when the pellets were immersed in infinitely large medium. Unlike conventional pellets or core-shell pellets with inert core, effectiveness factor (η) was affected by r2 as well as ratio of effective diffusivity in inert shell and active core (Υ), and η could be enhanced by increasing Υ. η increased in the order of sphere > cylinder > slab due to surface area per unit volume of medium, and η increased with increasing Biot number (Bi), because of decreased external film resistance. For series reaction, A→B→C, transient change of the concentration of starting reactant (CA) and intermittent product (CB) in bulk fluid of reactor could be predicted by solving material balance equations assuming pseudo-steady state approximation and equal value of Thiele modulus for both reactions in batch reactor and CSTR. When B is desirable product, CB increased by increasing Υ and decreasing r2. Similar to conventional pellets, CB in batch reactor decreased by increase of distribution coefficient of B on surface of pellet, KB, due to increased adsorption, and increased in the order of sphere < cylinder < slab, similar to conventional pellets. In CSTR, CB at steady state could be enhanced by decreasing amount of catalyst and Thiele modulus (Φ), implying that performance of reactor can be controlled by adjusting various reaction parameters. From numerical solutions of reaction diffusion equations for series reactions, the effect of r2 and Υ were also confirmed from transient value of CA and CB in exit stream calculated by finite element method.

A Chebyshev multidomain adaptive mesh method for reaction-diffusion equations

Reaction-diffusion equations can present solutions in the form of traveling waves with stiff wave fronts. Such solutions evolve in different spatial and temporal scales, and it is desired to construct numerical methods that can adopt spatial refinement at locations where the solution becomes stiff. In this work, we develop a high-order adaptive mesh method based on Chebyshev spectral methods with a multidomain approach for the traveling wave solutions of reaction-diffusion equations. The proposed method uses the non-conforming and non-overlapping spectral multidomain method with the temporal adaptation of the computational mesh. The spectral multidomain methods have been used for solving PDEs including the reaction-diffusion equations. However, the non-conformal, non-overlapping and adaptive mesh spectral methods have not been used in the reaction-diffusion community. The proposed multidomain spectral method solves the given reaction-diffusion equations in each subdomain locally first and the boundary and interface conditions are enforced in a global manner. In this way, the method can be parallelizable and is efficient for large reaction-diffusion systems. We show that the proposed method is stable and accurate for solving reaction-diffusion equations with stiff traveling wave solutions. We provide both one- and two-dimensional numerical results that show the efficacy of the proposed method. The application of the adaptive multidomain spectral methods to the reaction-diffusion equations, that yield stiff traveling wave solutions, is new and needs further investigation.

Multivariable Linear Algebraic Discretization of Nonlinear Parabolic Equations for Computational Analysis.

Since the nonlinear parabolic equation has many variables, its calculation process is mostly an algebraic operation, which makes it difficult to express the discrete process concisely, which makes it difficult to effectively solve the two grid algorithm problems and the convergence problem of reaction diffusion. The extended mixed finite element method is a common method for solving reaction-diffusion equations. By introducing intermediate variables, the discretized algebraic equations have great nonlinearity. To address this issue, the paper proposes a multivariable linear algebraic discretization method for NPEs. First, the NPE is discretized, the algebraic form of the nonlinear equation is transformed into the vector form, and the rough set (RS) and information entropy (IE) are constructed to allocate the weights of different variable attributes. According to the given variable attribute weight, the multiple variables in the equation are discretized by linear algebra. It can effectively solve the two grid algorithm problems and the convergence problem of reaction diffusion and has good adaptability in this field.

Modeling of autocatalytic degradation of polymer microparticles with various morphologies based on analytical solutions of reaction-diffusion equations

Analytical solutions of transient concentration of degraded components inside cylindrical and slab-type PLGA particles immersed in infinite medium were derived by solving reaction-diffusion equations of autocatalytic reaction using eigenfunction expansion method. The resulting average concentrations were compared with the modeling results of spherical PLGA particles by Versypt and her colleagues to study the effect of particle morphology on the autocatalytic reaction. Mass transfer resistance inside and outside of the particles was also considered using Biot number, and its effects on the concentration inside particles with various morphologies were also studied by solving reaction-diffusion equation. To predict transient concentration in surrounding medium, coupled differential equations were solved for the three shapes of PLGA particles by assuming finite volume of the decomposition system. Mathematical solutions were obtained by Laplace transform, and the results were compared for the PLGA particles with different shapes depending on Thiele modulus and particle volume fraction.

Network Connectivity Control of Mobile Robots by Fast Position Estimations and Laplacian Kernel

Together with wireless distributed sensor technologies, the connectivity control of mobile robot networks has widely expanded in recent years. Network connectivity has been greatly improved by theoretical frameworks based on graph theory. Most network connectivity studies have focused on algebraic connectivity and the Fiedler vector, which constitutes a network structure matrix eigenpair. Theoretical graph frameworks have popularly been adopted in robot deployment studies; however, the eigenpairs’ computation requires quite a lot of iterative calculations and is extremely time-intensive. In the present study, we propose a robot deployment algorithm that only requires a finite iterative calculation. The proposed algorithm rapidly estimates the robot positions by solving reaction-diffusion equations on the graph, and gradient methods using a Laplacian kernel. The effectiveness of the algorithm is evaluated in computer simulations of mobile robot networks. Furthermore, we implement the algorithm in the actual hardware of a two-wheeled robot.

A P0–P0 weak Galerkin finite element method for solving singularly perturbed reaction–diffusion problems

AbstractThis paper investigates the lowest‐order weak Galerkin finite element (WGFE) method for solving reaction–diffusion equations with singular perturbations in two and three space dimensions. The system of linear equations for the new scheme is positive definite, and one might readily get the well‐posedness of the system. Our numerical experiments confirmed our error analysis that our WGFE method of the lowest order could deliver numerical approximations of the order O(h1/2) and O(h) in H1 and L2 norms, respectively.

Transcendental Bernstein series for solving reaction–diffusion equations with nonlocal boundary conditions through the optimization technique

AbstractIn this paper, we apply transcendental Bernstein series (TBS) for solving reaction–diffusion equations with nonlocal boundary conditions which is the novel approximation tool. To carry out the method, we firstly expand the solution of the system in the term of TBS through the operational matrix scheme. To determine the unknown free coefficients and control parameters appeared in TBS expansion, we define an optimization problem which combines the reaction–diffusion equation with its nonlocal boundary conditions. Then we use the Lagrange multipliers technique for converting the problem under study into a system of algebraic equations. High accuracy and simplicity in reducing the integral boundary conditions are some privileges of the proposed scheme. We emphasize that Bernstein polynomials is the particular case of transcendental Bernstein series. Theoretical discussion about convergence confirms the reliability of the proposed method. Some test problems are chosen to investigate the applicability and computational efficiency. The experimental results confirm that the obtained results are in good agreement with the exact solutions with high rate of convergence.

Design neural network based upon decomposition approach for solving reaction diffusion equation

This paper proposed a new approach for designing artificial neural network based upon decomposition technique that is the problem decomposing into two sub problems. The first sub problem processed by an artificial neural network depending on the given data to get an estimate of the parameters in problem and then uses to solve the second sub problem by obtaining the differential terms in the formulation of the this sub problem. The proposed design is greatly simplifies the problem of parameter estimation, high computational requiring is avoided and instead two smaller problems which be easier solved. Also, the proposed design is useful for noisy data sets.

Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations

Computational costs of numerically solving multidimensional partial differential equations (PDEs) increase significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points. For multidimensional reaction-diffusion equations, stiffness of the system provides additional challenges for achieving efficient numerical simulations. In this paper, we propose a class of Krylov implicit integration factor (IIF) discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion equations on high spatial dimensions. The key ingredient of spatial DG discretization is the multiwavelet bases on nested sparse grids, which can significantly reduce the numbers of degrees of freedom. To deal with the stiffness of the DG spatial operator in discretizing reaction-diffusion equations, we apply the efficient IIF time discretization methods, which are a class of exponential integrators. Krylov subspace approximations are used to evaluate the large size matrix exponentials resulting from IIF schemes for solving PDEs on high spatial dimensions. Stability and error analysis for the semi-discrete scheme are performed. Numerical examples of both scalar equations and systems in two and three spatial dimensions are provided to demonstrate the accuracy and efficiency of the methods. The stiffness of the reaction-diffusion equations is resolved well and large time step size computations are obtained.

Approximation of continuous surface differential operators with the generalized moving least-squares (GMLS) method for solving reaction–diffusion equation

In this paper, a meshless approximation based on generalized moving least squares is applied to solve the reaction–diffusion equations on the sphere and red-blood cell surfaces. The proposed method is based on the projected gradient of the shape functions, and it approximates the Laplace operator defined on the surfaces that is called Laplace–Beltrami. This technique only requires nodes at locations on the surface and the corresponding normal vectors to the surface. To discretize the time variable, an explicit time technique based on the fourth-order Runge–Kutta is used. Some numerical results on Turing and Fitzhugh–Nagumo partial differential equations are given for showing patterns which are appeared in biological phenomena.

A second-order efficient L-stable numerical method for space fractional reaction–diffusion equations

ABSTRACTIn this article we are proposing a strongly stable computationally efficient method in time to numerically solve reaction–diffusion equations with space fractional derivative. The Riesz space fractional derivative is discretized using the second-order fractional centred difference method. Time stepping scheme is based on second-order exponential time differencing Runge–Kutta method. Second-order positivity preserving Padé approximations is used to develop the proposed L-stable method. The computation efficiency of the method is significantly enhanced by using partial fractions splitting technique. The method is shown to be stable and reliable. Solution profiles as well as convergence tables in time are presented for various values of diffusion rates α.

Nonlinear reaction-diffusion equations with delay: some theorems, test problems, exact and numerical solutions

The paper deals with nonlinear reaction-diffusion equations with one or several delays. We formulate theorems that allow constructing exact solutions for some classes of these equations, which depend on several arbitrary functions. Examples of application of these theorems for obtaining new exact solutions in elementary functions are provided. We state basic principles of construction, selection, and use of test problems for nonlinear partial differential equations with delay. Some test problems which can be suitable for estimating accuracy of approximate analytical and numerical methods of solving reaction-diffusion equations with delay are presented. Some examples of numerical solutions of nonlinear test problems with delay are considered.

Computational Simulation of Tumor Hypoxia Based on In Vivo Microvasculature Assessed in a Dorsal Skin Window Chamber.

Malignant growth usually leads to the depletion of oxygen (O2) supply in most solid tumors. Hypoxia can cause resistance to standard radiotherapy, some chemotherapy and immunotherapy. Furthermore, it can also trigger malignant progression by modulating gene expression and inducing genetic instability. The relationship between microvasculature, perfusion and tumor hypoxia has been intensively studied and many computational simulations have been developed to model tissue O2 transport. Usually simplified 2D phantoms are used to investigate tumor hypoxia and it is assumed that vessels are perpendicular to the region of interest and randomly distributed across the domain. Such idealistic topology overlooks vascular heterogeneity and is not accurate enough to approximate real scenarios. In addition, experimental verification of the spatial gradient of computational simulations is not directly feasible. Realistic vasculature obtained from fluorescence imaging imported as geometry for partial differential equations solving did not receive necessary attention so far. Therefore, we established a computational simulation of in vivo conditions using experimental data obtained from dorsal skin window chamber tumor preparations in nude rats for the verification of computational results. Tumor microvasculature was assessed by fluorescence microscopy. Since the conventional finite difference method can hardly satisfy the real measurements, we established a finite element method (FEM) for the experimental data in this study. Realistic 2D tumor microvasculature was reconstructed by segmenting fluorescence images and then translated into FEM topology. O2 distributions and the O2 gradients were obtained by solving reaction-diffusion equations. The simulation results show that the development of tumor hypoxia is greatly influenced by the irregular architecture and function of microvascular networks.

A Radial Basis Function (RBF) Compact Finite Difference (FD) Scheme for Reaction-Diffusion Equations on Surfaces

We present a new high-order, local meshfree method for numerically solving reaction diffusion equations on smooth surfaces of codimension 1 embedded in $\mathbb{R}^d$. The novelty of the method is in the approximation of the Laplace--Beltrami operator for a given surface using Hermite radial basis function (RBF) interpolation over local node sets on the surface. This leads to compact (or implicit) RBF generated finite difference (RBF-FD) formulas for the Laplace--Beltrami operator, which gives rise to sparse differentiation matrices. The method only requires a set of (scattered) nodes on the surface and an approximation to the surface normal vectors at these nodes. Additionally, the method is based on Cartesian coordinates and thus does not suffer from any coordinate singularities. We also present an algorithm for selecting the nodes used to construct the compact RBF-FD formulas that can guarantee the resulting differentiation matrices have desirable stability properties. The improved accuracy and computa...

Finite Element Simulations with Adaptively Moving Mesh for the Reaction Diffusion System

AbstractA moving mesh method is proposed for solving reaction-diffusion equations. The finite element method is used to solving the partial different equation system, and an efficient numerical scheme is applied to implement mesh moving. In the practical calculations, the moving mesh step and the problem equation solver are performed alternatively. Several numerical examples are presented, including the Gray-Scott, the Activator-Inhibitor and a case with a growing domain. It is illustrated numerically that the moving mesh methods costs much lower, compared with the numerical schemes on a fixed mesh. Even in the case of complex pattern dynamics described by the reaction-diffusion systems, the adapted meshes can capture the details successfully.

Discontinuous Legendre wavelet Galerkin method for reaction–diffusion equation

ABSTRACTThis paper proposes a novel numerical method, that is, discontinuous Legendre wavelet Galerkin technique for solving reaction–diffusion equation (RDE). Specifically, variational formulation and corresponding numerical fluxes of this type equation are devised by utilizing the advantages of both Legendre wavelet bases and discontinuous Galerkin approach. Furthermore, adaptive algorithm, stability and error analysis of this method have been discussed. Especially, the distinctive features of the presented approach are easy to cope with a variety of boundary conditions and able to effectively approximate solution of the RDE with less execution and storage space. Finally, numerical tests affirm better accuracy for a range of benchmark problems and demonstrate the validity and utility of this approach.

A locally adaptive time stepping algorithm for the solution to reaction diffusion equations on branched structures

In this paper, we present a numerical method for solving reaction-diffusion equations on one dimensional branched structures. Through the use of a simple domain decomposition scheme, the many branches are decoupled so that the equations can be solved as a system of smaller problems that are tri-diagonal. This technique allows for locally adaptive time stepping, in which the time step used in each branch is determined by local activity. Though the method is presented in the specific context of electrical activity in neural systems, it is sufficiently general that it can be applied to other classes of reaction-diffusion problems and higher dimensions. Information in neurons, which can be effectively modeled as one-dimensional branched structures, is carried in the form of electrical impulses called action potentials. The model equations, based on the Hodgkin-Huxley cable equations, are a set of reaction equations coupled to a single diffusion process. Locally adaptive time stepping schemes are well suited to neural simulations due to the spatial localization of activity. The algorithm significantly reduces the computational cost compared to existing methods, especially for large scale simulations.