Journal articles: 'Diffusion-reaction equation'

1

Seki, Kazuhiko, Mariusz Wojcik, and M. Tachiya. "Fractional reaction-diffusion equation." Journal of Chemical Physics 119, no. 4 (July 22, 2003): 2165–70. http://dx.doi.org/10.1063/1.1587126.

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Angstmann, Christopher N., and Bruce I. Henry. "Time Fractional Fisher–KPP and Fitzhugh–Nagumo Equations." Entropy 22, no. 9 (September 16, 2020): 1035. http://dx.doi.org/10.3390/e22091035.

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A standard reaction–diffusion equation consists of two additive terms, a diffusion term and a reaction rate term. The latter term is obtained directly from a reaction rate equation which is itself derived from known reaction kinetics, together with modelling assumptions such as the law of mass action for well-mixed systems. In formulating a reaction–subdiffusion equation, it is not sufficient to know the reaction rate equation. It is also necessary to know details of the reaction kinetics, even in well-mixed systems where reactions are not diffusion limited. This is because, at a fundamental level, birth and death processes need to be dealt with differently in subdiffusive environments. While there has been some discussion of this in the published literature, few examples have been provided, and there are still very many papers being published with Caputo fractional time derivatives simply replacing first order time derivatives in reaction–diffusion equations. In this paper, we formulate clear examples of reaction–subdiffusion systems, based on; equal birth and death rate dynamics, Fisher–Kolmogorov, Petrovsky and Piskunov (Fisher–KPP) equation dynamics, and Fitzhugh–Nagumo equation dynamics. These examples illustrate how to incorporate considerations of reaction kinetics into fractional reaction–diffusion equations. We also show how the dynamics of a system with birth rates and death rates cancelling, in an otherwise subdiffusive environment, are governed by a mass-conserving tempered time fractional diffusion equation that is subdiffusive for short times but standard diffusion for long times.

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Ipsen, M., F. Hynne, and P. G. Sørensen. "Amplitude Equations and Chemical Reaction–Diffusion Systems." International Journal of Bifurcation and Chaos 07, no. 07 (July 1997): 1539–54. http://dx.doi.org/10.1142/s0218127497001217.

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The paper discusses the use of amplitude equations to describe the spatio-temporal dynamics of a chemical reaction–diffusion system based on an Oregonator model of the Belousov–Zhabotinsky reaction. Sufficiently close to a supercritical Hopf bifurcation the reaction–diffusion equation can be approximated by a complex Ginzburg–Landau equation with parameters determined by the original equation at the point of operation considered. We illustrate the validity of this reduction by comparing numerical spiral wave solutions to the Oregonator reaction–diffusion equation with the corresponding solutions to the complex Ginzburg–Landau equation at finite distances from the bifurcation point. We also compare the solutions at a bifurcation point where the systems develop spatio-temporal chaos. We show that the complex Ginzburg–Landau equation represents the dynamical behavior of the reaction–diffusion equation remarkably well, sufficiently far from the bifurcation point for experimental applications to be feasible.

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Păuna, Alina-Maria. "The auxiliary equation approach for solving reaction-diffusion equations." Journal of Physics: Conference Series 2719, no. 1 (February 1, 2024): 012002. http://dx.doi.org/10.1088/1742-6596/2719/1/012002.

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Abstract 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.

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Wang, Yulan, Xiaojun Song, and Chao Ye. "Fujita Exponent for a Nonlinear Degenerate Parabolic Equation with Localized Source." Advances in Mathematical Physics 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/301747.

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This paper is devoted to understand the blow-up properties of reaction-diffusion equations which combine a localized reaction term with nonlinear diffusion. In particular, we study the critical exponent of ap-Laplacian equation with a localized reaction. We obtain the Fujita exponentqcof the equation.

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Wu, G., Eric Wai Ming Lee, and Gao Li. "Numerical solutions of the reaction-diffusion equation." International Journal of Numerical Methods for Heat & Fluid Flow 25, no. 2 (March 2, 2015): 265–71. http://dx.doi.org/10.1108/hff-04-2014-0113.

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Purpose – The purpose of this paper is to introduce variational iteration method (VIM) to construct equivalent integral equations for initial-boundary value problems of nonlinear partial differential equations. The Lagrange multipliers become the integral kernels. Design/methodology/approach – Using the discrete numerical integral formula, the general way is given to solve the famous reaction-diffusion equation numerically. Findings – With the given explicit solution, the results show the conveniences of the general numerical schemes and numerical simulation of the reaction-diffusion is finally presented in the cases of various coefficients. Originality/value – The method avoids the treatment of the time derivative as that in the classical finite difference method and the VIM is introduced to construct equivalent integral equations for initial-boundary value problems of nonlinear partial differential equations.

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Agom, E. U., F. O. Ogunfiditimi, E. V. Bassey, and C. Igiri. "REACTION-DIFFUSION FISHER’S EQUATIONS VIA DECOMPOSITION METHOD." Journal of Computer Science and Applied Mathematics 5, no. 2 (October 30, 2023): 145–53. http://dx.doi.org/10.37418/jcsam.5.2.7.

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The effect of the source, initial or boundary conditions in the use of Adomian decomposition method (ADM) on nonlinear partial differential equation or nonlinear equation in general is enormous. Sometimes the equation in question result to continuous exact solution in series form, other times it result to discrete approximate analytical solutions. In this paper, we show that continuous exact solitons can be obtained on application of ADM to the Fisher's equation with the deployment Taylor theorem to the terms(s) in question. And, the resulting series is split into the integral equations during the solution process. Resulting to multivariate Taylor's series of the exact solitons with the help of Adomian polynomials of the nonlinear reaction term correctly calculated. More physical results are further depicted in 2D, 3D and contour plots.

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Prokhorova, M. F. "Factorization of the reaction-diffusion equation, the wave equation, and other equations." Proceedings of the Steklov Institute of Mathematics 287, S1 (November 27, 2014): 156–66. http://dx.doi.org/10.1134/s0081543814090156.

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9

Huang, Qicai. "Adaptive Extraction of Oil Painting Texture Features Based on Reaction Diffusion Equation." Advances in Mathematical Physics 2021 (November 3, 2021): 1–11. http://dx.doi.org/10.1155/2021/4464985.

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The oil painting retrieval technology based on the reaction diffusion equation has attracted widespread attention in the fields of oil painting processing and pattern recognition. The description and extraction of oil painting information and the classification method of oil paintings are two important processes in content-based oil painting retrieval. Inspired by the restoration and decomposition functional model of equal oil painting, we propose a reaction diffusion equation model. The new model contains two reaction diffusion equations with different principal parts. One principal part is total variation diffusion, which is used to remove noise. The other main part is thermal diffusion, which is used to modify the source term of the denoising reaction-diffusion equation to achieve the effect of protecting the texture of the oil painting. The interaction of the two reaction-diffusion equations finally achieves denoising while maintaining the boundaries and textures. Under the framework of the above reaction diffusion equation model, we introduce Laplace flow to replace the original total variation flow, so that the new denoising reaction diffusion equation combines the isotropic diffusion and total variation flow of the thermal reaction diffusion equation to achieve the effect of adaptive theoretical research. Using regularization methods and methods, we, respectively, get the well-posedness of the two model solutions, which provides the necessary preparation for numerical calculations. Based on the statistical theory and classification principles of support vector machines, combined with the characteristics of oil painting classification, the research and analysis are carried out from the three important aspects of kernel function, training algorithm, and multiclass classifier algorithm that affect the classification effect and speed. Numerical experiments show that the given filter model has a better processing effect on images with different types and different degrees of noise pollution. On this basis, an oil painting classification system based on texture features is designed, combined with an improved gray-level cooccurrence matrix algorithm and a multiclass support vector machine classification model, to extract, train, and classify oil paintings. Experiments with three types of oil paintings prove that the system can achieve a good oil painting classification effect. Different from the original model, the new model is based on the framework of reaction-diffusion equations. In addition, the new model has good effects in removing step effects, maintaining boundaries and denoising, especially in maintaining texture.

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RODRIGO, M., and M. MIMURA. "ON SOME CLASSES OF LINEARIZABLE REACTION-CONVECTION-DIFFUSION EQUATIONS." Analysis and Applications 02, no. 01 (January 2004): 11–19. http://dx.doi.org/10.1142/s0219530504000266.

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In this paper, we consider the class of equations ut=[F(x,u)ux+G(x,u)]x+H(x,u). Using hodograph and dependent variable transformations, we determine sufficient conditions on F, G, and H such that this equation is linearizable. We also derive a general quasilinear equation, which includes the Clarkson–Fokas–Ablowitz equation (SIAM J. Appl. Math.49 (1989), 1188–1209), that can be transformed into semilinear form.

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Grasselli, M., and V. Pata. "A reaction-diffusion equation with memory." Discrete & Continuous Dynamical Systems - A 15, no. 4 (2006): 1079–88. http://dx.doi.org/10.3934/dcds.2006.15.1079.

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12

Isaacson, Samuel A. "A convergent reaction-diffusion master equation." Journal of Chemical Physics 139, no. 5 (August 7, 2013): 054101. http://dx.doi.org/10.1063/1.4816377.

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13

Krishnan, E. V. "Exact Solutions of Reaction-Diffusion Equation." Journal of the Physical Society of Japan 62, no. 3 (March 15, 1993): 1076–77. http://dx.doi.org/10.1143/jpsj.62.1076.

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14

Zhao, Xueqing, Keke Huang, Xiaoming Wang, Meihong Shi, Xinjuan Zhu, Quanli Gao, and Zhaofei Yu. "Reaction–diffusion equation based image restoration." Applied Mathematics and Computation 338 (December 2018): 588–606. http://dx.doi.org/10.1016/j.amc.2018.06.054.

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15

Wang, Liwen, and Qingyi Chen. "ON REACTION-DIFFUSION EQUATION WITH ABSORPTION." Acta Mathematica Scientia 13, no. 2 (1993): 147–52. http://dx.doi.org/10.1016/s0252-9602(18)30201-7.

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16

Finkelshtein, Dmitri, Yuri Kondratiev, Stanislav Molchanov, and Pasha Tkachov. "Global stability in a nonlocal reaction-diffusion equation." Stochastics and Dynamics 18, no. 05 (September 12, 2018): 1850037. http://dx.doi.org/10.1142/s0219493718500375.

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We study stability of stationary solutions for a class of nonlocal semilinear parabolic equations. To this end, we prove the Feynman–Kac formula for a Lévy processes with time-dependent potentials and arbitrary initial condition. We propose sufficient conditions for asymptotic stability of the zero solution, and use them to the study of the spatial logistic equation arising in population ecology. For this equation, we find conditions which imply that its positive stationary solution is asymptotically stable. We consider also the case when the initial condition is given by a random field.

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17

Jurado, Francisco, and Andrés A. Ramírez. "State Feedback Regulation Problem to the Reaction-Diffusion Equation." Mathematics 8, no. 11 (November 6, 2020): 1983. http://dx.doi.org/10.3390/math8111983.

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In this work, we explore the state feedback regulator problem (SFRP) in order to achieve the goal for trajectory tracking with harmonic disturbance rejection to one-dimensional (1-D) reaction-diffusion (R-D) equation, namely, a partial differential equation of parabolic type, while taking into account bounded input, output, and disturbance operators, a finite-dimensional exosystem (exogenous system), and the state of the exosystem as the state to the feedback law. As is well-known, the SFRP can be solved only if the so-called Francis (regulator) equations have solution. In our work, we try with the solution of the Francis equations from the 1-D R-D equation following given criteria to the eigenvalues from the exosystem and transfer function of the system, but the state operator is here defined in terms of the Sturm–Liouville differential operator (SLDO). Within this framework, the SFRP is then solved for the 1-D R-D equation. The numerical simulation results validate the performance of the regulator.

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18

Gong, Chunye, Weimin Bao, Guojian Tang, Yuewen Jiang, and Jie Liu. "A Domain Decomposition Method for Time Fractional Reaction-Diffusion Equation." Scientific World Journal 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/681707.

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The computational complexity of one-dimensional time fractional reaction-diffusion equation isO(N2M)compared withO(NM)for classical integer reaction-diffusion equation. Parallel computing is used to overcome this challenge. Domain decomposition method (DDM) embodies large potential for parallelization of the numerical solution for fractional equations and serves as a basis for distributed, parallel computations. A domain decomposition algorithm for time fractional reaction-diffusion equation with implicit finite difference method is proposed. The domain decomposition algorithm keeps the same parallelism but needs much fewer iterations, compared with Jacobi iteration in each time step. Numerical experiments are used to verify the efficiency of the obtained algorithm.

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Smaoui, Nejib. "Analyzing the dynamics of the forced Burgers equation." Journal of Applied Mathematics and Stochastic Analysis 13, no. 3 (January 1, 2000): 269–85. http://dx.doi.org/10.1155/s1048953300000241.

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We study numerically the long-time dynamics of a system of reaction-diffusion equations that arise from the viscous forced Burgers equation (ut+uux−vuxx=F). A nonlinear transformation introduced by Kwak is used to embed the scalar Burgers equation into a system of reaction diffusion equations. The Kwak transformation is used to determine the existence of an inertial manifold for the 2-D Navier-Stokes equation. We show analytically as well as numerically that the two systems have a similar, long-time dynamical, behavior for large viscosity v.

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Phongthanapanich, Sutthisak, and Pramote Dechaumphai. "A CHARACTERISTIC-BASED FINITE VOLUME ELEMENT METHOD FOR CONVECTION-DIFFUSION-REACTION EQUATION." Transactions of the Canadian Society for Mechanical Engineering 32, no. 3-4 (September 2008): 549–60. http://dx.doi.org/10.1139/tcsme-2008-0037.

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A two-dimensional convection-diffusion-reaction equation is discretized by the finite volume element method on triangular meshes. Time-dependent convection-diffusion-reaction equation is developed along the characteristic path using the characteristic-based scheme, while the finite volume method is employed for deriving the discretized equations. The concept of the finite element technique is applied to estimate the gradient quantities at the cell faces of the finite volume. Numerical test cases have shown that the method does not require any artificial diffusion to improve the solution stability. The robustness and the accuracy of the method have been evaluated by using available analytical and numerical solutions of the pure-convection, convection-diffusion and convection-diffusion-reaction problems.

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Mocenni, C., E. Sparacino, and J. P. Zubelli. "Effective rough boundary parametrization for reaction-diffusion systems." Applicable Analysis and Discrete Mathematics 8, no. 1 (2014): 33–59. http://dx.doi.org/10.2298/aadm140126002m.

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We address the problem of parametrizing the boundary data for reaction- diffusion partial differential equations associated to distributed systems that possess rough boundaries. The boundaries are modeled as fast oscillating periodic structures and are endowed with Neumann or Dirichlet boundary conditions. Using techniques from homogenization theory and multiple-scale analysis we derive the effective equation and boundary conditions that are satisfied by the homogenized solution. We present numerical simulations that validate our theoretical results and compare it with the alternative approach based on solving the same equation with a smoothed version of the boundary. The numerical tests show the accuracy of the homogenized solution to the effective system vis a vis the numerical solution of the original differential equation. The homogenized solution is shown undergoing dynamical regime shifts, such as anticipation of pattern formation, obtained by varying the diffusion coefficient.

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Lin, Xiaodong, Joseph W. H. So, and Jianhong Wu. "Centre manifolds for partial differential equations with delays." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 122, no. 3-4 (1992): 237–54. http://dx.doi.org/10.1017/s0308210500021090.

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SynopsisA centre manifold theory for reaction-diffusion equations with temporal delays is developed. Besides an existence proof, we also show that the equation on the centre manifold is a coupled system of scalar ordinary differential equations of higher order. As an illustration, this reduction procedure is applied to the Hutchinson equation with diffusion.

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23

del Razo, Mauricio J., Stefanie Winkelmann, Rupert Klein, and Felix Höfling. "Chemical diffusion master equation: Formulations of reaction–diffusion processes on the molecular level." Journal of Mathematical Physics 64, no. 1 (January 1, 2023): 013304. http://dx.doi.org/10.1063/5.0129620.

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The chemical diffusion master equation (CDME) describes the probabilistic dynamics of reaction–diffusion systems at the molecular level [del Razo et al., Lett. Math. Phys. 112, 49 (2022)]; it can be considered as the master equation for reaction–diffusion processes. The CDME consists of an infinite ordered family of Fokker–Planck equations, where each level of the ordered family corresponds to a certain number of particles and each particle represents a molecule. The equations at each level describe the spatial diffusion of the corresponding set of particles, and they are coupled to each other via reaction operators—linear operators representing chemical reactions. These operators change the number of particles in the system and, thus, transport probability between different levels in the family. In this work, we present three approaches to formulate the CDME and show the relations between them. We further deduce the non-trivial combinatorial factors contained in the reaction operators, and we elucidate the relation to the original formulation of the CDME, which is based on creation and annihilation operators acting on many-particle probability density functions. Finally, we discuss applications to multiscale simulations of biochemical systems among other future prospects.

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Li, Changhao, Jianfeng Li, and Yuliang Yang. "A Feynman Path Integral-like Method for Deriving Reaction–Diffusion Equations." Polymers 14, no. 23 (November 27, 2022): 5156. http://dx.doi.org/10.3390/polym14235156.

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This work is devoted to deriving a more accurate reaction–diffusion equation for an A/B binary system by summing over microscopic trajectories. By noting that an originally simple physical trajectory might be much more complicated when the reactions are incorporated, we introduce diffusion–reaction–diffusion (DRD) diagrams, similar to the Feynman diagram, to derive the equation. It is found that when there is no intermolecular interaction between A and B, the newly derived equation is reduced to the classical reaction–diffusion equation. However, when there is intermolecular interaction, the newly derived equation shows that there are coupling terms between the diffusion and the reaction, which will be manifested on the mesoscopic scale. The DRD diagram method can be also applied to derive a more accurate dynamical equation for the description of chemical reactions occurred in polymeric systems, such as polymerizations, since the diffusion and the reaction may couple more deeply than that of small molecules.

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Anguiano, María. "Reaction–Diffusion Equation on Thin Porous Media." Bulletin of the Malaysian Mathematical Sciences Society 44, no. 5 (March 13, 2021): 3089–110. http://dx.doi.org/10.1007/s40840-021-01103-0.

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Bessonov, Nick, Gennady Bocharov, Tarik Mohammed Touaoula, Sergei Trofimchuk, and Vitaly Volpert. "Delay reaction-diffusion equation for infection dynamics." Discrete & Continuous Dynamical Systems - B 24, no. 5 (2019): 2073–91. http://dx.doi.org/10.3934/dcdsb.2019085.

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Shoji, Tetsuya, Seiji Miura, and Tetsuo Mohri. "Reaction-Diffusion Equation for Dislocation Multiplication Process." Materials Transactions, JIM 41, no. 5 (2000): 585–88. http://dx.doi.org/10.2320/matertrans1989.41.585.

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28

Paster, A., D. Bolster, and D. A. Benson. "Particle tracking and the diffusion-reaction equation." Water Resources Research 49, no. 1 (January 2013): 1–6. http://dx.doi.org/10.1029/2012wr012444.

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Okoya, S. S., S. O. Ajadi, and A. B. Kolawole. "Thermal runaway for a reaction–diffusion equation." International Communications in Heat and Mass Transfer 30, no. 6 (August 2003): 845–50. http://dx.doi.org/10.1016/s0735-1933(03)00132-5.

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Ho, Choon-Lin, and Chih-Min Yang. "Convection-diffusion-reaction equation with similarity solutions." Chinese Journal of Physics 59 (June 2019): 117–25. http://dx.doi.org/10.1016/j.cjph.2019.02.030.

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Iovane, G., and O. V. Kapustyan. "Global Attractor for Impulsive Reaction-Diffusion Equation." Nonlinear Oscillations 8, no. 3 (July 2005): 318–28. http://dx.doi.org/10.1007/s11072-006-0004-7.

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Chen, Tsu-Fen, Howard A. Levine, and Paul E. Sacks. "Analysis of a convective reaction-diffusion equation." Nonlinear Analysis: Theory, Methods & Applications 12, no. 12 (December 1988): 1349–70. http://dx.doi.org/10.1016/0362-546x(88)90083-1.

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Zhao, Zhihong, and Erhua Rong. "Reaction diffusion equation with spatio-temporal delay." Communications in Nonlinear Science and Numerical Simulation 19, no. 7 (July 2014): 2252–61. http://dx.doi.org/10.1016/j.cnsns.2013.11.006.

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Ping, Wang, Hsieh Din-Yu, and Tang Shaoqiang. "Pattern selection in a reaction-diffusion equation." Acta Mechanica Sinica 18, no. 6 (December 2002): 652–60. http://dx.doi.org/10.1007/bf02487968.

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Ahmad, B., M. S. Alhothuali, H. H. Alsulami, M. Kirane, and S. Timoshin. "On a time fractional reaction diffusion equation." Applied Mathematics and Computation 257 (April 2015): 199–204. http://dx.doi.org/10.1016/j.amc.2014.06.099.

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Sharifi, Shokofeh, and Jalil Rashidinia. "Collocation method for Convection-Reaction-Diffusion equation." Journal of King Saud University - Science 31, no. 4 (October 2019): 1115–21. http://dx.doi.org/10.1016/j.jksus.2018.10.004.

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Choi, Jae Seok, Takayuki Yamada, Kazuhiro Izui, Shinji Nishiwaki, and Jeonghoon Yoo. "Topology optimization using a reaction–diffusion equation." Computer Methods in Applied Mechanics and Engineering 200, no. 29-32 (July 2011): 2407–20. http://dx.doi.org/10.1016/j.cma.2011.04.013.

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Zuo, Li, and Fengtai Mei. "Multivariable Linear Algebraic Discretization of Nonlinear Parabolic Equations for Computational Analysis." Computational Intelligence and Neuroscience 2022 (September 29, 2022): 1–8. http://dx.doi.org/10.1155/2022/6323418.

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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.

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Sene, Ndolane, and Aliou Niang Fall. "Homotopy Perturbation ρ-Laplace Transform Method and Its Application to the Fractional Diffusion Equation and the Fractional Diffusion-Reaction Equation." Fractal and Fractional 3, no. 2 (March 27, 2019): 14. http://dx.doi.org/10.3390/fractalfract3020014.

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In this paper, the approximate solutions of the fractional diffusion equations described by the fractional derivative operator were investigated. The homotopy perturbation Laplace transform method of getting the approximate solution was proposed. The Caputo generalized fractional derivative was used. The effects of the orders α and ρ in the diffusion processes was addressed. The graphical representations of the approximate solutions of the fractional diffusion equation and the fractional diffusion-reaction equation both described by the Caputo generalized fractional derivative were provided.

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El Kettani, Perla, Danielle Hilhorst, and Kai Lee. "A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion." Discrete & Continuous Dynamical Systems - A 38, no. 11 (2018): 5615–48. http://dx.doi.org/10.3934/dcds.2018246.

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HADELER, KARL P., THOMAS HILLEN, and FRITHJOF LUTSCHER. "THE LANGEVIN OR KRAMERS APPROACH TO BIOLOGICAL MODELING." Mathematical Models and Methods in Applied Sciences 14, no. 10 (October 2004): 1561–83. http://dx.doi.org/10.1142/s0218202504003726.

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In the Langevin or Ornstein–Uhlenbeck approach to diffusion, stochastic increments are applied to the velocity rather than to the space variable. The density of this process satisfies a linear partial differential equation of the general form of a transport equation which is hyperbolic with respect to the space variable but parabolic with respect to the velocity variable, the Klein–Kramers or simply Kramers equation. This modeling approach allows for a more detailed description of individual movement and orientation dependent interaction than the frequently used reaction diffusion framework.For the Kramers equation, moments are computed, the infinite system of moment equations is closed at several levels, and telegraph and diffusion equations are derived as approximations. Then nonlinearities are introduced such that the semi-linear reaction Kramers equation describes particles which move and interact on the same time-scale. Also for these nonlinear problems a moment approach is feasible and yields nonlinear damped wave equations as limiting cases.We apply the moment method to the Kramers equation for chemotactic movement and obtain the classical Patlak–Keller–Segel model. We discuss similarities between chemotactic movement of bacteria and gravitational movement of physical particles.

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ÖZUĞURLU, E. "A NOTE ON THE NUMERICAL APPROACH FOR THE REACTION–DIFFUSION PROBLEM WITH A FREE BOUNDARY CONDITION." ANZIAM Journal 51, no. 3 (January 2010): 317–30. http://dx.doi.org/10.1017/s1446181110000817.

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AbstractThe equation modelling the evolution of a foam (a complex porous medium consisting of a set of gas bubbles surrounded by liquid films) is solved numerically. This model is described by the reaction–diffusion differential equation with a free boundary. Two numerical methods, namely the fixed-point and the averaging in time and forward differences in space (the Crank–Nicolson scheme), both in combination with Newton’s method, are proposed for solving the governing equations. The solution of Burgers’ equation is considered as a special case. We present the Crank–Nicolson scheme combined with Newton’s method for the reaction–diffusion differential equation appearing in a foam breaking phenomenon.

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Rosa, M., and M. L. Gandarias. "Multiplier method and exact solutions for a density dependent reaction-diffusion equation." Applied Mathematics and Nonlinear Sciences 1, no. 2 (July 1, 2016): 311–20. http://dx.doi.org/10.21042/amns.2016.2.00026.

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AbstractReaction-diffusion equations have enjoyed a considerable amount of scientific interest. The reason for the large amount of work put into studying these equations is not only their practical relevance, but also interesting phenomena that can arise from such equations. Fisher equation is commonly used in biology for population dynamics models and in bacterial growth problems as well as development and growth of solid tumours. The physical aspects of this equation are not fully understood without getting deeper into the concept of conservation laws. In [4], Anco and Bluman gave a general treatment of a direct conservation law method for partial differential equations expressed in a standard Cauchy-Kovaleskaya form. In this work we study the well known density dependent diffusion-reaction equation. We derive conservation laws by using the direct method of the multipliers.

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Abrashina-Zhadaeva, Natalia, and Natalie Romanova. "ALGORITHMS FOR NUMERICAL SOLVING OF 2D ANOMALOUS DIFFUSION PROBLEMS." Mathematical Modelling and Analysis 17, no. 3 (June 1, 2012): 447–55. http://dx.doi.org/10.3846/13926292.2012.686123.

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Fractional analog of the reaction diffusion equation is used to model the subdiffusion process. Diffusion equation with fractional Riemann–Liouville operator is analyzed in this paper. We offer finite-difference methods that can be used to solve the initial-boundary value problems for some time-fractional order differential equations. Stability and convergence theorems are proved.

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45

Gabrick, Enrique C., Paulo R. Protachevicz, Ervin K. Lenzi, Elaheh Sayari, José Trobia, Marcelo K. Lenzi, Fernando S. Borges, Iberê L. Caldas, and Antonio M. Batista. "Fractional Diffusion Equation under Singular and Non-Singular Kernel and Its Stability." Fractal and Fractional 7, no. 11 (October 30, 2023): 792. http://dx.doi.org/10.3390/fractalfract7110792.

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The fractional reaction–diffusion equation has been used in many real-world applications in fields such as physics, biology, and chemistry. Motivated by the huge application of fractional reaction–diffusion, we propose a numerical scheme to solve the fractional reaction–diffusion equation under different kernels. Our method can be particularly employed for singular and non-singular kernels, such as the Riemann–Liouville, Caputo, Fabrizio–Caputo, and Atangana–Baleanu operators. Moreover, we obtained general inequalities that guarantee that the stability condition depends explicitly on the kernel. As an implementation of the method, we numerically solved the diffusion equation under the power-law and exponential kernels. For the power-law kernel, we solved by considering fractional time, space, and both operators. In another example, we considered the exponential kernel acting on the time derivative and compared the numerical results with the analytical ones. Our results showed that the numerical procedure developed in this work can be employed to solve fractional differential equations considering different kernels.

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46

RODRIGO, MARIANITO R. "BOUNDS ON THE CRITICAL TIMES FOR THE GENERAL FISHER–KPP EQUATION." ANZIAM Journal 63, no. 4 (October 2021): 448–68. http://dx.doi.org/10.1017/s1446181121000365.

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AbstractThe Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation is one of the prototypical reaction–diffusion equations and is encountered in many areas, primarily in population dynamics. An important consideration for the phenomena modelled by diffusion equations is the length of the diffusive process. In this paper, three definitions of the critical time are given, and bounds are obtained by a careful construction of the upper and lower solutions. The comparison functions satisfy the nonlinear, but linearizable, partial differential equations of Fisher–KPP type. Results of the numerical simulations are displayed. Extensions to some classes of reaction–diffusion systems and an application to a spatially heterogeneous harvesting model are also presented.

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Wang, Haoyu, and Ge Tian. "Propagating interface in reaction-diffusion equations with distributed delay." Electronic Journal of Differential Equations 2021, no. 01-104 (June 21, 2021): 54. http://dx.doi.org/10.58997/ejde.2021.54.

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This article concerns the limiting behavior of the solution to a reaction-diffusion equation with distributed delay. We firstly consider the quasi-monotone situation and then investigate the non-monotone situation by constructing two auxiliary quasi-monotone equations. The limit behaviors of solutions of the equation can be obtained from the sandwich technique and the comparison principle of the Cauchy problem. It is proved that the propagation speed of the interface is equal to the minimum wave speed of the corresponding traveling waves. This makes possible to observe the minimum speed of traveling waves from a new perspective. For more information see https://ejde.math.txstate.edu/Volumes/2021/54/abstr.html

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Rodrigo, Marianito. "Bounds on the critical times for the Fisher-KPP equation." ANZIAM Journal 63 (December 31, 2021): 448–68. http://dx.doi.org/10.21914/anziamj.v63.16588.

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The Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation is one of the prototypical reaction–diffusion equations and is encountered in many areas, primarily in population dynamics. An important consideration for the phenomena modelled by diffusion equations is the length of the diffusive process. In this paper, three definitions of the critical time are given, and bounds are obtained by a careful construction of the upper and lower solutions. The comparison functions satisfy the nonlinear, but linearizable, partial differential equations of Fisher–KPP type. Results of the numerical simulations are displayed. Extensions to some classes of reaction–diffusion systems and an application to a spatially heterogeneous harvesting model are also presented. doi:10.1017/S1446181121000365

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AHMED, SHAIMAA A. A. "APPROXIMATE TRAVELING WAVE SOLUTION OF AVIAN FLU TELEGRAPH REACTION DIFFUSION EQUATION." International Journal of Biomathematics 03, no. 04 (December 2010): 509–14. http://dx.doi.org/10.1142/s1793524510001057.

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It is known that, the telegraph equation is more suitable than ordinary diffusion equation in modeling reaction diffusion in several branches of sciences. In this paper we generalize the governing equation of distributed-infective model which represents the spread of avian flu to the telegraph reaction diffusion equation and presents its approximate traveling wave solution by using linear piecewise approximation.

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Iagar, Razvan Gabriel, and Ariel Sánchez. "Eternal solutions for a reaction-diffusion equation with weighted reaction." Discrete & Continuous Dynamical Systems 42, no. 3 (2022): 1465. http://dx.doi.org/10.3934/dcds.2021160.

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We prove existence and uniqueness of eternal solutions in self-similar form growing up in time with exponential rate for the weighted reaction-diffusion equation<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \partial_tu = \Delta u^m+|x|^{\sigma}u^p, $\end{document} </tex-math></disp-formula>posed in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>, with <inline-formula><tex-math id="M2">\begin{document}$ m>1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 0<p<1 $\end{document}</tex-math></inline-formula> and the critical value for the weight<disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \sigma = \frac{2(1-p)}{m-1}. $\end{document} </tex-math></disp-formula>Existence and uniqueness of some specific solution holds true when <inline-formula><tex-math id="M4">\begin{document}$ m+p\geq2 $\end{document}</tex-math></inline-formula>. On the contrary, no eternal solution exists if <inline-formula><tex-math id="M5">\begin{document}$ m+p<2 $\end{document}</tex-math></inline-formula>. We also classify exponential self-similar solutions with a different interface behavior when <inline-formula><tex-math id="M6">\begin{document}$ m+p>2 $\end{document}</tex-math></inline-formula>. Some transformations to reaction-convection-diffusion equations and traveling wave solutions are also introduced.

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Journal articles on the topic 'Diffusion-reaction equation'

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