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Journal articles on the topic 'Diffusion equations'

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Published: 4 June 2021
Last updated: 9 June 2025

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1

Slijepčević, Siniša. "Entropy of scalar reaction-diffusion equations." Mathematica Bohemica 139, no. 4 (2014): 597–605. http://dx.doi.org/10.21136/mb.2014.144137.

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2

Bögelein, Verena, Frank Duzaar, Paolo Marcellini, and Stefano Signoriello. "Nonlocal diffusion equations." Journal of Mathematical Analysis and Applications 432, no. 1 (December 2015): 398–428. http://dx.doi.org/10.1016/j.jmaa.2015.06.053.

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3

SOKOLOV, I. M., and A. V. CHECHKIN. "ANOMALOUS DIFFUSION AND GENERALIZED DIFFUSION EQUATIONS." Fluctuation and Noise Letters 05, no. 02 (June 2005): L275—L282. http://dx.doi.org/10.1142/s0219477505002653.

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Fractional diffusion equations are widely used to describe anomalous diffusion processes where the characteristic displacement scales as a power of time. The forms of such equations might differ with respect to the position of the corresponding fractional operator in addition to or instead of the whole-number derivative in the Fick's equation. For processes lacking simple scaling the corresponding description may be given by distributed-order equations. In the present paper different forms of distributed-order diffusion equations are considered. The properties of their solutions are discussed for a simple special case.
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4

Zubair, Muhammad. "Fractional diffusion equations and anomalous diffusion." Contemporary Physics 59, no. 4 (September 11, 2018): 406–7. http://dx.doi.org/10.1080/00107514.2018.1515252.

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5

Gurevich, Pavel, and Sergey Tikhomirov. "Systems of reaction-diffusion equations with spatially distributed hysteresis." Mathematica Bohemica 139, no. 2 (2014): 239–57. http://dx.doi.org/10.21136/mb.2014.143852.

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6

Fila, Marek, and Ján Filo. "Global behaviour of solutions to some nonlinear diffusion equations." Czechoslovak Mathematical Journal 40, no. 2 (1990): 226–38. http://dx.doi.org/10.21136/cmj.1990.102377.

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7

KOLTUNOVA, L. N. "ON AVERAGED DIFFUSION EQUATIONS." Chemical Engineering Communications 114, no. 1 (April 1992): 1–15. http://dx.doi.org/10.1080/00986449208936013.

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8

Kern, Peter, Svenja Lage, and Mark M. Meerschaert. "Semi-fractional diffusion equations." Fractional Calculus and Applied Analysis 22, no. 2 (April 24, 2019): 326–57. http://dx.doi.org/10.1515/fca-2019-0021.

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Abstract It is well known that certain fractional diffusion equations can be solved by the densities of stable Lévy motions. In this paper we use the classical semigroup approach for Lévy processes to define semi-fractional derivatives, which allows us to generalize this statement to semistable Lévy processes. A Fourier series approach for the periodic part of the corresponding Lévy exponents enables us to represent semi-fractional derivatives by a Grünwald-Letnikov type formula. We use this formula to calculate semi-fractional derivatives and solutions to semi-fractional diffusion equations numerically. In particular, by means of the Grünwald-Letnikov type formula we provide a numerical algorithm to compute semistable densities.
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9

Wei, G. W. "Generalized reaction–diffusion equations." Chemical Physics Letters 303, no. 5-6 (April 1999): 531–36. http://dx.doi.org/10.1016/s0009-2614(99)00270-5.

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10

Freidlin, Mark. "Coupled Reaction-Diffusion Equations." Annals of Probability 19, no. 1 (January 1991): 29–57. http://dx.doi.org/10.1214/aop/1176990535.

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11

Krishnan, E. V. "On Some Diffusion Equations." Journal of the Physical Society of Japan 63, no. 2 (February 15, 1994): 460–65. http://dx.doi.org/10.1143/jpsj.63.460.

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12

Calvo, J., A. Marigonda, and G. Orlandi. "Anisotropic tempered diffusion equations." Nonlinear Analysis 199 (October 2020): 111937. http://dx.doi.org/10.1016/j.na.2020.111937.

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13

Saxena, R. K., A. M. Mathai, and H. J. Haubold. "Fractional Reaction-Diffusion Equations." Astrophysics and Space Science 305, no. 3 (November 9, 2006): 289–96. http://dx.doi.org/10.1007/s10509-006-9189-6.

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14

Perumal, Muthiah, and Kittur G. Ranga Raju. "Approximate Convection-Diffusion Equations." Journal of Hydrologic Engineering 4, no. 2 (April 1999): 160–64. http://dx.doi.org/10.1061/(asce)1084-0699(1999)4:2(160).

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15

Feo, Filomena, Juan Luis Vázquez, and Bruno Volzone. "Anisotropic fast diffusion equations." Nonlinear Analysis 233 (August 2023): 113298. http://dx.doi.org/10.1016/j.na.2023.113298.

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16

Coville, Jérôme, Changfeng Gui, and Mingfeng Zhao. "Propagation acceleration in reaction diffusion equations with anomalous diffusions." Nonlinearity 34, no. 3 (March 1, 2021): 1544–76. http://dx.doi.org/10.1088/1361-6544/abe17c.

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17

Karamzin, Y. N., T. A. Kudryashova, and S. V. Polyakov. "On a class of flux schemes for convection-diffusion equations." Computational Mathematics and Information Technologies 2 (2017): 169–79. http://dx.doi.org/10.23947/2587-8999-2017-2-169-179.

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18

Yarmolenko, M. V. "Analytically Solvable Differential Diffusion Equations Describing the Intermediate Phase Growth." METALLOFIZIKA I NOVEISHIE TEKHNOLOGII 40, no. 9 (December 5, 2018): 1201–7. http://dx.doi.org/10.15407/mfint.40.09.1201.

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19

Truman, A., and H. Z. Zhao. "On stochastic diffusion equations and stochastic Burgers’ equations." Journal of Mathematical Physics 37, no. 1 (January 1996): 283–307. http://dx.doi.org/10.1063/1.531391.

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20

Gladkov, A. V., V. V. Dmitrieva, and R. A. Sharipov. "Some nonlinear equations reducible to diffusion-type equations." Theoretical and Mathematical Physics 123, no. 1 (April 2000): 436–45. http://dx.doi.org/10.1007/bf02551049.

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21

Stephenson, John. "Some non-linear diffusion equations and fractal diffusion." Physica A: Statistical Mechanics and its Applications 222, no. 1-4 (December 1995): 234–47. http://dx.doi.org/10.1016/0378-4371(95)00201-4.

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22

Tarasov, Vasily E. "Fractional Diffusion Equations for Lattice and Continuum: Grünwald-Letnikov Differences and Derivatives Approach." International Journal of Statistical Mechanics 2014 (December 8, 2014): 1–7. http://dx.doi.org/10.1155/2014/873529.

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Fractional diffusion equations for three-dimensional lattice models based on fractional-order differences of the Grünwald-Letnikov type are suggested. These lattice fractional diffusion equations contain difference operators that describe long-range jumps from one lattice site to another. In continuum limit, the suggested lattice diffusion equations with noninteger order differences give the diffusion equations with the Grünwald-Letnikov fractional derivatives for continuum. We propose a consistent derivation of the fractional diffusion equation with the fractional derivatives of Grünwald-Letnikov type. The suggested lattice diffusion equations can be considered as a new microstructural basis of space-fractional diffusion in nonlocal media.
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23

Hasal, Pavel, and Vladimír Kudrna. "Certain Problems with the Application of Stochastic Diffusion Processes for the Description of Chemical Engineering Phenomena. Numerical Simulation of One-Dimensional Diffusion Process." Collection of Czechoslovak Chemical Communications 61, no. 4 (1996): 512–35. http://dx.doi.org/10.1135/cccc19960512.

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Some problems are analyzed arising when a numerical simulation of a random motion of a large ensemble of diffusing particles is used to approximate the solution of a one-dimensional diffusion equation. The particle motion is described by means of a stochastic differential equation. The problems emerging especially when the diffusion coefficient is a function of spatial coordinate are discussed. The possibility of simulation of various kinds of stochastic integral is demonstrated. It is shown that the application of standard numerical procedures commonly adopted for ordinary differential equations may lead to erroneous results when used for solution of stochastic differential equations. General conclusions are verified by numerical solution of three stochastic differential equations with different forms of the diffusion coefficient.
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24

Carrillo, J. A., M. G. Delgadino, and F. S. Patacchini. "Existence of ground states for aggregation-diffusion equations." Analysis and Applications 17, no. 03 (May 2019): 393–423. http://dx.doi.org/10.1142/s0219530518500276.

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We analyze free energy functionals for macroscopic models of multi-agent systems interacting via pairwise attractive forces and localized repulsion. The repulsion at the level of the continuous description is modeled by pressure-related terms in the functional making it energetically favorable to spread, while the attraction is modeled through nonlocal forces. We give conditions on general entropies and interaction potentials for which neither ground states nor local minimizers exist. We show that these results are sharp for homogeneous functionals with entropies leading to degenerate diffusions while they are not sharp for fast diffusions. The particular relevant case of linear diffusion is totally clarified giving a sharp condition on the interaction potential under which the corresponding free energy functional has ground states or not.
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25

Aziz, Imran, and Imran Khan. "Numerical Solution of Diffusion and Reaction–Diffusion Partial Integro-Differential Equations." International Journal of Computational Methods 15, no. 06 (September 2018): 1850047. http://dx.doi.org/10.1142/s0219876218500470.

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In this paper, a collocation method based on Haar wavelet is developed for numerical solution of diffusion and reaction–diffusion partial integro-differential equations. The equations are parabolic partial integro-differential equations and we consider both one-dimensional and two-dimensional cases. Such equations have applications in several practical problems including population dynamics. An important advantage of the proposed method is that it can be applied to both linear as well as nonlinear problems with slide modification. The proposed numerical method is validated by applying it to various benchmark problems from the existing literature. The numerical results confirm the accuracy, efficiency and robustness of the proposed method.
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26

Kordyumov, G. D. "DERIVATIVES IN THE MEAN OF RANDOM PROCESSES AND DIFFUSION MODELS IN ECONOMICS." Bulletin of the South Ural State University series "Mathematics. Mechanics. Physics" 13, no. 3 (2021): 26–30. http://dx.doi.org/10.14529/mmph210303.

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The article is devoted to diffusion models. The authors discuss the theoretical and methodological foundations of diffusion models in financial mathematics. Like the economic system, the modern world is developing rapidly. It seems impossible to predict what will happen tomorrow, how the emergence of new technologies will affect the market, and how changes in random factors will affect the product and the market as a whole. Diffusion models are one of the main methods for studying economic objects and processes. This is why it is so important to develop a diffusion model. The authors propose extending the applicability of the models by passing from Itô type stochastic equations to equations with so-called derivatives in the mean. For this, following E. Nelson, the authors introduce the concept of derivatives in the mean on the right and on the left. The equation with the derivative in the mean does not involve the Wiener process, therefore, it is not assumed in advance that the solution is diffusional. The article describes some well-known diffusion models, in which the transition from equations like an Itô type stochastic differential equation to equations satisfying a system of equations with derivatives in the mean leads to an expansion of the set of possible solutions. The authors also consider a generalization of geometric Brownian motion that satisfies a system of stochastic equations with derivatives in the mean and can cover a wider class of problems.
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27

Altınbaşak, Sevda Üsküplü. "Highly Oscillatory Diffusion-Type Equations." Journal of Computational Mathematics 31, no. 6 (June 2013): 549–72. http://dx.doi.org/10.4208/jcm.1307-m3955.

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28

Philibert, Jean. "Adolf Fick and Diffusion Equations." Defect and Diffusion Forum 249 (January 2006): 1–6. http://dx.doi.org/10.4028/www.scientific.net/ddf.249.1.

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29

Polyanin, A. D., A. I. Zhurov, and A. V. Vyazmin. "Time-Delayed Reaction-Diffusion Equations." Vestnik Tambovskogo gosudarstvennogo tehnicheskogo universiteta 21, no. 1 (2015): 071–77. http://dx.doi.org/10.17277/vestnik.2015.01.pp.071-077.

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30

INOUE, Akihiko. "Path integral for diffusion equations." Hokkaido Mathematical Journal 15, no. 1 (February 1986): 71–99. http://dx.doi.org/10.14492/hokmj/1381518221.

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31

Bakunin, O. G. "Diffusion equations and turbulent transport." Plasma Physics Reports 29, no. 11 (November 2003): 955–70. http://dx.doi.org/10.1134/1.1625992.

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32

Bocharov, G. A., V. A. Volpert, and A. L. Tasevich. "Reaction–Diffusion Equations in Immunology." Computational Mathematics and Mathematical Physics 58, no. 12 (December 2018): 1967–76. http://dx.doi.org/10.1134/s0965542518120059.

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33

Paripour, M., E. Babolian, and J. Saeidian. "Analytic solutions to diffusion equations." Mathematical and Computer Modelling 51, no. 5-6 (March 2010): 649–57. http://dx.doi.org/10.1016/j.mcm.2009.10.043.

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34

Anikin, V. M., Yu A. Barulina, and A. F. Goloubentsev. "Regression equations modelling diffusion processes." Applied Surface Science 215, no. 1-4 (June 2003): 185–90. http://dx.doi.org/10.1016/s0169-4332(03)00290-3.

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35

Tasevich, A., G. Bocharov, and V. Wolpert. "Reaction-diffusion equations in immunology." Журнал вычислительной математики и математической физики 58, no. 12 (December 2018): 2048–59. http://dx.doi.org/10.31857/s004446690003551-7.

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36

Ninomiya, Hirokazu. "Separatrices of competition-diffusion equations." Journal of Mathematics of Kyoto University 35, no. 3 (1995): 539–67. http://dx.doi.org/10.1215/kjm/1250518709.

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37

Cahn, John W., Shui-Nee Chow, and Erik S. Van Vleck. "Spatially Discrete Nonlinear Diffusion Equations." Rocky Mountain Journal of Mathematics 25, no. 1 (March 1995): 87–118. http://dx.doi.org/10.1216/rmjm/1181072270.

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38

Schneider, W. R., and W. Wyss. "Fractional diffusion and wave equations." Journal of Mathematical Physics 30, no. 1 (January 1989): 134–44. http://dx.doi.org/10.1063/1.528578.

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39

Matuszak, Daniel, and Marc D. Donohue. "Inversion of multicomponent diffusion equations." Chemical Engineering Science 60, no. 15 (August 2005): 4359–67. http://dx.doi.org/10.1016/j.ces.2005.02.071.

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40

Shah, Jayant. "Reaction–Diffusion Equations and Learning." Journal of Visual Communication and Image Representation 13, no. 1-2 (March 2002): 82–93. http://dx.doi.org/10.1006/jvci.2001.0478.

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41

Constantin, Peter. "Nonlocal nonlinear advection-diffusion equations." Chinese Annals of Mathematics, Series B 38, no. 1 (January 2017): 281–92. http://dx.doi.org/10.1007/s11401-016-1071-4.

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42

Li, Chenkuan, Reza Saadati, Safoura Rezaei Aderyani, and Min-Jie Luo. "On the Generalized Fractional Convection–Diffusion Equation with an Initial Condition in Rn." Fractal and Fractional 9, no. 6 (May 27, 2025): 347. https://doi.org/10.3390/fractalfract9060347.

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Time-fractional convection–diffusion equations are significant for their ability to model complex transport phenomena that deviate from classical behavior, with numerous applications in anomalous diffusion, memory effects, and nonlocality. This paper derives, for the first time, a unique series solution to a multiple time-fractional convection–diffusion equation with a non-homogenous source term, based on an inverse operator, a newly-constructed space, and the multivariate Mittag–Leffler function. Several illustrative examples are provided to show the power and simplicity of our main theorems in solving certain fractional convection–diffusions equations. Additionally, we compare these results with solutions obtained using the AI model DeepSeek-R1, highlighting the effectiveness and validity of our proposed methods and main theorems.
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43

Malinowski, Marek T. "Bipartite Fuzzy Stochastic Differential Equations with Global Lipschitz Condition." Mathematical Problems in Engineering 2016 (2016): 1–13. http://dx.doi.org/10.1155/2016/3830529.

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We introduce and analyze a new type of fuzzy stochastic differential equations. We consider equations with drift and diffusion terms occurring at both sides of equations. Therefore we call them the bipartite fuzzy stochastic differential equations. Under the Lipschitz and boundedness conditions imposed on drifts and diffusions coefficients we prove existence of a unique solution. Then, insensitivity of the solution under small changes of data of equation is examined. Finally, we mention that all results can be repeated for solutions to bipartite set-valued stochastic differential equations.
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44

Gautam, Pushpa Nidhi, Buddhi Prasad Sapkota, and Kedar Nath Uprety. "A brief review on the solutions of advection-diffusion equation." Scientific World 15, no. 15 (June 14, 2022): 4–9. http://dx.doi.org/10.3126/sw.v15i15.45668.

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In this work both linear and nonlinear advection-diffusion equations are considered and discussed their analytical solutions with different initial and boundary conditions. The work of Ogata and Banks, Harleman and Rumer, Cleary and Adrian, Atul Kumar et al., Mojtabi and Deville are reviewed for linear advection-diffusion equations and for nonlinear, we have chosen the work of Sakai and Kimura. Some enthusiastic functions used in the articles, drawbacks and applications of the results are discussed. Reduction of the advection-diffusion equations into diffusion equations make the governing equation solvable by using integral transform method for analytical solution. For nonlinear advection-diffusion equations, the Cole-Hopf transformation is used to reduce into the diffusion equation. Different dispersion phenomena in atmosphere, surface and subsurface area are outlined.
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45

Gomez, Francisco, Victor Morales, and Marco Taneco. "Analytical solution of the time fractional diffusion equation and fractional convection-diffusion equation." Revista Mexicana de Física 65, no. 1 (December 31, 2018): 82. http://dx.doi.org/10.31349/revmexfis.65.82.

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In this paper, we obtain analytical solutions for the time-fractional diffusion and time-fractional convection-diffusion equations. These equations are obtained from the standard equations by replacing the time derivative with a fractional derivative of order $\alpha$. Fractional operators of type Liouville-Caputo, Atangana-Baleanu-Caputo, fractional conformable derivative in Liouville-Caputo sense and Atangana-Koca-Caputo were used to model diffusion and convection-diffusion equation. The Laplace and Fourier transforms were applied to obtain the analytical solutions for the fractional order diffusion and convection-diffusion equations. The solutions obtained can be useful to understand the modeling of anomalous diffusive, subdiffusive systems and super-diffusive systems, transport processes, random walk and wave propagation phenomenon.
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46

Goto, Shin-itiro, and Hideitsu Hino. "Diffusion equations from master equations—A discrete geometric approach." Journal of Mathematical Physics 61, no. 11 (November 1, 2020): 113301. http://dx.doi.org/10.1063/5.0003656.

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47

Othmer, Hans G., and Thomas Hillen. "The Diffusion Limit of Transport Equations II: Chemotaxis Equations." SIAM Journal on Applied Mathematics 62, no. 4 (January 2002): 1222–50. http://dx.doi.org/10.1137/s0036139900382772.

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48

Bertrand, Nicolas, Jocelyn Sabatier, Olivier Briat, and Jean-Michel Vinassa. "An Implementation Solution for Fractional Partial Differential Equations." Mathematical Problems in Engineering 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/795651.

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The link between fractional differentiation and diffusion equation is used in this paper to propose a solution for the implementation of fractional diffusion equations. These equations permit us to take into account species anomalous diffusion at electrochemical interfaces, thus permitting an accurate modeling of batteries, ultracapacitors, and fuel cells. However, fractional diffusion equations are not addressed in most commercial software dedicated to partial differential equations simulation. The proposed solution is evaluated in an example.
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49

Xie, Longjie, and Li Yang. "Diffusion approximation for multi-scale stochastic reaction-diffusion equations." Journal of Differential Equations 300 (November 2021): 155–84. http://dx.doi.org/10.1016/j.jde.2021.07.039.

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50

Philip, J. R. "Some exact solutions of convection-diffusion and diffusion equations." Water Resources Research 30, no. 12 (December 1994): 3545–51. http://dx.doi.org/10.1029/94wr01329.

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