Journal articles: 'Diffusion equations'

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

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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3

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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4

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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5

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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6

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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7

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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8

Abrashina-Zhadaeva, Natali. "A SPLITTING TYPE ALGORITHM FOR NUMERICAL SOLUTION OF PDES OF FRACTIONAL ORDER." Mathematical Modelling and Analysis 12, no. 4 (December 31, 2007): 399–408. http://dx.doi.org/10.3846/1392-6292.2007.12.399-408.

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Fractional order diffusion equations are generalizations of classical diffusion equations, treating super‐diffusive flow processes. In this paper, we examine a splitting type numerical methods to solve a class of two‐dimensional initial‐boundary value fractional diffusive equations. Stability, consistency and convergence of the methods are investigated. It is shown that both schemes are unconditionally stable. A numerical example is presented.

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9

Scheel, Arnd, and Erik S. Van Vleck. "Lattice differential equations embedded into reaction–diffusion systems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 139, no. 1 (February 2009): 193–207. http://dx.doi.org/10.1017/s0308210507000248.

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We show that lattice dynamical systems naturally arise on infinite-dimensional invariant manifolds of reaction–diffusion equations with spatially periodic diffusive fluxes. The result connects wave-pinning phenomena in lattice differential equations and in reaction–diffusion equations in inhomogeneous media. The proof is based on a careful singular perturbation analysis of the linear part, where the infinite-dimensional manifold corresponds to an infinite-dimensional centre eigenspace.

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10

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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11

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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12

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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13

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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14

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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15

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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16

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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17

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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18

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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19

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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20

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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21

Guo, Jong-Shenq, and Yoshihisa Morita. "Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations." Discrete & Continuous Dynamical Systems - A 12, no. 2 (2005): 193–212. http://dx.doi.org/10.3934/dcds.2005.12.193.

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22

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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23

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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24

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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25

Nec, Y., and A. A. Nepomnyashchy. "Amplitude equations for a sub-diffusive reaction–diffusion system." Journal of Physics A: Mathematical and Theoretical 41, no. 38 (August 18, 2008): 385101. http://dx.doi.org/10.1088/1751-8113/41/38/385101.

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26

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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27

Ban, H., S. Venkatesh, and K. Saito. "Convection-Diffusion Controlled Laminar Micro Flames." Journal of Heat Transfer 116, no. 4 (November 1, 1994): 954–59. http://dx.doi.org/10.1115/1.2911471.

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Small laminar diffusion flames (flame height ≃2–3 mm) established by a fuel jet issuing into a quiescent medium are investigated. It was found that for these flames buoyancy effects disappeared as the flame size decreased (Fr≫1), and diffusive transport of the fuel was comparable to the convective transport of the fuel. The effect of buoyancy on these flames was studied by examining the flame shape for horizontally oriented burners. A phenomenological model was developed (based on experimentally determined flame shapes) to compare diffusion and convection transport effects. Finally, the flame shapes were theoretically determined by solving the conservation equations using similarity methods. It was seen that when the axial diffusion (in momentum and species equations) terms are included in the conservation equations, the calculated flame shape is in better agreement (as compared to without the axial diffusion term) with the experimentally measured flame shape.

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28

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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29

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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30

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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31

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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32

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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33

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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34

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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35

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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36

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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37

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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38

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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39

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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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

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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43

Su, Lijuan, and Pei Cheng. "A High-Accuracy MOC/FD Method for Solving Fractional Advection-Diffusion Equations." Journal of Applied Mathematics 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/648595.

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Fractional-order diffusion equations are viewed as generalizations of classical diffusion equations, treating super-diffusive flow processes. In this paper, in order to solve the fractional advection-diffusion equation, the fractional characteristic finite difference method is presented, which is based on the method of characteristics (MOC) and fractional finite difference (FD) procedures. The stability, consistency, convergence, and error estimate of the method are obtained. An example is also given to illustrate the applicability of theoretical results.

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44

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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45

Nec, Y., A. A. Nepomnyashchy, and A. A. Golovin. "Oscillatory instability in super-diffusive reaction – diffusion systems: Fractional amplitude and phase diffusion equations." EPL (Europhysics Letters) 82, no. 5 (May 27, 2008): 58003. http://dx.doi.org/10.1209/0295-5075/82/58003.

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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

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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49

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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50

Le Roux, Marie-Noëlle. "Numerical solution of fast diffusion or slow diffusion equations." Journal of Computational and Applied Mathematics 97, no. 1-2 (September 1998): 121–36. http://dx.doi.org/10.1016/s0377-0427(98)00106-x.

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Journal articles on the topic 'Diffusion equations'

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