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Journal articles on the topic 'Advection-diffusion equation'

Author: Grafiati

Published: 4 June 2021

Last updated: 28 January 2023

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1

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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2

Hadadian Nejad Yousefi, Mohsen, Seyed Hossein Ghoreishi Najafabadi, and Emran Tohidi. "A new spectral integral equation method for solving two-dimensional unsteady advection-diffusion equations via Chebyshev polynomials." Engineering Computations 36, no. 7 (August 12, 2019): 2327–68. http://dx.doi.org/10.1108/ec-02-2018-0063.

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Purpose The purpose of this paper is to develop an efficient and reliable spectral integral equation method for solving two-dimensional unsteady advection-diffusion equations. Design/methodology/approach In this study, the considered two-dimensional unsteady advection-diffusion equations are transformed into the equivalent partial integro-differential equations via integrating from the considered unsteady advection-diffusion equation. After this stage, by using Chebyshev polynomials of the first kind and the operational matrix of integration, the integral equation would be transformed into the system of linear algebraic equations. Robustness and efficiency of the proposed method were illustrated by six numerical simulations experimentally. The numerical results confirm that the method is efficient, highly accurate, fast and stable for solving two-dimensional unsteady advection-diffusion equations. Findings The proposed method can solve the equations with discontinuity near the boundaries, the advection-dominated equations and the equations in irregular domains. One of the numerical test problems designed specially to evaluate the performance of the proposed method for discontinuity near boundaries. Originality/value This study extends the intention of one dimensional Chebyshev approximate approaches (Yuksel and Sezer, 2013; Yuksel et al., 2015) for two-dimensional unsteady advection-diffusion problems and the basic intention of our suggested method is quite different from the approaches for hyperbolic problems (Bulbul and Sezer, 2011).

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3

Sene, Ndolane, and Karima Abdelmalek. "Nonlinear sub-diffusion and nonlinear sub-diffusion dispersion equations and their proposed solutions." Applied Mathematics and Nonlinear Sciences 5, no. 1 (March 31, 2020): 221–36. http://dx.doi.org/10.2478/amns.2020.1.00020.

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AbstractMany investigations related to the analytical solutions of the nonlinear sub-diffusion equation exist. In this paper, we investigate the conditions under which the analytical and the approximate solutions of the nonlinear sub-diffusion equation and the nonlinear sub-advection dispersion equation exist. In other words, the problems of existence and uniqueness of the solutions the fractional diffusion equations have been addressed. We use the Banach fixed Theorem. After proving the existence and uniqueness, we propose the analytical and the approximate solutions of the nonlinear sub-diffusion, and the nonlinear sub-advection dispersion equations. We analyze the impact of the sub-diffusion coefficient, the advection coefficient and the dispersion coefficient in the diffusion processes. The homotopy perturbation Laplace transform method has been used in this paper. Some numerical examples are provided to illustrate the main results of the article.

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4

Chang Fu-Xuan, Chen Jin, and Huang Wei. "Anomalous diffusion and fractional advection-diffusion equation." Acta Physica Sinica 54, no. 3 (2005): 1113. http://dx.doi.org/10.7498/aps.54.1113.

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5

Avci, Derya, Eroğlu İskender, and Necati Özdemir. "The Dirichlet problem of a conformable advection-diffusion equation." Thermal Science 21, no. 1 Part A (2017): 9–18. http://dx.doi.org/10.2298/tsci160421235a.

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The fractional advection-diffusion equations are obtained from a fractional power law for the matter flux. Diffusion processes in special types of porous media which has fractal geometry can be modelled accurately by using these equations. However, the existing nonlocal fractional derivatives seem complicated and also lose some basic properties satisfied by usual derivatives. For these reasons, local fractional calculus has recently been emerged to simplify the complexities of fractional models defined by nonlocal fractional operators. In this work, the conformable, a local, well-behaved and limit-based definition, is used to obtain a local generalized form of advection-diffusion equation. In addition, this study is devoted to give a local generalized description to the combination of diffusive flux governed by Fick?s law and the advection flux associated with the velocity field. As a result, the constitutive conformable advection-diffusion equation can be easily achieved. A Dirichlet problem for conformable advection-diffusion equation is derived by applying fractional Laplace transform with respect to time t and finite sin-Fourier transform with respect to spatial coordinate x. Two illustrative examples are presented to show the behaviours of this new local generalized model. The dependence of the solution on the fractional order of conformable derivative and the changing values of problem parameters are validated using graphics held by MATLcodes.

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6

Wu, Jiankang. "Wave equation model for solving advection-diffusion equation." International Journal for Numerical Methods in Engineering 37, no. 16 (August 30, 1994): 2717–33. http://dx.doi.org/10.1002/nme.1620371603.

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7

Dhawan, S., and S. Kapoor. "Numerical simulation of advection-diffusion equation." International Journal of Mathematical Modelling and Numerical Optimisation 2, no. 1 (2011): 13. http://dx.doi.org/10.1504/ijmmno.2011.037197.

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8

Grant, John, and Michael Wilkinson. "Advection–Diffusion Equation with Absorbing Boundary." Journal of Statistical Physics 160, no. 3 (April 30, 2015): 622–35. http://dx.doi.org/10.1007/s10955-015-1257-2.

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9

CANUTO, C., and A. RUSSO. "ON THE ELLIPTIC-HYPERBOLIC COUPLING I: THE ADVECTION-DIFFUSION EQUATION VIA THE χ-FORMULATION." Mathematical Models and Methods in Applied Sciences 03, no. 02 (April 1993): 145–70. http://dx.doi.org/10.1142/s0218202593000096.

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An advection-diffusion equation is considered, for which the solution is advection-dominated in most of the domain. A domain decomposition method based on a self-adaptive, smooth coupling of the reduced advection equation and the full advection-diffusion equation is proposed. The convergence of an iteration-by-subdomain method is investigated.

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10

Appadu, Appanah Rao. "Performance of UPFD scheme under some different regimes of advection, diffusion and reaction." International Journal of Numerical Methods for Heat & Fluid Flow 27, no. 7 (July 3, 2017): 1412–29. http://dx.doi.org/10.1108/hff-01-2016-0038.

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Purpose An unconditionally positive definite finite difference scheme termed as UPFD has been derived to approximate a linear advection-diffusion-reaction equation which models exponential travelling waves and the coefficients of advection, diffusion and reactive terms have been chosen as one (Chen-Charpentier and Kojouharov, 2013). In this work, the author tests UPFD scheme under some other different regimes of advection, diffusion and reaction. The author considers the case when the coefficient of advection, diffusion and reaction are all equal to one and also cases under which advection or diffusion or reaction is more important. Some errors such as L1 error, dispersion, dissipation errors and relative errors are tabulated. Moreover, the author compares some spectral properties of the method under different regimes. The author obtains the variation of the following quantities with respect to the phase angle: modulus of exact amplification factor, modulus of amplification factor of the scheme and relative phase error. Design/methodology/approach Difficulties can arise in stability analysis. It is important to have a full understanding of whether the conditions obtained for stability are sufficient, necessary or necessary and sufficient. The advection-diffusion-reaction is quite similar to the advection-diffusion equation, it has an extra reaction term and therefore obtaining stability of numerical methods discretizing advection-diffusion-reaction equation is not easy as is the case with numerical methods discretizing advection-diffusion equations. To avoid difficulty involved with obtaining region of stability, the author shall consider unconditionally stable finite difference schemes discretizing advection-diffusion-reaction equations. Findings The UPFD scheme is unconditionally stable but not unconditionally consistent. The scheme was tested on an advection-diffusion-reaction equation which models exponential travelling waves, and the author computed various errors such as L1 error, dispersion and dissipation errors, relative errors under some different regimes of advection, diffusion and reaction. The scheme works best for very small values of k as k → 0 (for instance, k = 0.00025, 0.0005) and performs satisfactorily at other values of k such as 0.001 for two regimes; a = 1, D = 1, κ = 1 and a = 1, D = 1, κ = 5. When a = 5, D = 1, κ = 1, the scheme performs quite well at k = 0.00025 and satisfactorily at k = 0.0005 but is not efficient at larger values of k. For the diffusive case (a = 1, D = 5, κ = 1), the scheme does not perform well. In general, the author can conclude that the choice of k is very important, as it affects to a great extent the performance of the method. Originality/value The UPFD scheme is effective to solve advection-diffusion-reaction problems when advection or reactive regime is dominant and for the case, a = 1, D = 1, κ = 1, especially at low values of k. Moreover, the magnitude of the dispersion and dissipation errors using UPFD are of the same order for all the four regimes considered as seen from Tables 1 to 4. This indicates that if the author is to optimize the temporal step size at a given value of the spatial step size, the optimization function must consist of both the AFM and RPE. Some related work on optimization can be seen in Appadu (2013). Higher-order unconditionally stable schemes can be constructed for the regimes for which UPFD is not efficient enough for instance when advection and diffusion are dominant.

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11

Sari, Murat, and Huseyin Tunc. "Finite element based hybrid techniques for advection-diffusion-reaction processes." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 8, no. 2 (February 4, 2018): 127–36. http://dx.doi.org/10.11121/ijocta.01.2018.00452.

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In this paper, numerical solutions of the advection-diffusion-reaction (ADR) equation are investigated using the Galerkin, collocation and Taylor-Galerkin cubic B-spline finite element method in strong form of spatial elements using an ?-family optimization approach for time variation. The main objective of this article is to capture effective results of the finite element techniques with B-spline basis functions under the consideration of the ADR processes. All produced results are compared with the exact solution and the literature for various versions of problems including pure advection, pure diffusion, advection-diffusion, and advection-diffusion-reaction equations. It is proved that the present methods have good agreement with the exact solution and the literature.

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12

Alikakos, Nicholas D., Peter W. Bates, and Christopher P. Grant. "Blow up for a diffusion-advection equation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 113, no. 3-4 (1989): 181–90. http://dx.doi.org/10.1017/s0308210500024057.

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SynopsisThese results describe the asymptotic behaviour of solutions to a certain non-linear diffusionadvection equation on the unit interval. The “no flux” boundary conditions prescribed result in mass being conserved by solutions and the existence of a mass-parametrised family of equilibria. A natural question is whether or not solutions stabilise to equilibria and if not, whether they blow up in finite time. Here it is shown that for non-linearities which characterise “fast association” there is a criticalmass such that initial data which have supercritical mass must lead to blow up in finite time. It is also shown that there exist initial data with arbitrarily small mass which also lead to blow up in finite time.

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13

Dağ, İdris, Aynur Canivar, and Ali Şahin. "Taylor‐Galerkin method for advection‐diffusion equation." Kybernetes 40, no. 5/6 (June 14, 2011): 762–77. http://dx.doi.org/10.1108/03684921111142304.

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14

Heinrichs, Wilhelm. "Defect correction for the advection-diffusion equation." Computer Methods in Applied Mechanics and Engineering 119, no. 3-4 (December 1994): 191–97. http://dx.doi.org/10.1016/0045-7825(94)90088-4.

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15

Sun, Yubiao, Amitesh S. Jayaraman, and Gregory S. Chirikjian. "Approximate solutions of the advection–diffusion equation for spatially variable flows." Physics of Fluids 34, no. 3 (March 2022): 033318. http://dx.doi.org/10.1063/5.0084789.

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The advection–diffusion equation (ADE) describes many important processes in hydrogeology, mechanics, geology, and biology. The equations model the transport of a passive scalar quantity in a flow. In this paper, we have developed a new approach to solve incompressible advection–diffusion equations (ADEs) with variable convective terms, which are essential to study species transport in various flow scenarios. We first reinterpret advection diffusion equations on a microscopic level and obtain stochastic differential equations governing the behavior of individual particles of the species transported by the flow. Then, simplified versions of ADEs are derived to approximate the original ADEs governing concentration evolution of species. The approximation is effectively a linearization of the spatially varying coefficient of the advective term. These simplified equations are solved analytically using the Fourier transform. We have validated this new method by comparing our results to solutions obtained from the canonical stochastic sampling method and the finite element method. This mesh-free algorithm achieves comparable accuracy to the results from discrete stochastic simulation of spatially resolved species transport in a Lagrangian frame of reference. The good consistency shows that our proposed method is efficient in simulating chemical transport in a convective flow. The proposed method is computationally efficient and quantitatively reliable, providing an alternative technique to investigate various advection–diffusion processes.

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16

Kaur, Navjot, and Kavita Goyal. "Hybrid Hermite polynomial chaos SBP-SAT technique for stochastic advection-diffusion equations." International Journal of Modern Physics C 31, no. 09 (August 8, 2020): 2050128. http://dx.doi.org/10.1142/s0129183120501284.

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The study of advection–diffusion equation has relatively became an active research topic in the field of uncertainty quantification (UQ) due to its numerous real life applications. In this paper, Hermite polynomial chaos is united with summation-by-parts (SBP) – simultaneous approximation terms (SATs) technique to solve the advection–diffusion equations with random Dirichlet boundary conditions (BCs). Polynomial chaos expansion (PCE) with Hermite basis is employed to separate the randomness, then SBP operators are used to approximate the differential operators and SATs are used to enforce BCs by ensuring the stability. For each chaos coefficient, time integration is performed using Runge–Kutta method of fourth order (RK4). Statistical moments namely mean and variance are computed using polynomial chaos coefficients without any extra computational effort. The method is applied on three test problems for validation. The first two test problems are stochastic advection equations on [Formula: see text] without any boundary and third problem is stochastic advection–diffusion equation on [0,2] with Dirichlet BCs. In case of third problem, we have obtained a range of permissible parameters for a stable numerical solution.

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17

Lorimer, Shelley, Ryan Boehnke, and Brigida Meza. "Mass Transfer Behaviour in Hybrid Solvent Oil Recovery Processes." Defect and Diffusion Forum 367 (April 2016): 77–85. http://dx.doi.org/10.4028/www.scientific.net/ddf.367.77.

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The mechanisms of mass transfer in thermal solvent enhanced oil recovery processes and the influence of grid size in the numerical simulation of these processes is not well understood [1, 2]. The literature has indicated that, experimentally, solvent fronts progress more rapidly that what can be predicted using current approximations [3]. It has also been shown that under certain modelling conditions with coarser grid meshes, the influence of numerical errors can be substantial. The equations that govern thermal/solvent multiphase flow through porous media are extremely complex and it is very difficult to decouple the contribution of the mass transfer mechanisms from the thermal effects. This paper was written to increase the understanding of the mass transfer mechanisms in hybrid thermal solvent recovery processes through sensitivity study using a numerical solution of the linear one dimensional advection diffusion/dispersion (ADD) equation. This equation was modeled using finite difference methods. The effects of grid size were examined to verify the use of this method, and the results were then used to examine the sensitivity of the equation to the parameters that govern the mass transfer mechanisms (advection velocity, diffusion and dispersion coefficients).In particular, a range of values for diffusion and dispersion coefficients were selected for the sensitivity study, and one advection velocity. These values were then used to numerically solve the ADD equation to assess the impact of each mechanism (advection, diffusion and dispersion) and their contribution to the movement of the solvent front. The parameters chosen for this study were based on values obtained from the literature for advection velocity, diffusion and dispersion coefficients consistent with a gravity drainage thermal/solvent oil recovery process. This sensitivity study has indicated that all three mechanisms (advection, diffusion and dispersion) must be included to have the solvent front progress at rates that are consistent with experimental solvent front advance rates published in the literature to date [1]. This result suggests that diffusion alone cannot account for the movement of the solvent front within the ranges of values that have been studied.

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18

Gurarslan, Gurhan. "Sixth-Order Combined Compact Finite Difference Scheme for the Numerical Solution of One-Dimensional Advection-Diffusion Equation with Variable Parameters." Mathematics 9, no. 9 (May 1, 2021): 1027. http://dx.doi.org/10.3390/math9091027.

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A high-accuracy numerical method based on a sixth-order combined compact difference scheme and the method of lines approach is proposed for the advection–diffusion transport equation with variable parameters. In this approach, the partial differential equation representing the advection-diffusion equation is converted into many ordinary differential equations. These time-dependent ordinary differential equations are then solved using an explicit fourth order Runge–Kutta method. Three test problems are studied to demonstrate the accuracy of the present methods. Numerical solutions obtained by the proposed method are compared with the analytical solutions and the available numerical solutions given in the literature. In addition to requiring less CPU time, the proposed method produces more accurate and more stable results than the numerical methods given in the literature.

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19

Huang, F., and F. Liu. "The time fractional diffusion equation and the advection-dispersion equation." ANZIAM Journal 46, no. 3 (January 2005): 317–30. http://dx.doi.org/10.1017/s1446181100008282.

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AbstractThe time fractional diffusion equation with appropriate initial and boundary conditions in an n-dimensional whole-space and half-space is considered. Its solution has been obtained in terms of Green functions by Schneider and Wyss. For the problem in whole-space, an explicit representation of the Green functions can also be obtained. However, an explicit representation of the Green functions for the problem in half-space is difficult to determine, except in the special cases α = 1 with arbitrary n, or n = 1 with arbitrary α. In this paper, we solve these problems. By investigating the explicit relationship between the Green functions of the problem with initial conditions in whole-space and that of the same problem with initial and boundary conditions in half-space, an explicit expression for the Green functions corresponding to the latter can be derived in terms of Fox functions. We also extend some results of Liu, Anh, Turner and Zhuang concerning the advection-dispersion equation and obtain its solution in half-space and in a bounded space domain.

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20

Andallah, LS, and MR Khatun. "Numerical solution of advection-diffusion equation using finite difference schemes." Bangladesh Journal of Scientific and Industrial Research 55, no. 1 (April 21, 2020): 15–22. http://dx.doi.org/10.3329/bjsir.v55i1.46728.

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This paper presents numerical simulation of one-dimensional advection-diffusion equation. We study the analytical solution of advection diffusion equation as an initial value problem in infinite space and realize the qualitative behavior of the solution in terms of advection and diffusion co-efficient. We obtain the numerical solution of this equation by using explicit centered difference scheme and Crank-Nicolson scheme for prescribed initial and boundary data. We implement the numerical scheme by developing a computer programming code and present the stability analysis of Crank-Nicolson scheme for ADE. For the validity test, we perform error estimation of the numerical scheme and presented the numerical features of rate of convergence graphically. The qualitative behavior of the ADE for different choice of the advection and diffusion co-efficient is verified. Finally, we estimate the pollutant in a river at different times and different points by using these numerical scheme. Bangladesh J. Sci. Ind. Res.55(1), 15-22, 2020

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21

Hwang, Sooncheol, Sangyoung Son, and Patrick J. Lynett. "A GPU-ACCELERATED MODELING OF SCALAR TRANSPORT BASED ON BOUSSINESQ-TYPE EQUATIONS." Coastal Engineering Proceedings, no. 36v (December 28, 2020): 11. http://dx.doi.org/10.9753/icce.v36v.waves.11.

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This paper describes a two-dimensional scalar transport model solving advection-diffusion equation based on GPU-accelerated Boussinesq model called Celeris. Celeris is the firstly-developed Boussinesq-type model that is equipped with an interactive system between user and computing unit. Celeris provides greatly advantageous user-interface that one can change not only water level, topography but also model parameters while the simulation is running. In this study, an advection-diffusion equation for scalar transport was coupled with extended Boussinesq equations to simulate scalar transport in the nearshore.Recorded Presentation from the vICCE (YouTube Link): https://youtu.be/aHvMmdz3wps

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22

Szymkiewicz, Romuald, and Dariusz Gąsiorowski. "Adaptive method for the solution of 1D and 2D advection–diffusion equations used in environmental engineering." Journal of Hydroinformatics 23, no. 6 (September 30, 2021): 1290–311. http://dx.doi.org/10.2166/hydro.2021.062.

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Abstract The paper concerns the numerical solution of one-dimensional (1D) and two-dimensional (2D) advection–diffusion equations. For the numerical solution of the 1D advection–diffusion equation, a method, originally proposed for the solution of the 1D pure advection equation, has been developed. A modified equation analysis carried out for the proposed method allowed increasing of the resulting solution accuracy and, consequently, to reduce the numerical dissipation and dispersion. This is achieved by proper choice of the involved weighting parameter being a function of the Courant number and the diffusive number. The method is adaptive because for uniform grid point and for uniform flow velocity, the weighting parameter takes a constant value, whereas for non-uniform grid and for varying flow velocity, its value varies in the region of solution. For the solution of the 2D transport equation, the dimensional decomposition in the form of Strang splitting technique is used. Consequently, this equation is reduced to a series of the 1D equations with regard to x- and y-directions which next are solved using the aforementioned method. The results of computational experiments compared with the exact solutions confirmed that the proposed approaches ensure high solution accuracy of the transport equations.

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23

Kusuma, Jeffry, Agustinus Ribal, and Andi Galsan Mahie. "On FTCS Approach for Box Model of Three-Dimension Advection-Diffusion Equation." International Journal of Differential Equations 2018 (November 1, 2018): 1–9. http://dx.doi.org/10.1155/2018/7597861.

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This paper describes a numerical solution for mathematical model of the transport equation in a simple rectangular box domain. The model of street tunnel pollution distribution using two-dimension advection and three-dimension diffusion is solved numerically. Because of the nature of the problem, the model is extended to become three-dimension advection and three-dimension diffusion to study the sea-sand mining pollution distribution. This model with various advection and diffusion parameters and the boundaries conditions is also solved numerically using a finite difference (FTCS) method.

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24

Mahmoud, Elsayed I., and Temirkhan S. Aleroev. "Boundary Value Problem of Space-Time Fractional Advection Diffusion Equation." Mathematics 10, no. 17 (September 2, 2022): 3160. http://dx.doi.org/10.3390/math10173160.

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In this article, the analytical and numerical solution of a one-dimensional space-time fractional advection diffusion equation is presented. The separation of variables method is used to carry out the analytical solution, the basis of the system eigenfunction and their corresponding eigenvalue for basic equation is determined, and the numerical solution is based on constructing the Crank-Nicolson finite difference scheme of the equivalent partial integro-differential equations. The convergence and unconditional stability of the solution are investigated. Finally, the numerical and analytical experiments are given to verify the theoretical analysis.

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Essa, Khaled S. M., Sawsan E. M. Elsaid, and Fawzia Mubarak. "Time Dependent Advection Diffusion Equation in Two Dimensions." Journal of Atmosphere 1, no. 1 (2015): 8–16. http://dx.doi.org/10.18488/journal.94/2015.1.1/94.1.8.16.

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26

Taigbenu, Akpofure, and James A. Liggett. "An Integral Solution for the Diffusion-Advection Equation." Water Resources Research 22, no. 8 (August 1986): 1237–46. http://dx.doi.org/10.1029/wr022i008p01237.

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27

WRIGHT, DANIEL G. "Finite difference approximations to the advection-diffusion equation." Tellus A 44, no. 3 (May 1992): 261–69. http://dx.doi.org/10.1034/j.1600-0870.1992.t01-2-00005.x.

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28

de Loubens, R., and T. S. Ramakrishnan. "Asymptotic solution of a nonlinear advection-diffusion equation." Quarterly of Applied Mathematics 69, no. 2 (March 10, 2011): 389–401. http://dx.doi.org/10.1090/s0033-569x-2011-01214-x.

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29

Tang, X. Z., and A. H. Boozer. "Finite time Lyapunov exponent and advection-diffusion equation." Physica D: Nonlinear Phenomena 95, no. 3-4 (September 1996): 283–305. http://dx.doi.org/10.1016/0167-2789(96)00064-4.

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Wright, Daniel G. "Finite difference approximations to the advection-diffusion equation." Tellus A: Dynamic Meteorology and Oceanography 44, no. 3 (January 1992): 261–69. http://dx.doi.org/10.3402/tellusa.v44i3.14958.

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31

Su, Fang, Zhan Xu, Jun-Zhi Cui, Xin-Peng Du, and Hao Jiang. "Multiscale computation method for an advection–diffusion equation." Applied Mathematics and Computation 218, no. 14 (March 2012): 7369–74. http://dx.doi.org/10.1016/j.amc.2011.12.001.

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32

Ginting, Victor, and Yulong Li. "On the fractional diffusion-advection-reaction equation in ℝ." Fractional Calculus and Applied Analysis 22, no. 4 (August 27, 2019): 1039–62. http://dx.doi.org/10.1515/fca-2019-0055.

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Abstract We present an analysis of existence, uniqueness, and smoothness of the solution to a class of fractional ordinary differential equations posed on the whole real line that models a steady state behavior of a certain anomalous diffusion, advection, and reaction. The anomalous diffusion is modeled by the fractional Riemann-Liouville differential operators. The strong solution of the equation is sought in a Sobolev space defined by means of Fourier Transform. The key component of the analysis hinges on a characterization of this Sobolev space with the Riemann-Liouville derivatives that are understood in a weak sense. The existence, uniqueness, and smoothness of the solution is demonstrated with the assistance of several tools from functional and harmonic analyses.

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Tran, Dung Anh, Hang Thi Chu, and Long Ta Bui. "Application of the Bessel function to compute the air pollutant with the stratification of the atmospheric." Science and Technology Development Journal 18, no. 2 (June 30, 2015): 14–20. http://dx.doi.org/10.32508/stdj.v18i2.1067.

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The Bessel differential equation with the Bessel function of solution has been applied. Bessel functions are the canonical solutions of Bessel's differential equation. Bessel's equation arises when finding separable solutions to Laplace's equation in cylindrical or spherical coordinates. Bessel functions are important for many problems of advection–diffusion progress and wave propagation. In this paper, authors present the analytic solutions of the atmospheric advection-diffusion equation with the stratification of the boundary condition. The solution has been found by applied the separation of variable method and Bessel’s equation.

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FA, KWOK SAU, and K. G. WANG. "INTEGRO-DIFFERENTIAL EQUATIONS ASSOCIATED WITH CONTINUOUS-TIME RANDOM WALK." International Journal of Modern Physics B 27, no. 12 (April 29, 2013): 1330006. http://dx.doi.org/10.1142/s0217979213300065.

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The continuous-time random walk (CTRW) model is a useful tool for the description of diffusion in nonequilibrium systems, which is broadly applied in nature and life sciences, e.g., from biophysics to geosciences. In particular, the integro-differential equations for diffusion and diffusion-advection are derived asymptotically from the decoupled CTRW model and a generalized Chapmann–Kolmogorov equation, with generic waiting time probability density function (PDF) and external force. The advantage of the integro-differential equations is that they can be used to investigate the entire diffusion process i.e., covering initial-, intermediate- and long-time ranges of the process. Therefore, this method can distinguish the evolution detail for a system having the same behavior in the long-time limit but with different initial- and intermediate-time behaviors. An integro-differential equation for diffusion-advection is also presented for the description of the subdiffusive and superdiffusive regime. Moreover, the methods of solving the integro-differential equations are developed, and the analytic solutions for PDFs are obtained for the cases of force-free and linear force.

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Alkhasawneh, Raed Ali. "A stage-structured delayed advection reaction-diffusion model for single species." International Journal of Electrical and Computer Engineering (IJECE) 10, no. 6 (December 1, 2020): 6260. http://dx.doi.org/10.11591/ijece.v10i6.pp6260-6267.

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In this paper, we derived a delay advection reaction-diffusion equation with linear advection term from a stage-structured model, then the derived equation is used under the homogeneous Dirichlet boundary conditions u_m (0,t)=0, u_m (L,t)=0, and the initial condition u_m (x,0)=u_m^0 (x)>0,x∈[-τ,0] with u_m^0 (0)>0 in order to find the minimum value of domain L that prevents extinction of the species under the effect of advection reaction diffusion equation. Finally, for the measurement the time lengths from birth to the development of the species population, time delays are integrated.

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Yan, Sheng, Zhili Zou, and Zaijin You. "Eulerian Description of Wave-Induced Stokes Drift Effect on Tracer Transport." Journal of Marine Science and Engineering 10, no. 2 (February 12, 2022): 253. http://dx.doi.org/10.3390/jmse10020253.

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The wave-induced Stokes drift plays a significant role on mass/tracer transport in the ocean and the evolution of coastal morphology. The tracer advection diffusion equation needs to be modified for Eulerian ocean models to properly account for the surface wave effects. The Eulerian description of Stokes drift effect on the tracer transport is derived in this study to show that this effect can be accounted for automatically in the wave-averaged advection-diffusion equation. The advection term in this equation is the wave-averaged concentration flux produced by the interaction between fluctuations of linear wave orbital velocity and tracer concentration, and the advection velocity is the same as the Stokes drift velocity. Thus, the effective dispersion of tracers by surface gravity waves is calculated due to the Stokes drift effect and the corresponding dispersion coefficient in the depth-integrated equation is then derived. The Eulerian description of Stokes drift effect of tracer concentration is illustrated by the direct numerical simulation of the advection–diffusion equation under simple linear waves. The equivalence between both the Eulerian and Lagrangian descriptions is also verified by particle tracking method. The theoretical analysis is found to agree well with the wave-induced dye drift velocity observed outside the surf zone in a longshore current experiment.

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Meng, Zhijuan, Xiaofei Chi, and Lidong Ma. "A Hybrid Interpolating Meshless Method for 3D Advection–Diffusion Problems." Mathematics 10, no. 13 (June 27, 2022): 2244. http://dx.doi.org/10.3390/math10132244.

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A hybrid interpolating meshless (HIM) method is established for dealing with three-dimensional (3D) advection–diffusion equations. To improve computational efficiency, a 3D equation is changed into correlative two-dimensional (2D) equations. The improved interpolating moving least-squares (IIMLS) method is applied in 2D subdomains to obtain the required approximation function with interpolation property. The finite difference method (FDM) is utilized in time domain and the splitting direction. Setting diagonal elements to one in the coefficient matrix is chosen to directly impose Dirichlet boundary conditions. Using the HIM method, difficulties created by the singularity of the weight functions, such as truncation error and calculation inconvenience, are overcome. To prove the advantages of the new method, some advection–diffusion equations are selected and solved by HIM, dimension splitting element-free Galerkin (DSEFG), and improved element-free Galerkin (IEFG) methods. Comparing and analyzing the calculation results of the three methods, it can be shown that the HIM method effectively improves computation speed and precision. In addition, the effectiveness of the HIM method in the nonlinear problem is verified by solving a 3D Richards’ equation.

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Hosseininia, M., M. H. Heydari, Z. Avazzadeh, and F. M. Maalek Ghaini. "Two-Dimensional Legendre Wavelets for Solving Variable-Order Fractional Nonlinear Advection-Diffusion Equation with Variable Coefficients." International Journal of Nonlinear Sciences and Numerical Simulation 19, no. 7-8 (December 19, 2018): 793–802. http://dx.doi.org/10.1515/ijnsns-2018-0168.

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AbstractThis article studies a numerical scheme for solving two-dimensional variable-order time fractional nonlinear advection-diffusion equation with variable coefficients, where the variable-order fractional derivative is in the Caputo type. The main idea is expanding the solution in terms of the 2D Legendre wavelets (2D LWs) where the variable-order time fractional derivative is discretized. We describe the method using the matrix operators and then implement it for solving various types of fractional advection-diffusion equations. The experimental results show the computational efficiency of the new approach.

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Leite, Jefferson, Moiseis Cecconello, Jackellyne Leite, and R. C. Bassanezi. "On Fuzzy Solutions for Diffusion Equation." Journal of Applied Mathematics 2015 (2015): 1–10. http://dx.doi.org/10.1155/2015/874931.

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Our main goal is to define a fuzzy solution for problems involving diffusion. To this end, the solution of fuzzy diffusion-reaction-advection equation will be defined as Zadeh’s extension of deterministic solution of the associated problem. Important aspects such as unity and stability of these solutions will also be studied. Graphical representations of these solutions will be presented.

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Gómez, Francisco, Enrique Escalante, Celia Calderón, Luis Morales, Mario González, and Rodrigo Laguna. "Analytical solutions for the fractional diffusion-advection equation describing super-diffusion." Open Physics 14, no. 1 (January 1, 2016): 668–75. http://dx.doi.org/10.1515/phys-2016-0074.

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AbstractThis paper presents the alternative construction of the diffusion-advection equation in the range (1; 2). The fractional derivative of the Liouville-Caputo type is applied. Analytical solutions are obtained in terms of Mittag-Leffler functions. In the range (1; 2) the concentration exhibits the superdiffusion phenomena and when the order of the derivative is equal to 2 ballistic diffusion can be observed, these behaviors occur in many physical systems such as semiconductors, quantum optics, or turbulent diffusion. This mathematical representation can be applied in the description of anomalous complex processes.

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Çenesiz, Y., and A. Kurt. "New Fractional Complex Transform for Conformable Fractional Partial Differential Equations." Journal of Applied Mathematics, Statistics and Informatics 12, no. 2 (December 1, 2016): 41–47. http://dx.doi.org/10.1515/jamsi-2016-0007.

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Abstract Conformable fractional complex transform is introduced in this paper for converting fractional partial differential equations to ordinary differential equations. Hence analytical methods in advanced calculus can be used to solve these equations. Conformable fractional complex transform is implemented to fractional partial differential equations such as space fractional advection diffusion equation and space fractional telegraph equation to obtain the exact solutions of these equations.

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Loeck, Jaqueline Fischer, Juliana Schramm, and Bardo Bodmann. "Modelo para dispersão de poluentes com condições de contorno reflexivas e simulação de dados no CALPUFF." Ciência e Natura 40 (April 18, 2018): 257. http://dx.doi.org/10.5902/2179460x30764.

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The present work is an attempt to simulate the pollutants dispersion in the surroundings of the thermoelectric power plant located in Linhares from a new mathematical model based on reflective boundaries in the deterministic advection-diffusion equation. In addition to the advection-diffusion equation with reflective boundaries, it was used data simulated with the CALPUFF model. The exposed model was validated previously with the Hanford and Copenhagen experiments.

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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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Puigferrat, Albert, Miguel Masó, Ignasi de-Pouplana, Guillermo Casas, and Eugenio Oñate. "Semi-Lagrangian formulation for the advection–diffusion–absorption equation." Computer Methods in Applied Mechanics and Engineering 380 (July 2021): 113807. http://dx.doi.org/10.1016/j.cma.2021.113807.

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GHAREHBAGHI, Amin, Birol KAYA, and Gökmen TAYFUR. "Comparative Analysis of Numerical Solutions of Advection-Diffusion Equation." Cumhuriyet Science Journal 38, no. 1 (February 16, 2017): 49. http://dx.doi.org/10.17776/csj.53808.

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ASAI, Koji, Toshimitsu KOMATSU, and Koichiro OHGUSHI. "Simple and Highly-Accurate Scheme for Advection-Diffusion Equation." Journal of applied mechanics 1 (1998): 273–81. http://dx.doi.org/10.2208/journalam.1.273.

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Das, Subir, Anup Singh, and Seng Ong. "Numerical solution of fractional order advection-reaction diffusion equation." Thermal Science 22, Suppl. 1 (2018): 309–16. http://dx.doi.org/10.2298/tsci170624034d.

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In this paper, the Laplace transform method is used to solve the advection-diffusion equation having source or sink term with initial and boundary conditions. The solution profile of normalized field variable for both conservative and non-conservative systems are calculated numerically using the Bellman method and the results are presented through graphs for different particular cases. A comparison of the numerical solution with the existing analytical solution for standard order conservative system clearly exhibits that the method is effective and reliable. The important part of the study is the graphical presentations of the effect of the reaction term on the solution profile for the non-conservative case in the fractional order as well as standard order system. The salient feature of the article is the exhibition of stochastic nature of the considered fractional order model.

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Bizjan, Benjamin, Alen Orbanić, Brane Širok, Tom Bajcar, Lovrenc Novak, and Boštjan Kovač. "Flow Image Velocimetry Method Based on Advection-Diffusion Equation." Strojniški vestnik – Journal of Mechanical Engineering 60, no. 7-8 (July 15, 2014): 483–94. http://dx.doi.org/10.5545/sv-jme.2013.1614.

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Buske, Daniela, Marco T. Vilhena, Everson J. G. Silva, and Tiziano Tirabassi. "A solution of the time-dependent advection-diffusion equation." International Journal of Environment and Pollution 65, no. 1/2/3 (2019): 211. http://dx.doi.org/10.1504/ijep.2019.10023408.

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Tirabassi, Tiziano, Everson J. G. Silva, Daniela Buske, and Marco T. Vilhena. "A solution of the time-dependent advection-diffusion equation." International Journal of Environment and Pollution 65, no. 1/2/3 (2019): 211. http://dx.doi.org/10.1504/ijep.2019.101842.

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