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  4. Mathematical modeling of impurity diffusion process under given statistics of a point mass sources system. I

Mathematical modeling of impurity diffusion process under given statistics of a point mass sources system. I

MMC.
2024;
Volume 11, Number 2
: pp. 385–393
Received: January 25, 2024
Revised: April 12, 2024
Accepted: April 15, 2024

Pukach P. Y., Chernukha Y. A. Mathematical modeling of impurity diffusion process under given statistics of a point mass sources system. I. Mathematical Modeling and Computing. Vol. 11, No. 2, pp. 385–393 (2024)

Authors:
  1. P. Y. Pukach
  2. Y. A. Chernukha
1
Lviv Polytechnic National University
2
Lviv Polytechnic National University

The model of the impurity diffusion process in the layer where a system of random point mass sources acts, is proposed.  Mass sources of various power are uniformly distributed in a certain internal interval of the body.  Statistics of random sources are given.  The solution of the initial-boundary value problem is constructed as a sum of the homogeneous problem solution and the convolution of the Green's function and the system of the random point mass sources.  The solution is averaged over both certain internal subinterval and the entire body region.  Simulation units are designed for modeling of the behavior of the averaged concentration function with acting system of point mass sources of various power.  On this basis, the averaged concentration field is investigated depending on the internal interval length, power and number of sources in the system as well as the concentration values at the layer boundaries.

mathematical modeling
diffusion
random point source
Green's function
uniform distribution
software
  1. Chen J., Zhang X., Li S.  Heteroskedastic linear regression model with compositional response and covariates.  Journal of Applied Statistics.  45 (12), 2164–2181 (2018).
  2. Dyvak M., Porplytsya N., Maslyak Y., Shynkaryk M.  Method of parametric identification for interval discrete dynamic models and the computational scheme of its implementation.  Advances in Intelligent Systems and Computing.  689, 101–112 (2018).
  3. Gauthierab B., Pronzatob L.  Convex relaxation for IMSE optimal design in random-field models.  Computational Statistics & Data Analysis.  113, 375–394 (2017).
  4. Moffatt J., Scarf P.  Sequential regression measurement error models with application.  Statistical Modelling.  16 (6), 454–476 (2016).
  5. Sikaroudi A. E., Park C.  A mixture of linear-linear regression models for a linear-circular regression.  Statistical Modelling.  21 (3), 220–243 (2019).
  6. Wilcox R. R.  Linear regression: robust heteroscedastic confidence bands that have some specified simultaneous probability coverage.  Journal of Applied Statistics.  44 (14), 2564–2574 (2017).
  7. Gong Y., Li P., Wang X., Xu X.  Numerical solution of an inverse random source problem for the time fractional diffusion equation via Phase Lift.  Inverse Problems.  37 (4), 045001 (2021).
  8. Diaz-Adame R., Jerez S.  Convergence of time-splitting approximations for degenerate convection-diffusion equations with a random source.  Journal of Numerical Mathematics.  29 (1), 23–38 (2021).
  9. Zhang R., Bai H., Zhao F.  L1-Finite Difference Method for Inverse Source Problem of Fractional Diffusion Equation.  Journal of Physics: Conference Series.  1624 (3), 032001 (2020).
  10. Niu P., Helin T., Zhang Z.  An inverse random source problem in a stochastic fractional diffusion equation.  Inverse Problems.  36 (4), 045002 (2020).
  11. Reis F. D. A. A., Voller V. R.  Models of infiltration into homogeneous and fractal porous media with localized sources.  Physical Review E.  99 (4), 042111 (2019).
  12. Thach T. N., Huy T. N., Tam P. T. M., Minh M. N., Can N. H.  Identification of an inverse source problem for time-fractional diffusion equation with random noise.  Mathematical Methods in the Applied Sciences.  42 (1), 204–218 (2019).
  13. Le Vot F., Abad E., Yuste S. B.  Continuous-time random-walk model for anomalous diffusion in expanding media.  Physical Review E.  96 (3), 032117 (2017).
  14. Budhiraja A., Chen J., Dupuis P.  Large deviations for stochastic partial differential equations driven by a Poisson random measure.  Stochastic Processes and their Applications.  123 (2), 523–560 (2013).
  15. Krasowska M., Rybak A., Dudek G., Strzelewicz A., Pawelek K., Grzywna Z. J.  Structure morphology problems in the air separation by polymer membranes with magnetic particles.  Journal of Membrane Science.  415–416, 864–870 (2012).
  16. Uffink G., Elfeki A., Dekking M., Bruining J., Kraaikamp C.  Understanding the Non-Gaussian Nature of Linear Reactive Solute Transport in 1D and 2D: From Particle Dynamics to the Partial Differential Equations.  Transport in Porous Media.  91 (2), 547–571 (2012).
  17. Delfs J.-O., Park C.-H., Kolditz O.  An Euler-Lagrange approach to transport modelling in coupled hydrosystems.  IAHS-AISH Publication.  341, 166–171 (2011).
  18. Sen S., Kundu S., Absi R., Ghoshal K.  A Model for Coupled Fluid Velocity and Suspended Sediment Concentration in an Unsteady Stratified Turbulent Flow through an Open Channel.  Journal of Engineering Mechanics.  149 (1), 04022088 (2023).
  19. Yu Q., Turner I., Liu F., Moroney T.  A study of distributed-order time fractional diffusion models with continuous distribution weight functions.  Numerical Methods for Partial Differential Equations.  39 (1), 383–420 (2023).
  20. Hoang V. H., Quek J. H., Schwab C.  Multilevel Markov chain monte carlo for bayesian inversion of parabolic partial differential equations under gaussian prior.  SIAM-ASA Journal on Uncertainty Quantification.  9 (2), 384–419 (2021).
  21. Abramowitz M., Stegun I. A.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.  New York, Dover (1972).
  22. Evans L. C.  Partial Differential Equations.  Province, Rhode Island, American Mathematical Society (2010).
  23. Crank J.  The mathematics of diffusion.  Oxford, Clarendon Press; 2nd edition (1975).
  24. Sneddon I. N.  Fourier transforms.  New York, Dover Publications, Inc. (1995).
  25. Korn G. A., Korn T. M.  Mathematical handbook for scientists and engineers.  Mineola, Dover Publications, Inc; 2nd edition (2000).
  26. Mood A. M., Graybill F. A., Boes D. C.  Introduction to the theory of statistics.  McGraw-Hill (1974).
  27. Pukach P., Chernukha Y.  Simulation of Diffusion of Impurity Substance from a Random Point Source of Mass.  Modeling, Control and Information Technologies: Proceedings of International Scientific and Practical Conference.  6, 139–143 (2010).