Numerical Simulation for Unsteady Anisotropic-Diffusion Convection Equation of Spatially Variable Coefficients and Incompressible Flow

Authors

  • Moh.Ivan Azis Hasanuddin University

DOI:

https://doi.org/10.5556/j.tkjm.54.2023.4069

Keywords:

variable coefficients, anisotropic functionally graded materials, unsteady diffusion convection equation, Laplace transform, boundary element method

Abstract

The anisotropic-diffusion convection equation of spatially
variable coefficients which is relevant for functionally graded media
is discussed in this paper to find numerical solutions by using a
combined Laplace transform and boundary element method. The variable
coefficients equation is transformed to a constant coefficients equation.
The constant coefficients equation is then Laplace-transformed so
that the time variable vanishes. The Laplace-transformed equation
is consequently written in a pure boundary integral equation which
involves a time-free fundamental solution. The boundary integral equation
is therefore employed to find numerical solutions using a standard
boundary element method. Finally the results obtained are inversely
transformed numerically using the Stehfest formula to get solutions
in the time variable. The combined Laplace transform and boundary
element method is easy to be implemented, efficient and accurate for
solving unsteady problems of anisotropic functionally graded media
governed by the diffusion convection equation.

References

M. A. H. Assagaf, A. Massinai, A. Ribal, S. Toaha S, M. I. Azis, Numerical simulation for steady anisotropic-diffusion convection problems of compressible flow in exponentially graded media, Journal of Physics: Conference Series, 1341(8) (2019), 082016.

M. I. Azis, BEM solutions to exponentially variable coefficient Helmholtz equation of anisotropic media, Journal of Physics: Conference Series, 1277 (2019), 012036.

M. I. Azis, Numerical solutions for the Helmholtz boundary value problems of anisotropic homogeneous media, Journal of Physics: Conference Series, 381 (2019), 42–51.

M. I. Azis, Standard-BEM solutions to two types of anisotropic-diffusion convection re- action equations with variable coefficients, Engineering Analysis with Boundary Elements, 105 (2019), 87–93.

M. I. Azis, I. Solekhudin I, M. H. Aswad, A. R. Jalil, Numerical simulation of two- dimensional modified Helmholtz problems for anisotropic functionally graded materials, Journal of King Saud University - Science, 32(3) (2020), 2096–2102.

S. Baja, S. Arif, Fahruddin, N. Haedar, M. I. Azis, Boundary element method solutions for steady anisotropic-diffusion convection problems of incompressible flow in quadratically graded media, Journal of Physics: Conference Series, 1341(8) (2019), 062019.

Fendoğlu H, Bozkaya C, Tezer-Sezgin M, DBEM and DRBEM solutions to 2D transient convection-diffusion-reaction type equations, Engineering Analysis with Boundary Ele- ments, 93 (2018), 124–134.

A. Haddade, M. I. Azis, Z. Djafar, St. N. Jabir, B. Nurwahyu, Numerical solutions to a class of scalar elliptic BVPs for anisotropic, IOP Conf. Ser.: Earth Environ. Sci., 279 (2019), 012007.

A. Haddade, E. Syamsuddin, M. F. I. Massinai, M. I. Azis, A. I. Latunra, Numerical solu- tions for anisotropic-diffusion convection problems of incompressible flow in exponentially graded media, Journal of Physics: Conference Series, 1341(8) (2019), 082015.

S. Hamzah, M. I. Azis, A. Haddade, A. K. Amir, Numerical solutions to anisotropic BVPs for quadratically graded media governed by a Helmholtz equation, IOP Conf. Ser.: Mater. Sci. Eng., 619 (2019), 012060.

S. Hamzah, A. Haddade, A. Galsan, M. I. Azis, A. M. Abdal, Numerical solution to diffusion convection-reaction equation with trigonometrically variable coefficients of incompressible flow, Journal of Physics: Conference Series, 1341(8) (2019), 082005.

Hernandez-Martinez E, Puebla H, Valdes-Parada F, Alvarez-Ramirez J, Nonstandard fi- nite difference schemes based on Green’s function formulations for reaction–diffusion– convection systems, Chemical Engineering Science, 94 (2013), 245–255.

St. N. Jabir, M. I. Azis, Z. Djafar, B. Nurwahyu, BEM solutions to a class of elliptic BVPs for anisotropic trigonometrically graded media, IOP Conference Series: Materials Science and Engineering, 619 (2019), 012059.

A. R. Jalil, M. I. Azis, S. Amir, M. Bahri, S. Hamzah, Numerical simulation for anisotropic- diffusion convection reaction problems of inhomogeneous media, Journal of Physics: Con- ference Series, 1341(8) (2019), 082013.

Khaeruddin, A. Galsan, M. I. Azis, N. Ilyas, Paharuddin, Boundary value problems governed by Helmholtz equation for anisotropic trigonometrically graded materials: A boundary el- ement method solution, Journal of Physics: Conference Series, 1341 (2019), 062007.

N. Lanafie, N. Ilyas, M. I. Azis, A. K. Amir, A class of variable coefficient elliptic equations solved using BEM, IOP Conference Series: Materials Science and Engineering, 619 (2019), 012025.

N. Lanafie, P. Taba, A. I. Latunra, Fahruddin, M. I. Azis, On the derivation of a boundary element method for diffusion convection-reaction problems of compressible flow in expo- nentially inhomogeneous media, Journal of Physics: Conference Series, 1341(6) (2019), 062013.

Q. Li, Z. Chai, B. Shi, Lattice Boltzmann model for a class of convection–diffusion equations with variable coefficients, Computers and Mathematics with Applications, 70 (2015), 548– 561.

M. Meenal, T. I. Eldho, Two-dimensional contaminant transport modeling using mesh free point collocation method (PCM), Engineering Analysis with Boundary Elements, 36 (2012), 551–561.

B. Nurwahyu, B. Abdullah, A. Massinai, M. I. Azis, Numerical solutions for BVPs governed by a Helmholtz equation of anisotropic FGM, IOP Conference Series: Earth and Environ- mental Science, 279 (2019), 012008.

Paharuddin, Sakka, P. Taba, S. Toaha, M. I. Azis, Numerical solutions to Helmholtz equation of anisotropic functionally graded materials, Journal of Physics: Conference Series, 1341 (2019), 082012.

R. Pettres, L. A. de Lacerda, Numerical analysis of an advective diffusion domain coupled with a diffusive heat source, Engineering Analysis with Boundary Elements, 84 (2017), 129– 140.

A. Rap, L. Elliott, D. B. Ingham, D. Lesnic, X. Wen, DRBEM for Cauchy convection-diffusion problems with variable coefficients, Engineering Analysis with Boundary Elements, 28 (2004), 1321–1333.

N. Rauf, H. Halide, A. Haddade, D. A. Suriamihardja, M. I. Azis, A numerical study on the effect of the material’s anisotropy in diffusion convection reaction problems, Journal of Physics: Conference Series, 1341(8) (2019), 082014.

J. Ravnik, L. Škerget, A gradient free integral equation for diffusion–convection equation with variable coefficient and velocity, Engineering Analysis with Boundary Elements, 37 (2013), 683–690.

J. Ravnik, L. Škerget, Integral equation formulation of an unsteady diffusion–convection equation with variable coefficient and velocity, Computers and Mathematics with Applications, 66 (2014), 2477–2488.

I. Raya, Firdaus, M. I. Azis, Siswanto, A. R. Jalil, Diffusion convection-reaction equation in exponentially graded media of incompressible flow: Boundary element method solutions, Journal of Physics: Conference Series, 1341(8) (2019), 082004.

Sakka, E. Syamsuddin, B. Abdullah, M. I. Azis, A. M. A. Siddik, On the derivation of a boundary element method for steady anisotropic-diffusion convection problems of incom- pressible flow in trigonometrically graded media, Journal of Physics: Conference Series, 1341(8) (2019), 062020.

N. Salam, A. Haddade, D. L. Clements, M. I. Azis, A boundary element method for a class of elliptic boundary value problems of functionally graded media, Engineering Analysis with Boundary Elements, 84 (2017), 186–190.

N. Salam, D. A. Suriamihardja, D. Tahir, M. I. Azis, E. S. Rusdi, A boundary element method for anisotropic-diffusion convection-reaction equation in quadratically graded media of in- compressible flow, Journal of Physics: Conference Series, 1341(8) (2019), 082003.

S. Suryani, J. Kusuma, N. Ilyas, M. Bahri, M. I. Azis, A boundary element method solution to spatially variable coefficients diffusion convection equation of anisotropic media, Journal of Physics: Conference Series, 1341(6) (2019), 062018.

R. Syam, Fahruddin, M. I. Azis, A. Hayat, Numerical solutions to anisotropic FGM BVPs governed by the modified Helmholtz type equation, IOP Conference Series: Materials Sci- ence and Engineering, 619(1) (2019), 012061.

F. Wang, W. Chen, A. Tadeu, C. G. Correia, Singular boundary method for transient convection–diffusion problems with time-dependent fundamental solution, International Journal of Heat and Mass Transfer, 114 (2017), 1126–1134.

X-H. Wu, Z-J. Chang, Y-L. Lu, W-Q. Tao, S-P. Shen, Ananalysis of the convection–diffusion problems using meshless and mesh based methods, Engineering Analysis with Boundary Elements, 36 (2012), 1040–1048.

H. Yoshida, M. Nagaoka, Multiple-relaxation-time lattice Boltzmann model for the con- vection and anisotropic diffusion equation, Journal of Computational Physics, 229 (2010), 7774–7795.

C. Zoppou, J. H. Knight, Analytical solution of a spatially variable coefficient advection- diffusion equation in up to three dimensions, Applied Mathematical Modelling, 23 (1999), 667–685.

Downloads

Published

2021-11-10

How to Cite

Azis, M. (2021). Numerical Simulation for Unsteady Anisotropic-Diffusion Convection Equation of Spatially Variable Coefficients and Incompressible Flow. Tamkang Journal of Mathematics, 54. https://doi.org/10.5556/j.tkjm.54.2023.4069

Issue

Section

Papers