Browse > Article
http://dx.doi.org/10.5666/KMJ.2022.62.3.557

A Boundary Integral Equation Formulation for an Unsteady Anisotropic-Diffusion Convection Equation of Exponentially Variable Coefficients and Compressible Flow  

Azis, Mohammad Ivan (Department of Mathematics, Hasanuddin University)
Publication Information
Kyungpook Mathematical Journal / v.62, no.3, 2022 , pp. 557-581 More about this Journal
Abstract
The anisotropic-diffusion convection equation with exponentially variable coefficients is discussed in this paper. Numerical solutions are found using a combined Laplace transform and boundary element method. The variable coefficients equation is usually used to model problems of functionally graded media. First 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 as a 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 are easy to implement and accurate for solving unsteady problems of anisotropic exponentially graded media governed by the diffusion convection equation.
Keywords
variable coefficients; anisotropic functionally graded materials; unsteady diffusion convection equation; Laplace transform; boundary element method;
Citations & Related Records
연도 인용수 순위
  • Reference
1 S. Hamzah, A. Haddade, A. Galsan, M. I. Azis and A. M. Abdal, Numerical solution to diffusion convection-reaction equation with trigonometrically variable coefficients of incompressible flow, J. Phys. Conf. Ser., 1341(8)(2019), 082005.   DOI
2 N. Lanafie, P. Taba, A. I. Latunra, Fahruddin and M. I. Azis, On the derivation of a boundary element method for diffusion convection-reaction problems of compressible flow in exponentially inhomogeneous media, J. Phys. Conf. Ser., 1341(6)(2019), 062013.   DOI
3 C. Zoppou and J. H. Knight, Analytical solution of a spatially variable coefficient advection-diffusion equation in up to three dimensions, Appl. Math. Model., 23(1999), 667-685.   DOI
4 M. I. Azis, I. Solekhudin I, M. H. Aswad and A. R. Jalil, Numerical simulation of two-dimensional modified Helmholtz problems for anisotropic functionally graded materials, J. King Saud Univ. Sci., 32(3)(2020), 2096-2102.   DOI
5 Q. Li, Z. Chai and B. Shi, Lattice Boltzmann model for a class of convection- diffusion equations with variable coefficients, Comput. Math. Appl., 70(2015), 548-561.   DOI
6 R. Pettres and L. A. de Lacerda, Numerical analysis of an advective diffusion domain coupled with a diffusive heat source, Eng. Anal. Bound. Elem., 84(2017), 129-140.   DOI
7 A. Rap, L. Elliott, D. B. Ingham, D. Lesnic and X. Wen, DRBEM for Cauchy convection-diffusion problems with variable coefficients, Eng. Anal. Bound. Elem., 28(2004), 1321-1333.   DOI
8 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs and Mathematical Tables, Dover Publications, Washington(1972).
9 M. I. Azis, BEM solutions to exponentially variable coefficient Helmholtz equation of anisotropic media, J. Phys. Conf. Ser., 1277(1)(2019), 012036.   DOI
10 S. Baja, S. Arif, Fahruddin, N. Haedar and M. I. Azis, Boundary element method solutions for steady anisotropic-diffusion convection problems of incompressible flow in quadratically graded media, J. Phys. Conf. Ser., 1341(6)(2019), 062019.   DOI
11 Fendoglu H, Bozkaya C and Tezer-Sezgin M, DBEM and DRBEM solutions to 2D transient convection-diffusion-reaction type equations, Eng. Anal. Bound. Elem., 93(2018), 124-134.   DOI
12 S. Hamzah, M. I. Azis, A. Haddade and A. K. Amir, Numerical solutions to anisotropic BVPs for quadratically graded media governed by a Helmholtz equation, IOP Conf. Ser.: Mater. Sci. Eng., 619(1)(2019), 012060.   DOI
13 M. I. Azis, Standard-BEM solutions to two types of anisotropic-diffusion convection reaction equations with variable coefficients, Eng. Anal. Bound. Elem., 105(2019), 87-93.   DOI
14 A. R. Jalil, M. I. Azis, S. Amir, M. Bahri and S. Hamzah, Numerical simulation for anisotropic-diffusion convection reaction problems of inhomogeneous media, J. Phys. Conf. Ser., 1341(8)(2019), 082013.   DOI
15 B. Nurwahyu, B. Abdullah, A. Massinai and M. I. Azis, Numerical solutions for BVPs governed by a Helmholtz equation of anisotropic FGM, IOP Conference Series: Earth and Environmental Science, 279(1)(2019), 012008.   DOI
16 M. A. H. Assagaf, A. Massinai, A. Ribal, S. Toaha S and M. I. Azis, Numerical simulation for steady anisotropic-diffusion convection problems of compressible flow in exponentially graded media, J. Phys. Conf. Ser., 1341(8)(2019), 082016.   DOI
17 M. I. Azis, Numerical solutions for the Helmholtz boundary value problems of anisotropic homogeneous media, J. Comput. Phys., 381(2019), 42-51.   DOI
18 R. Syam, Fahruddin, M. I. Azis and A. Hayat, Numerical solutions to anisotropic FGM BVPs governed by the modified Helmholtz type equation, IOP Conference Series: Materials Science and Engineering, 619(1)(2019), 012061.   DOI
19 A. Haddade, E. Syamsuddin, M. F. I. Massinai, M. I. Azis and A. I. Latunra, Numerical solutions for anisotropic-diffusion convection problems of incompressible flow in exponentially graded media, J. Phys. Conf. Ser., 1341(8)(2019), 082015.   DOI
20 N. Salam, A. Haddade, D. L. Clements and M. I. Azis, A boundary element method for a class of elliptic boundary value problems of functionally graded media, Eng. Anal. Bound. Elem., 84(2017), 186-190.   DOI
21 N. Salam, D. A. Suriamihardja, D. Tahir, M. I. Azis and E. S. Rusdi, A boundary element method for anisotropic-diffusion convection-reaction equation in quadratically graded media of incompressible flow, J. Phys. Conf. Ser., 1341(8)(2019), 082003.   DOI
22 S. Suryani, J. Kusuma, N. Ilyas, M. Bahri and M. I. Azis, A boundary element method solution to spatially variable coefficients diffusion convection equation of anisotropic media, J. Phys. Conf. Ser., 1341(6)(2019), 062018.   DOI
23 X-H. Wu, Z-J. Chang, Y-L. Lu, W-Q. Tao and S-P. Shen, An analysis of the convection-diffusion problems using meshless and mesh based methods, Eng. Anal. Bound. Elem., 36(2012), 1040-1048.   DOI
24 H. Yoshida and M. Nagaoka, Multiple-relaxation-time lattice Boltzmann model for the convection and anisotropic diffusion equation, J. Comput. Phys, 229(2010), 7774-7795.   DOI
25 F. Wang, W. Chen, A. Tadeu and C. G. Correia, Singular boundary method for transient convection- diffusion problems with time-dependent fundamental solution, Int. J. Heat Mass Transf., 114(2017), 1126-1134.   DOI
26 I. Raya, Firdaus, M. I. Azis, Siswanto and A. R. Jalil, Diffusion convection-reaction equation in exponentially graded media of incompressible flow: Boundary element method solutions, J. Phys. Conf. Ser., 1341(8)(2019), 082004.   DOI
27 Sakka, E. Syamsuddin, B. Abdullah, M. I. Azis and A. M. A. Siddik, On the derivation of a boundary element method for steady anisotropic-diffusion convection problems of incompressible flow in trigonometrically graded media, J. Phys. Conf. Ser., 1341(6)(2019), 062020.   DOI
28 H. Stehfest, Algorithm 368: Numerical inversion of Laplace transforms [D5], Communications of the ACM, 13(1)(1970), 47-49.   DOI
29 J. Ravnik and L. ASkerget, A gradient free integral equation for diffusion-convection equation with variable coefficient and velocity, Eng. Anal. Bound. Elem., 37(2013), 683-690.   DOI
30 A. Haddade, M. I. Azis, Z. Djafar, St. N. Jabir and B. Nurwahyu, Numerical solutions to a class of scalar elliptic BVPs for anisotropic, IOP Conf. Ser.: Earth Environ. Sci., 279(1)(2019), 012007.   DOI
31 N. Khaeruddin, A. Galsan, M. I. Azis, N. Ilyas and P. Paharuddin, Boundary value problems governed by Helmholtz equation for anisotropic trigonometrically graded materials: A boundary element method solution, J. Phys. Conf. Ser., 1341(6)(2019), 062007.   DOI
32 H. Hassanzadeh and M. Pooladi-Darvish, Comparison of different numerical Laplace inversion methods for engineering applications, Appl. Math. Comput., 189(2007), 1966-1981.   DOI
33 E. Hernandez-Martinez, H. Puebla, F. Valdes-Parada and J. Alvarez-Ramirez, Nonstandard finite difference schemes based on Green's function formulations for reactionadiffusionaconvection systems, Chemical Engineering Science, 94(2013), 245-255.   DOI
34 St. N. Jabir, M. I. Azis, Z. Djafar and B. Nurwahyu, BEM solutions to a class of elliptic BVPs for anisotropic trigonometrically graded media, IOP Conference Series: Materials Science and Engineering, 619(1)(2019), 012059.   DOI
35 N. Lanafie, N. Ilyas, M. I. Azis and A. K. Amir, A class of variable coefficient elliptic equations solved using BEM, IOP Conference Series: Materials Science and Engineering, 619(1)(2019), 012025.   DOI
36 M. Meenal and T. I. Eldho, Two-dimensional contaminant transport modeling using mesh free point collocation method (PCM), Eng. Anal. Bound. Elem., 36(2012), 551-561.   DOI
37 M. I. Azis, Fundamental solutions to two types of 2D boundary value problems of anisotropic materials, Far East J. Math. Sci., 101(11)(2017), 2405-2420.
38 J. Ravnik and L. ASkerget, Integral equation formulation of an unsteady diffusion - convection equation with variable coefficient and velocity, Comput. Math. Appl., 66(2014), 2477-2488.   DOI
39 Paharuddin, Sakka, P. Taba, S. Toaha and M. I. Azis, Numerical solutions to Helmholtz equation of anisotropic functionally graded materials, J. Phys. Conf. Ser., 1341(8)(2019), 082012.   DOI
40 N. Rauf, H. Halide, A. Haddade, D. A. Suriamihardja and M. I. Azis, A numerical study on the effect of the material's anisotropy in diffusion convection reaction problems, J. Phys. Conf. Ser., 1341(8)(2019), 082014.   DOI