Sains Malaysiana 54(9)(2025): 2315-2326

http://doi.org/10.17576/jsm-2025-5409-17

 

A Numerical Simulation of Diffusion-Convection Problems in Heterogeneous Media with a Point Source

(Simulasi Berangka Masalah Resapan-Perolakan dalam Media Heterogen dengan Punca Titik)

 

IMAM SOLEKHUDIN^1,*, NOORMA YULIA MEGAWATI¹, SOLIKHATUN¹, INDIRA ANGGRIANI^1,2, MOHAMMAD IVAN AZIS³, IKHA MAGDALENA⁴ & WIDOWATI⁵

 

¹Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Gadjah Mada, Yogyakarta 55281, Indonesia
²Institut Teknologi Kalimantan, Balikpapan, Kalimantan Timur 76127, Indonesia
³Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Hasanuddin, Makassar 90245, Indonesia
⁴Faculty of Mathematics and Natural Sciences, Bandung Institute of Technology, Bandung 40132, Indonesia
⁵Department of Mathematics, Faculty of Science and Mathematics, Universitas Diponegoro, Semarang
50275, Indonesia

 

Received: 3 December 2024/Accepted: 18 July 2025

 

Abstract

In this paper, some time-independent diffusion-convection problems in anisotropic media are considered. To study the problems, the governing equation of the problems is transformed into a diffusion-convection equation in a homogeneous isotropic media. The transformed equation with respect to transformed boundary conditions are solved numerically using Dual Reciprocity Boundary Element Method (DRBEM). The method is tested using several problems involving time-independent diffusion-convection. Two of the problems are with analytical solution, and the other problems are with point sources without known analytical solution.

 

Keywords: Anisotropic media; DRBEM; point source; time-independent diffusion-convection

 

Abstrak

Dalam makalah ini, beberapa masalah resapan-konveksi bebas masa dalam media anisotropik dipertimbangkan. Untuk mengkaji masalah, persamaan yang mengawal masalah diubah menjadi persamaan resapan-konveksi dalam media homogen isotropik. Persamaan berkenaan dengan keadaan sempadan yang diubah diselesaikan secara berangka menggunakan Kaedah Unsur Dual Kesalingan Sempadan (DRBEM). Kaedah ini diuji menggunakan beberapa masalah yang melibatkan resapan-konveksi bebas masa. Dua daripada masalah adalah dengan penyelesaian analitik dan masalah lain adalah dengan sumber mata tanpa penyelesaian analitik yang diketahui.

 

Kata kunci: DRBEM; media anisotropik; resapan-konveksi bebas masa; sumber mata

 

REFERENCES

Ang, W.T. 2007. A Beginner’s Course in Boundary Element Methods. Florida: Universal Publishers.

Ang, W.T. & Clements, D.L. 2009. Nonlinear heat equation for nonhomogeneous anisotropic materials: A dual-reciprocity boundary element solution. Numerical Methods for Partial Differential Equations 26: 771-784.

Ashar, N.Y. & Solekhudin, I. 2021. A numerical study of steady pollutant spread in water from a point source. Engineering Letters 29(3): 840-848.

Atangana, A. 2018. Fractional Operators with Constant and Variable Order with Application to Geo-Hydrology. Massachusetts: Academic Press.

Azis, M.I., Solekhudin, I., Aswad, M.H., Hamzah, S. & Jalil, A.R. 2021. A combined laplace transform and boundary element method for unsteady laplace problems of several classes of anisotropic functionally graded materials. Engineering Letters 29: 534-542.

Capinski, W.S., Maris, H.J., Ruf, T., Cardona, M., Ploog, K. & Katzer, D.S. 1999. Thermal-conductivity measurements of GaAs/AlAs superlattices using a picosecond optical pump-and-probe technique. Phys. Rev. B. 59(12): 8105-8113.

Clements, D.L. & Lobo, M. 2010. A BEM for time dependent infiltration from an irrigation channel. Engineering Analysis with Boundary Elements 34: 1100-1104.

Huxtable, S., Cahill, D.G., Fauconnier, V., White, J.O. & Zhao, J.C.  2004. Thermal conductivity imaging at micrometre-scale resolution for combinatorial studies of materials. Nature Mater. 3: 298-301.

Munadi, Solekhudin, I., Sumardi & Zulijanto, A. 2020. A numerical study of steady infiltration from a single irrigation channel with an impermeable soil layer. Engineering Letters 28(3): 1-8.

Munadi, Solekhudin, I., Sumardi & Zulijanto, A. 2019. Steady water flow from different types of single irrigation channel.  JP Journal of Heat and Mass Transfer 16(1): 95-106.

Norris, P.M., Caffrey, A.P., Stevens, R.J., Klopf, J.M., James, J., McLeskey, T. & Smith, A.N.     2003. Femtosecond pump–probe nondestructive examination of materials. Rev. Sci.   Instrum. 74(1): 400-406.

Paddock, C.A. & Eesley, G.L. 1986. Transient thermos reflectance from thin metal films. J. Appl. Phys. 60(1): 285-290.

Pramesthi, A.A.N., Solekhudin, I. & Azis, M.I. 2021. Implementation of dual reciprocity boundary element method for heat conduction problems in anisotropic solid. IAENG International Journal of Applied Mathematics 52(1): 122-130.

Solekhudin, I. 2020. Boundary interface water infiltration into layered soils using dual reciprocity methods. Engineering Analysis with Boundary Elements 119: 280-292.

Solekhudin, I. 2018. A numerical method for time-dependent infiltration from periodic trapezoidal channels with different types of root-water uptake. IAENG International Journal of Applied Mathematics 48(1): 84-89.

Solekhudin, I., Sari, A.K & Makhrus, F.  2024. An iterative dual reciprocity method for a class of infiltration problems in two-layered soils with different types of root-water uptake. IAENG International Journal of Applied Mathematics 54(7): 1390-1399.

Solekhudin, I., Purnama, D., Malysa, N.H. & Sumardi. 2018. Characteristic of water flow in heterogeneous soils. JP Journal of Heat and Mass Transform 15(3): 597-608.

 

*Corresponding author; email: imams@ugm.ac.id