ORIGINAL_ARTICLE
Meshfree method for solving mathematical fractional order model of capillary formation in tumor
angiogenesis
This paper was devoted to numerical solution of capillary formation in tumor angiogenesis with time fractional derivative. A time discretization approach based on the θ-weighted fractional finite difference scheme was employed for time fractional derivative and a mesh free process was applied by using radial basis functions (RBFs). Stability analysis of the method was also investigated and some numerical cases were studied.
https://jamm.scu.ac.ir/article_11494_06008e2daa0af9adda77b35645c5bb8b.pdf
2015-11-22
1
18
10.22055/jamm.2015.11494
Meshfree method
Radial basis functions
Fractional derivative
Tumor angiogenesis
Bahman
Ghazanfari
bahman_ghazanfari@yahoo.com
1
Department of Mathematics, Lorestan University
LEAD_AUTHOR
Amin
Shahkarami
shahkarami67@gmail.com
2
Department of Mathematics, Lorestan University
AUTHOR
[1] Kilbas, A.A., Strvastava, H.M. and Trujilloو J.J. (2006). Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Vol. 204, Elsevier, Amsterdam.
1
[2] Lakshmikantham, V. and Leela, J.D. (2009). Theory of fractional dynamics systems, Cambridge Scientific Publishers, Cambridge, UK.
2
[3] Podlubny, I., (2002). Geometric and physical interpretation of fractionalintegration and fractional differentiation, Fract. Calc. Appl. Anal. 5, 367-386.
3
[4] Baleanu, D., Diethelm, K., Scalas, E. and Trujillo J.J. (2012). Fractional calculua models and numerical methods, World Scientific, Singapore.
4
[5] Pudlubny, I. (1999). Fractional differential equations, Academic Press, New York.
5
[6] Hilfr, R. (2000). Applications of fractional calculus in physics, Publishing Company Singapore, World scientific.
6
[7] Carpinteri, A. and. Mainardi, F. (1997). Fractional calculus: "Some basic problems in continuum and statistical mechanics", Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, New York, 291-348.
7
[8] He,J.H., (1999). Some applications Of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol. 15, 86-90.
8
[9] Luchko,A. and Gorenflo, R. (1998). The initial value problem for some fractional differential equations with the Caputo derivative, Preprint series A08-98.
9
[10] Miller,K.S. and Ross, B. (1993). An Introduction to the fractional calculus and fractional differential equations, John Wiley and Sons, Inc., New York.
10
[11] Caputo, M. (1969). Elasticita e dissipazione, Bologna, Italy, Zanichelli.
11
[12] Jumarie, G. (2006). Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions further results, Comput. Math. Appl., 51, 1367–1376.
12
[13] Momani, S. and Odibat, A. (2007). Numerical comparison of methods for solving linear differential equations of fractional order, Appl. Math. Comput. 31, 1248-1255.
13
[14] Levine, H.A., Pamuk, S., Sleeman, B.D. and Nilsen-Hamilton, M (2001). Mathematical model of capillary formation and development in tumor angiogenesis: penetration into the stroma, Bull. Math. Biol., 63, 801-863.
14
[15] Pamuk, S. and Endern, A. (2007). The method of lines for the numerical solution of a mathematical model for capillary formation: the role of endothelial cells in the capillary, Appl. Math. Comput. 186, 831–835.
15
[16] Saadatmandi, A. and Dehghan, M. (2008). Numerical solution of a mathematical model for capillary formation in tumor angiogenesis via the tau method,Commun. Number. Math. Eng. 24, 1467–1474.
16
[17] Abbasbandy, S., Roohani, H.G. and Hashim, I. (2012). Numerical analysis of a mathematical model for capillary formation in tumor angiogenesis using a mesh free method based on the radial basis function, Engineering Analysis with Boundary Elements: 36, 1811-1818.
17
[18] Buhmann, M.D. (2004). Radial basis functions: theory and implementation, Cambridge University Press.
18
[19] Madych, W.R. (1992). Miscellaneous error bounds for multiquadric and related interpolators, Comput. Math. Appl., 24, 121-138.
19
[20] Micchelli, C.A. (1986). Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constr. Approx.
20
[21] Horn, R.A. and Johnson, C.R. (2013). Matrix Analysis, second edition, Cambridge University Press.
21
[22] Hardy, R.L. (1971). Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res., 176, 1905-1915.
22
ORIGINAL_ARTICLE
General progressive joint Type-II censoring scheme and inference for parameters of two Weibull populations under this scheme
In this paper, a generalization of the progressive joint Type-II censoring scheme is introduced. Application of this scheme is in the case that the lifetimes of first units of two samples are missing or lost and also because of preventing of long time of the test, some units are removed during the test. After introducing the scheme, for parameters of two Weibull populations, maximum likelihood estimators and confidence interval using procedures such as asymptotic normality and bootstrap methods, under the scheme, are obtained. Finally, by means a simulation study these estimations are evaluated and also all confidence intervals are compared in terms of coverage probabilities.
https://jamm.scu.ac.ir/article_11490_280c0260feab337854182be4e94381af.pdf
2015-11-22
19
38
10.22055/jamm.2015.11490
Asymptotic normality
Bootstrap confidence interval
Coverage probabilities
General progressive joint progressive Type-II censoring
Weibull distribution
Hamzeh
Torabi
htorabi@yazd.ac.ir
1
Department of Statistics, Yazd University
LEAD_AUTHOR
Saeedeh
Bafekri Fadafen
sa.bafekri67@gmail.com
2
Department of Statistics, Yazd University
AUTHOR
Hossein
Nadeb
honadeb@yahoo.com
3
Department of Statistics, Yazd University
AUTHOR
[1] Balakrishnan, N. and Rasouli, A. (2008). Exact likelihood inference for two exponential population under joint Type-II censoring, Computational Statistics and Data Analysis, 52, 2738-2752.
1
[2] Rasouli, A. and Balakrishnan, N. (2010). Exact likelihood inference for two exponential populations under joint progressive Type-II censoring, Communication in Statistics-Theory and Methods, 39, 2172-2191.
2
[3] Viveros, R. and Balakrishnan, N. (1994). Interval estimation of parameters of life from progressively censored data., Technometrics, 36, 84-91.
3
[4] Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics, New York: Wiley.
4
[5] Parsi, S., Ganjali, M. and Sanjari, N. (2011). Conditional maximum likelihood and interval estimation for two Weibull populations under joint Type-II censoring, Communication in Statistics-Theory and Methods, 40, 2117-2135.
5
[6] Wang, B. X. (2012). Exact inference estimation for the scale family under general progressive Type-II censoring, Communication in Statistics-Theory and Methods, 41, 4444-4452.
6
[7] Ferguson, T.S. (1996). A Course in Large Sample Theory, 41, New York: Chapman and Hall/CRC Press.
7
[8] Efron, B. and Tibshirani, R.J. (1994). An Introduction to the Bootstrap, New York: Chapman and Hall/CRC Press.
8
[9] Smith, R. L. and Naylor, J. C. (1987). A Comparison of Maximum Likelihood and Bayesian Estimators for the Three- Parameter Weibull Distribution, Journal of the Royal Statistical Society. Series C (Applied Statistics), 36, 358-369.
9
ORIGINAL_ARTICLE
Maximum M-Entropy model- type ΙΙ for obtaining the ordered weighted averaging operator weights
One key issue in the theory of the OWA operator is to determine its associated weights. In this paper, based upon the M-Entropy measures, new models for obtaining the ordered weighted averaging (OWA) operators are proposed. In the models it is assumed, according to available information that the OWA weights are in decreasing or increasing order. Some properties of the models are analyzed and the method of Lagrange multipliers is used to provide a direct way to find these weights. The models are solved with specific level of Orness comparing the results with some other related models and with the other maximum M-entropy model. The results demonstrate the efficiency of the M-Entropy models in generating the OWA operator weights. Also, the obtained weights of the two M-entropy models confirm the difference between two types of the models. Finally, an applied example is presented to illustrate the applications of the proposed model.
https://jamm.scu.ac.ir/article_11491_051fc0f1f6b9c9bdfd03ac0cc26d80e8.pdf
2015-11-22
39
58
10.22055/jamm.2015.11491
OWA operator
Operator weights
Orness
Maximum entropy
Entropy
Bayesian entropy
Alireza
Chaji
chaji85ms@yahoo.com
1
Shohadaie Hovaizeh University of Technology, Susangerd, Iran
LEAD_AUTHOR
Gholamhoussain
Yari
yari@iust.ac.ir
2
Department of Mathematics, Iran University of Science and Technology
AUTHOR
[1] Yager, R. R. (1988). On ordered weighted averaging aggregation operators in multi-criteria decision- making. IEEE Transactions on Systems, Man and Cybernetics, 18, 183–190.
1
[2] Merigó.José, M and Casanovas, Montserrat. (2010). The Fuzzy generalized OWA operator and its application in strategic decision making, Cybernetics and Systems, 41, 359-370.
2
[3] Yager, R. R. (2004). OWA aggregation over a continuous interval argument with applications to decision making, IEEE Transactions on Systems, Man, and Cybernetics, Part B 34, 1952-1963.
3
[4] Yager, R. R. (2009). Weighted Maximum Entropy OWA Aggregation with Applications to Decision Making Under Risk, IEEE Transactions on Systems, Man, and Cybernetics, Part A: Systems and Humans, 39(3), 555-564.
4
[5] Torra, V. (2004). OWA operators in data modeling and reidentication, IEEE Trans. Fuzzy Systems, 12, 652-660.
5
[6] Yager, R. R. and Filew, D.P. (1992). Fuzzy logic controllers with flexible structures. In: Proc Second IntConf on Fuzzy Sets and Neural Networks, Izuka, Japan; 317–320.
6
[7] Kacprzyk, J. and Zadrozny, S. (2001). Computing with words in intelligent database querying: standalone and internet-based applications, Information Sciences, 134, 71-109.
7
[8] Peláez, J.I. and Doña, J.M. (2003). Majority additive-ordered weighting averaging: a new neat ordered weighting averaging operator based on the majority process, international Journal of Intelligent Systems, 18, 469-481.
8
[9] Fuller, R. (2007). On obtaining OWA operator weights: a short survey of recent developments, in: Proceedings of the 5-th IEEE International Conference on Computational Cybernetics (ICCC)
9
[10]Yager, R.R. (1998). Including importances in OWA aggregations using fuzzy systems modeling. IEEE Trans Fuzzy Syst, 6(1); 286–294.
10
[11] Herrera-Viedma, E, Cordón. O, Luque. M, Lopez. A.G and Muñoz. A.M. (2003). A model of fuzzy linguistic IRS based on multi-granular linguistic information, International Journal of Approximate Reasoning, 34, 221239.
11
[12] Herrera-Viedma. E, Cordón. O, Luque. M, Lopez. A.G and Muñoz. A.M. (2007). A Model of Information Retrieval System with Unbalanced Fuzzy Linguistic Information, International Journal of Intelligent Systems, 22(11), 1197-1214.
12
[13] Herrera-Viedma, E and Pasi, G. (2003). Evaluating the Informative Quality of Documents in SGML-Format Using Fuzzy Linguistic Techniques Based on Computing with Words, Information Processing and Management, 39(2), 233-249.
13
[14] O’Hagan, M. (1988). Aggregating template rule antecedents in real-time expert systems with fuzzy set logic. In Proceedings of 22nd annual IEEE Asilomar conference on signals, systems, and computers Pacific Grove, CA, 681–689.
14
[15] Majlender, P. (2005). OWA operators with maximal Rényi entropy. Fuzzy Sets andSystems, 155, 340–360.
15
[16] Jian, Wu., Bo-Liang Sun, Chang-Yong Liang and Shan-Lin Yang. (2009). A linear programming model for determining ordered weighted averaging operator weights with maximal Yager’s entropy. Computers & Industrial Engineering, 57, 742–747
16
[17] Kapur, J. N and Sharma, S. (1999). On measures of M-Entropy, Indian J. pure appl. Math, 30(2), 129-145.
17
[18] Kapur, J. N and Sharma, S. (2002). Some new measures of M-Entropy, Indian J. pure appl. Math, 33(6), 869-893.
18
[19] Yari, G and Chaji, A. (2012). Determination of Ordered Weighted Averaging Operator Weights Based on the M-Entropy Measures. International Journal of Intelligent Systems, 27(12),1020–1033.
19
[20] Yager, R. R. (1995). Measures of entropy and fuzziness related to aggregation operators, Information Sciences, 82, 147-166.
20
[21] Filev, D. and Yager, R. (1995) Analytic Properties of Maximum Entropy OWA Operators. Information Sciences,85, 11 – 27.
21
[22] Carlsson, C., Fuller, R. and Fuller, S. (1997) OWA Operators for Doctoral Student Selection Problem, in: R.R. Yager, J. Kacprzyk (Eds.), The Ordered Weighted Averaging Operators: Theory, Methodology and Applications, Kluwer Academic Publishers, Boston, 167–178.
22
[23] Bordogna, G. Fedrizzi, M. and Pasi, G. (1997). A Linguistic Modeling of Consensus in Group Decision Making Based on OWA Operators, IEEE Trans. Systems, Man, Cybernet.-Pt. A: Systems Humans, 27(1), 126–132.
23
[24] Davey, A., Olson, D. and Wallenius, J. (1994). The Process of Multi Attribute Decision Making: a Case Study of Selecting Applicants for a Ph.D. Program, European J. Oper. Res.(72), 469–484,
24
[25] Smolíková, R. and Wachowiak, M.P. (2002) Aggregation Operators for Selection Problems, Fuzzy Sets and Systems. 131 (1).,23-34
25
]26[ میان آبادی، حجت؛ افشار، عباس (1386). تصمیمگیری گروهی فازی، محاسبه وزن نسبی تصمیمگیرندگان؛ مطالعه کاربردی: انتخاب دانشجویان مقطع دکترا. فصلنامه آموزش مهندسی ایران، شماره 35، سال نهم، صص 53-31.
26
ORIGINAL_ARTICLE
A 3-D Mathematical Modeling in Solving Analytical Consolidation Equation in a Homogeneous Saturated Soil
The drainage length can be artificially decreased using vertical drains where need to accelerate in the consolidation settlement, so that the flow can be drained radially and vertically. In this research, the three dimensional consolidation equation with appropriate boundary conditions was used in cylindrical coordinates, and then was solved analytically. After the analytical solution, analysis of the results was done on using MATLAB software. Moreover, the average degree of consolidation versus time was obtained and compared with the results of one-dimensional method. Findings showed that, the analytical method in this research is in accordance with another numerical method in the previous researches.
https://jamm.scu.ac.ir/article_11487_7fbcc843b3c1f123cc749e819196b871.pdf
2015-11-22
59
74
10.22055/jamm.2015.11487
Three-dimensional consolidation
Cylindrical coordinates
Boundary condition
Radial-vertical drainage
Reza
Poursaki
r-poursaki@mscstu.scu.ac.ir
1
Department of Water Sciences Engineering, Shahid Chamran University
LEAD_AUTHOR
Java
Ahadian
ja_ahadiyan@yahoo.com
2
Department of Water Sciences Engineering, Shahid Chamran University
AUTHOR
Mansour
Seraj
seraj.a@scu.ac.ir
3
Department of Mathematics, Shahid Chamran University
AUTHOR
[1] Nova, R. (2012), Soil mechanics, John Wiley & Sons.
1
[2] McKinley, J.D. (1998), Coupled consolidation of a solid infinite cylinder using a Terzaghi formulation, Journal of Computers and Geotechnics, 23, 193-204.
2
[3] Abbasi, N., Rahimi, H., Javadi, A.A. and Fakher A. (2007), Finite difference approach for consolidation with variable compressibility and permeability, Journal of Computers and Geotechnics, 34, 41-52.
3
[4] Oliaei, M. and Pak, A. (2009), Element free Galerkin meshless method for fully coupled analysis of consolidation process, Scientia Iranica Transaction, A, 16, 65-77.
4
[5] Gibson, R., Schiffman, R.and Pu, S. (1970), Plane strain and axially symmetric consolidation of a clay layer on a smooth impervious base, The Quarterly Journal of Mechanics and Applied Mathematics, 23, 505-520.
5
[6] Zheng, J.J., Lu, Y.E., Yin, J.H. and Guo, J. (2010), Radial consolidation with variable compressibility and permeability following pile installation. Computers and Geotechnics, 37, 408-412.
6
[7] Qin, A., Sun, D., Yang, L. and Weng, Y. (2010), A semi-analytical solution to consolidation of unsaturated soils with the free drainage well, Journal of Computers and Geotechnics, 37, 867–875.
7
[8] Rani, S., Kumar, R. and Singh, S.J. (2011), Consolidation of an Anisotropic Compressible Poroelastic Clay Layer by Axisymmetric Surface Loads, International Journal of Geomechanics, 11, 65-71.
8
[9] Wan-Huan, Z. and Shuai, T. (2012), Unsaturated Consolidation in a Sand Drain Foundation by Differential Quadrature Method, Procedia Earth and Planetary Science, 5, 52-57.
9
[10] Yi, J.T., Lee, F.H., Goh, S.H., Zhang, X.Y. and Wu, J.F. (2012), Eulerian finite element analysis of excess pore pressure generated by spudcan installation into soft clay, Journal of Computers and Geotechnics, 42, 157–170.
10
[11] Wang, C. (2012), Semi-analytical solution of consolidation for composite ground, Mechanics Research Communications, 48, 24– 31.
11
[12] Lu, M., Xie, K., Wang, S. and Li, C. (2013), Analytical solution for the consolidation of a composite foundation reinforced by an impervious column with an arbitrary stress increment, International Journal of Geomechanics, 13, 33-40.
12
[13] Tang, X., Niu, B., Cheng, G. and Shen, H. (2013), Closed-form solution for consolidation of three-layer soil with a vertical drain system, Journal of Geotextiles and Geomembranes, 36, 81–91.
13
[14] Tewatia, S. (2013), Equation of 3D consolidation in cartesian coordinates, International Journal of Geotechnical Engineering, 7, 105-108.
14
[15] Carillo, N. (1942), Simple two and three dimensional cases in the theory of consolidation of soils, Journal of Mathematics and Physics, 21, 11–18.
15
[16] Das, B.M. (2010), Principles of geotechnical engineering, C1 Engineering.
16
[17] Jeffrey, A. (2001), Advanced engineering mathematics, Academic Press, USA.
17
ORIGINAL_ARTICLE
Probabilistic Modeling of Wind Direction Data Related to Meteorological Stations of Kurdistan Province Using Skew Circular Distributions
In studying some phenomena, the researchers are usually encountered with data that are not Euclidean in nature. To investigate the properties of these data, it is required to use some new statistical tools. Statistical methods to analysis these typical data are called non-linear statistics. Circular statistics is an example of this field. After explaining some circular distributions that are able to model skew circular data, probabilistic modeling of wind direction data related to meteorological stations of Kurdistan province is studied in this paper.
https://jamm.scu.ac.ir/article_11492_820d3507e2ec5add039ce410983e84f8.pdf
2015-11-22
75
88
10.22055/jamm.2015.11492
Non-linear statistics
Circular distributions
Skewness
Wind direction data
Mousa
Golalizadeh
golalizadeh@modares.ac.ir
1
Department of Statistics, University of Tarbiat Modares
LEAD_AUTHOR
Hamidreza
Mosaferi Ghomikolaee
hamidreza.mosaferi@modares.ac.ir
2
Department of Statistics, University of Tarbiat Modares
AUTHOR
[1] Mardia, K. V. (1972). Statistics of directional data, Academic Press, London.
1
[2] Mardia, K. V. and Jupp, P. E. (2000). Directional statistics, Wiley, Chichester.
2
[3] Jammalamadaka, S. R. and SenGupta, A. (2001). Topics in Circular Statistics, World Scientific, New Jersey.
3
[4] Azzalini, A. (1985). A Class of distributions which includes the normal ones, Scandinavian Journal of Statistics, 12, 171-178.
4
[5] Pewsey, A. (2000). Problems of inference for Azzalini's skew-normal distribution, Journal of Applied Statistics, 27, 859-870.
5
[6] Pewsey, A. (2000). The Wrapped skew-normal distribution on the circle, Communications in Statistics Theory and Methods, 29, 2459-2472.
6
[7] Pewsey, A. (2006). Modelling asymmetrically distributed circular data using the wrapped skew-normal distribution, Environmental and Ecological Statistics, 13, 257-269.
7
[8] Hernández-Sánchez, E., and Scarpa, B. (2012). A Wrapped flexible generalized skew-normal model for a bimodal circular distribution of wind directions. Chil. J. Statist, 3, 131-143.
8
[9] Gatto, R., and Jammalamadaka, S. R. (2007). The Generalized von Mises distribution, Statistical Methodology, 4, 341-353.
9
[10] Gatto, R. (2008). Some computational aspects of the generalized von Mises distribution, Statistics and Computing, 18, 321-331.
10
[11] Umbach, D., and Jammalamadaka, S. R. (2009). Building asymmetry into circular distributions, Statistics and Probability Letters, 79, 659-663.
11
[12] Abe, T., and Pewsey, A. (2011). Sine-Skewed circular distributions, Statistical Papers, 52, 683-707.
12
[13] نجیبی، س. م؛ و گلعلیزاده، م. (۱۳۸۹). بررسی آماری زوایای جهت وزش باد، نشریه ندا، سال هشتم، شماره دوم، 49-42.
13
ORIGINAL_ARTICLE
Numerical simulation of natural convection heat transfer in a horizontal annulus with discrete heating using mesh-free lattice-Boltzmann method
This study investigates the natural convection heat transfer in a horizontal annulus with discrete heating using a mesh-free lattice Boltzmann method. The lattice Boltzmann method has become an alternative to the conventional computational fluid dynamics methods for simulation of complex fluid flows. The major advantages of the lattice Boltzmann method are the explicit feature of the governing equation, easy for parallel computation, and simple implementation of boundary conditions on curved boundaries. Despite well viability of standard LBM for the uniform mesh, it cannot be directly applied to problems with complex geometry and non-uniform mesh. An efficient method for removing this limitation is to use Taylor series expansion and least squares-based LBM (TLLBM). The final form of the TLLBM is an algebraic formulation with no limitation on the mesh structure. This method can also be applied to any lattice velocity model. In the present work, the TLLBM with D2Q9 lattice model is used to simulate natural convection heat transfer in a horizontal annulus with discrete heating. The effects of Rayleigh number and different arrangement of two heat source-sink pairs on the fluid flow and heat transfer characteristics are investigated.
https://jamm.scu.ac.ir/article_11493_d168589705c02397bb7d94455389f480.pdf
2015-11-22
89
112
10.22055/jamm.2015.11493
Natural convection
Discrete heating
Rayleigh number
Horizontal annulus
Mesh-free lattice-Boltzmann method
Abdolrahman
Dadvand
a.dadvand@mee.uut.ac.ir
1
Department of Mechanical Engineering, Urmia University of Technology
AUTHOR
Ahmad
Haghighi
ah.haghighi@gmail.com
2
Department of Mathematics, Urmia University of Technology
LEAD_AUTHOR
Hamideh
Hoseini Ghejlo
h.hoseini5761@gmail.com
3
Department of Mathematics, Urmia University of Technology
AUTHOR
[1] Basak T., Roy, S., and Pop, I. (2009), Heat flow analysis for natural convection within trapezoidal enclosure based on heat line concept, International Journal of Heat and Mas Transfer, 52(11), 3818-3828.
1
[2] Deng, Q. (2008), Fluid flow and heat transfer characteristics of natural convection in square cavities due to discrete source-sink pairs, International Journal of Heat and Mas Transfer, 51(25), 5949-5957.
2
[3] Abu-Nada, E., Masoud, Z., Oztop, H.F. and Campo, A. (2010), Effect of nanofluid variable properties on natural convection in enclosures, International Journal of Thermal Sciences, 49(3), 479-491.
3
[4] Soleimani, S., Qajarjazi, A., Bararnia, H., Barari, A. and Domairry, G. (2011), Entropy generation due to natural convection in a partially heated cavity by local RBF-DQ method, Meccanica, 46(5), 1023-1033.
4
[5] Oztop, H. F., Abu-Nada, E., Varol, Y. and Chamkha, A. (2011), Natural convection in wavy enclosures with volumetric heat sources, International Journal of Thermal Sciences, 50(4), 502-514.
5
[6] Chen, S., Liu, Z., Bao, S. and Zheng, C. (2010), Natural convection and entropy generation in a vertically concentric annular space, International Journal of Thermal Sciences, 49(12), 2439-2452.
6
[7] Refai, A. G. and Yovanovich, M. M. (1991), Influence of discrete heat source location on natural convection heat transfer in a vertical square enclosure, Journal of Electronic Packaging, 113(3), 268-274.
7
[8] Nelson, E. B., Balakrishnan, A. R. and Murthy, S. S. (1999), Experiments on stratified chilled water tank, Int. J. Refrig., 22(3), 216-234.
8
[9] Ho, C. J. and Lin, Y. H. (1989), A numerical study of natural convection in concentric and eccentric horizontal cylindrical annuli with mixed boundary conditions, International Journal of heat and Fluid Flow, 10(1), 40-47.
9
[10] Asan, H. (2000), Natural convection in an annulus between two isothermal concentric square ducts, International Communication Heat Mass Transfer, 27(3), 367-376.
10
[11] Kumar, De A. and Dalal, A. (2006), A numerical study of natural convection around a square, horizontal, heated cylinder placed in an enclosure, International Journal of Heat and Mass Transfer, 49(23), 4608-4623.
11
[12] Sheikhzadeh, G. A., Ehteram, H. and Aghaei, A. (2013), Numerical study of natural convection in a nanofluid filled enclosure with central heat source and presenting correlations for Nusselt number, Modares Mechanical Engineering, 13(10), 62-74.
12
[13] Dixit, H. N. and Babu, V. (2006), Simulation of high Rayleigh number natural convection in a square cavity using the lattice Boltzmann method, International Journal of Heat and Mass Transfer, 49(3), 727-739.
13
[14] Mohamad, A. A., El-Ganaoui, M. and Bennacer, R. (2009), Lattice Boltzmann and simulation of natural convection in an open ended cavity, International Journal of Thermal Sciences, 48(10), 1870-1875.
14
[15] Bararnia, H., Soleimani, S. and Ganji, D. D. (2011), Lattice Boltzmann simulation of natural convection around a horizontal elliptic cylinder inside a square enclosure, International Communications in Heat and Mass Transfer, 38(10), 1436-1442.
15
[16] Fattahi, E., Farhadi, M. and Sedighi, K. (2011), Lattice Boltzmann simulation of mixed convection heat transfer in eccentric annulus, International Communications in Heat and Mass Transfer, 38(8), 1135-1141.
16
[17] Shi Y., Zhao T. S., Guo Z. L. (2006), Finite difference-based lattice Boltzmann simulation of natural convection heat transfer in a horizontal concentric annulus, Computers and Fluids, 35(1), 1-15.
17
[18] Nazari, M., Kayhani, M. H. and Bagheri A. A. H. (2013), Comparison of heat transfer in a cavity between vertical and horizontal porous layers using LBM, Modares Mechanical Engineering, 13(8), 93-107.
18
[19] Nazari, M. and Shokri, H. (2011), Natural convection in semi-ellipse cavities with variable aspect ratios using lattice Boltzmann method, Modares Mechanical Engineering, 13(10), 1-13.
19
[20] Nazari, M. and Ramazani, S. (2013), Natural convection in a square cavity with a heated obstacle using lattice Boltzmann method, Modares Mechanical Engineering, 11(2), 119-133.
20
[21] Cao, N., Chen, S., Jin, S. and Martinez, D. (1997), Physical symmetry and lattice symmetry in the lattice Boltzmann method, Phys. Rev. E, 55(1), R21.
21
[22] Succi, S., Amati, G. and Benzi, R. (1995), Challenges in lattice Boltzmann computing, J. Stat. Phys. 81(1-2), 5-16.
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