Abstract
In this research paper, the authors aim to establish a novel algorithm in the finite difference method (FDM). The novel idea is proposed in the mesh generation process, the process to generate random grids. The FDM over a randomly generated grid enables fast convergence and improves the accuracy of the solution for a given problem; it also enhances the quality of precision by minimizing the error. The FDM involves uniform grids, which are commonly used in solving the partial differential equation (PDE) and the fractional partial differential equation. However, it requires a higher number of iterations to reach convergence. In addition, there is still no definite principle for the discretization of the model to generate the mesh. The newly proposed method, which is the SM method, employed randomly generated grids for mesh generation. This method is compared with the uniform grid method to check the validity and potential in minimizing the computational time and error. The comparative study is conducted for the first time by generating meshes of different cell sizes, i.e.,
Abbreviations
 2D

twodimensional
 CGM

conjugate gradient method
 FDM

finite difference method
 GS

Gauss–Siedel
 PDE

partial differential equation
 SM’s method

Sanaullah Mastoi’s method
1 Introduction
The numerical solution of the partial differential equation (PDE) is mostly solved by the finite difference method (FDM). The FDM is an approximate numerical method to find the approximate solutions for the problems arising in mathematical physics [1], engineering, and wideranging phenomenon, including transient, linear, nonlinear and steady state or nontransient cases [2,3,4]. This method is applied to various geometries, distinct boundary conditions and bodies composed of different materials [5]. FDM applications can easily solve the complex discretization and mathematical modeling computational codes [6]. Therefore, FDM has been considered to be the critical method for solving practical models [7]. Many numerical solution techniques [8] are now available to solve PDEs after highperformance[9] computing development with extensive storage capability [10,11,12,13,14]. However, the most desirable method for the solution of PDEs is the FDM [6,15].
The heat equation model is an elliptic type PDE [16,17]. The engineering model is commonly solved using the FDM on uniform grids [18]. However, very few studies have been reported that have applied random step sizes [5,11]. The heat equation theory was developed for modeling how heat diffuses through a given region [19,20,21]. The homogeneous heat equation plays a vital role in studying heat conduction and other diffusion processes [6,22].
PDEs can be used to solve various engineering, mathematical, and medical problems involving multiple unknown variables [23,24,25,26]. The numerical schemes with theoretical analysis focus on the convergence of numerical solution and it is observed that the behavior of randomness was found better during random walk [27,28,29]. The various numerical systems are only consistent with limited Fourier steps, spatial steps and the converging steps during the continuous time and solution. The simulation was repeated several times and it was observed that random meshes could reduce the iteration due to their random behavior in cell sizes [30,31].
The numerical solution of PDE using the FDM over uniform meshes has been extensively studied. However, randomly generated grids are also proposed [32,33,34]. To the best of the authors’ knowledge, no study is found in the open literature that discusses the validity and potential of randomly generated grids over uniform grids. The solution of fractional PDEs [35,36], Laplace transform, the inverse Laplace transform method through Bernoulli polynomials [36,37,38], PDE solutions compared with the domain decomposition method, the
The geometric representation for the grids (mesh) on the larger domain is discretized in smaller grids. Mesh types such as moving mesh and coupled moving mesh are applied to quasilinear PDEs for the solution of problems such as magnetohydrodynamics and the freesurface viscoelastic flow [47,48,49,50,51,52]. The mesh generation process needs advancement in engineering models for accuracy and less computational cost [53]. Additionally, strand meshes have also been used to solve problems involving the flow of multiphase, viscous, and lowspeed fluids. However, the mesh generation process is just a past practice. Typically, selecting a suitable mesh generation process for a particular problem becomes difficult, as no established principle is available and various studies have discussed the mesh sizes used for solving scientific problems [33,54–58].
The FDM relies on the discretization of function on grids or meshes. There is no principle for mesh generations. Usually, the FDM uses uniform or regular grids, but the SM method focuses on randomly generated grids. A special treatment in solving the SM (Sanaullah Mastoi) method compared to the FDM is the discretization process of regular grids to random grids. In this method, we use mathematical software programs like MATLAB, ANSYS, etc.
The implementation of the SM method is by analyzing the effect of samples of random meshes on the convergence of the numerical solution of the differential equation.
In this study, we investigated the potential of using randomly generated grids. Furthermore, a detailed comparison with the uniformly generated grids is also performed to check the method’s validity.
2 Numerical methodology
2.1 Heat conduction model
A simple demonstration of the 2D heat condition model is shown in Figure 1 [59]. In this model, the left bar behaves as a heat source and the right bar as a heat sink. Due to the temperature gradient between them, the heat flows from the heat source to the heat sink. Considering the two opposite faces (each of crosssectional area A), the heat source is at temperature 100°C and the heat sink is at temperature 25°C, while the heat covers the length (m, n) in t seconds during the thermal transition phase.
2.2 Finite difference formulation using onedimensional (1D) differential equation
Consider the onedimensional (1D) steadystate heat conduction as in Figure 2(a) by internal heat generation using Eq. (1):
At node
Figures 2(b) and (c) show the numerical solution versus the exact solution [19]. The mathematical steadystate heat conduction model used the grids at
Now the finitedifference approximation of the heat conduction equations is
We use the random step size for the randomly generated grids to solve onedimensional heat equations to get random realizations:
where
3 Finite difference formulation using twodimensional (2D) differential equation
The twodimensional PDE (heat equation) with initial and boundary conditions are
And the given conditions are
where
3.1 Discretization of the heat equation using the FDM
Eq. (6) is discretized with the help of FDM:
Figure 4 specifies the domain, which is discretized accordingly as an FDM mesh generation.
3.2 Uniform grids
Meshes are generated with the help of MATLAB. Codes were employed individually on the interior nodes by the Gauss–Seidel (GS) iterative method explicitly, as shown in Eq. (6):
where
After solving Eq. (7), sufficient uniform numerical results were obtained with
S. no.  Mesh sizes

Step sizes

Nodes  No. of cells  Avg cell size  Sd (standard deviation) 

1 

0.100  125 

0.100  0 
2  20 × 20  0.050  442 

0.050  0 
3  30 × 30  0.033  962 

0.033  0 
4  40 × 40  0.025  1,682 

0.025  0 
As shown in Table 1, uniform grids are used to solve 2D PDEs, and sample realization is used to test random grids’ practicability over uniform grids. The sample realizations are shown in Figure 5(a)–(d). Then, the numerical solution was executed on each mesh GS iterative technique. This method is known as the Liebmann method or the method of successive displacement, which is an iterative method for the approximation of the system of linear equations. However, it can be applied on the system or matrix with nonzero components on the diagonals; convergence is always guaranteed if and only if the matrix is either strictly symmetric or diagonally dominant [32] and positive significantly.
3.3 Random grids
The random grids (random samples) or randomly generated grids are generated with each having grid sizes of 10 × 10, 20 × 20, 30 × 30, 40 × 40, 60 × 60, 70 × 70, 80 × 80, 90 × 90 and 100 × 100. Grids are generated with the help of MATLAB, with the builtin “rand” function (specified mesh size). In each realization, various cell sizes are used, namely h and k, as shown in Table 2.
Mesh size  Cell size  Standard deviation (cell size)  No. of iterations  

Maximum  Average  Random grids  Uniform grids  
10 × 10  0.14137  0.01  0.01269  114  125 
20 × 20  0.03398  0.00250  0.00239  382  390 
30 × 30  0.01301  0.00111  0.00079  646  732 
40 × 40  0.02386  0.00063  0.00092  987  1116 
The numerical solution for uniform meshes employed by Eq. (6), the condition changed for random grids due to the sudden change of step size (h in x directions and k in y directions), is shown in Figure 6. Different approaches are possible in this discretization; as seen in Figure 6, we have various neighboring grids
4 Statistical significance (uniform vs random meshes)
The randomly generated grids reduced the number of iterations and computational time compared to uniform grids. Therefore, the relationship between uniform grids and randomly generated grids justifies the significance of randomly generated grids. However, the number of iterations is established in the eighthdegree polynomial regression equation, where the parameters lie within the 95% confidence interval test. The numerical results of the goodness of fit for the uniformly and randomly generated grids on each realization are presented in Figure 7(a)–(j). The findings conclude that randomly generated grids will converge with decreased computational cost and converging iteration.
The regression fit for the random grids shows that the linear regression consists of finding the bestfitting straight line through the uniform meshes versus random grid points.
5 Mathematical solutions
5.1 Analytical solution of PDEs
Analytical solutions of 2D PDEs are obtained through the separation of variables. Figure 8 represents an exact solution of 2D heat equations.
5.2 Numerical solution
5.2.1 Numerical solution profiles over uniform meshes
The numerical solution of twodimensional PDEs applied over uniform grids is defined in Figure 9(a)–(j). The cell sizes are
The heat (or thermal) energy of a body with uniform properties is presented in the local profile solutions, where the temperature varies from 25 to 100°C.
5.2.2 Numerical solution profiles over random meshes
The numerical method for solving 2D PDEs using the FDM over random meshes is described in Figure 10(a)–(d) (i = first realization and ii = second realization). In this solution, we presented the solution based on two realizations of randomly generated grids of size
The local profile solutions on the random mesh size with random realizations in the heat equation deviate from the temperature.
5.3 Computational time and percentage deductions in computational time
The randomly generated grid over uniform grids is analyzed, and the key output parameters (i.e., converging iterations, computational time (s) and percentage (%) reduction in computational time) are compared in the numerical solution. The improving iterations and the computational time (uniform versus randomly generated grids) are presented in Table 3. The computational time for the cell sizes could reduce up to 43% from uniformly generated grids to randomly generated grids
S. no  Cell size  Converging iteration  Computational time (s)  Percentage reduction (%)  

Uniform grids  Random grids  Uniform grids  Random grids  
1 

125  114  0.0189  0.0147  22.22 
2 

382  390  0.1246  0.1102  11.55 
3 

646  732  0.3154  0.2652  15.9 
4 

987  1,116  0.6094  0.4401  27.78 
6 Conclusion
This research compares the numerical solutions of the heat conduction equation and computational convergence over uniform and random meshes. The numerical solution uses the FDM over both random grids or randomly generated grids and uniformly generated grids. It has been observed that the numerical solutions obtained through randomly generated grids have shown fast convergence compared to the solutions achieved by uniform meshes. Therefore, the idea of getting a numerical solution using the randomly generated finite difference meshes has been tested and found feasible and practicable. Furthermore, the computational time for the cell sizes could reduce up to 43%, which is the achievement from uniformly generated grids to randomly generated grids. The idea of SM’s method has been extended and implemented in fractional calculus.
This research can further be extended in different directions to explore the randomly generated grids’ practicability over other methods. The sensitivity of random mesh parameters can also be analyzed. This study can be expanded in the CGM to use randomly generated grids. This research area needs further investigation into computational time and pointwise comparison of the numerical solution with uniform vs random meshes because sufficient research gaps are available.
Acknowledgements
The authors thank QUEST Pakistan and the Institute of Mathematical Science, Faculty of Science, University of Malaya, Malaysia, for providing an environment and facilities to fulfill the research goal.

Funding information: The authors thank QUEST University Pakistan for providing funding Grant Number (QUEST)/NH/FDP(ECL)/67/07032014 under the project of the faculty development program.

Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Conflict of interest: The authors state no conflict of interest.
References
[1] Ahmad H, Seadawy AR, Khan TA, Thounthong P. Analytic approximate solutions for some nonlinear Parabolic dynamical wave equations. J Taibah Univ Sci. 2020;14:346–58.10.1080/16583655.2020.1741943Search in Google Scholar
[2] Lei N, Zheng X, Luo Z, Luo F, Gu X. Quadrilateral mesh generation II: meromorphic quartic differentials and Abel–Jacobi condition. Computer Methods Appl Mech Eng. 2020;366:112980.10.1016/j.cma.2020.112980Search in Google Scholar
[3] AlShawba AA, Gepreel K, Abdullah F, Azmi A. Abundant closed form solutions of the conformable time fractional SawadaKoteraIto equation using (G′/G)expansion method. Results Phys. 2018;9:337–43.10.1016/j.rinp.2018.02.012Search in Google Scholar
[4] Saeed ST, Riaz MB, Baleanu D, Akgül A, Husnine SM. Exact analysis of second grade fluid with generalized boundary conditions. Intell Autom Soft Comput. 2021;28:547–59.10.32604/iasc.2021.015982Search in Google Scholar
[5] Shih AM, Yu TY, Gopalsamy S, Ito Y, Soni B. Geometry and mesh generation for high fidelity computational simulations using nonuniform rational Bsplines. Appl Numer Math. 2005;55:368–81.10.1016/j.apnum.2005.04.036Search in Google Scholar
[6] Thomas JW. Numerical partial differential equations: finite difference methods. Springer Science & Business Media; 2013. https://link.springer.com/book/10.1007/9781489972781.Search in Google Scholar
[7] Yang WY, Cao W, Kim J, Park KW, Park HH, Joung J et al. Applied numerical methods using MATLAB. John Wiley & Sons; 2020. https://onlinelibrary.wiley.com/doi/book/10.1002/0471705195.10.1002/9781119626879Search in Google Scholar
[8] Rashid S, Ahmad H, Khalid A, Chu YM. On discrete fractional integral inequalities for a class of functions. Complexity. 2020;2020:1–13.10.1155/2020/8845867Search in Google Scholar
[9] Abouelregal AE, Moustapha MV, Nofal TA, Rashid S, Ahmad HH. Generalized thermoelasticity based on higherorder memorydependent derivative with time delay. Results Phys. 2021;20:103705.10.1016/j.rinp.2020.103705Search in Google Scholar
[10] Samaniego E, Anitescu C, Goswami S, NguyenThanh VM, Guo H, Hamdia K, et al. An energy approach to the solution of partial differential equations in computational mechanics via machine learning: concepts, implementation and applications. Computer Methods Appl Mech Eng. 2020;362:112790.10.1016/j.cma.2019.112790Search in Google Scholar
[11] Lenz S, Geier M, Krafczyk M. An explicit gas kinetic scheme algorithm on nonuniform Cartesian meshes for GPGPU architectures. Computers Fluids. 2019;186:58–73.10.1016/j.compfluid.2019.04.011Search in Google Scholar
[12] Khan D, Yan DM, Wang Y, Hu K, Ye J, Zhang X. Highquality 2D mesh generation without obtuse and small angles. Computers Math Appl. 2018;75:582–95.10.1016/j.camwa.2017.09.041Search in Google Scholar
[13] Imani G. Lattice Boltzmann method for conjugate natural convection with heat generation on nonuniform meshes. Computers Math Appl. 2020;79:1188–207.10.1016/j.camwa.2019.08.021Search in Google Scholar
[14] Yu F, Zeng Y, Guan Z, Lo S. A robust DelaunayAFT based parallel method for the generation of largescale fully constrained meshes. Computers Struct. 2020;228:106170.10.1016/j.compstruc.2019.106170Search in Google Scholar
[15] Gu Y, Wang L, Chen W, Zhang C, He X. Application of the meshless generalized finite difference method to inverse heat source problems. Int J Heat Mass Transf. 2017;108:721–9.10.1016/j.ijheatmasstransfer.2016.12.084Search in Google Scholar
[16] Ali U, Ahmad H, Baili J, Botmart T, Aldahlan MA. Exact analytical wave solutions for spacetime variableorder fractional modified equal width equation. Results Phys. 2022;12:105216.10.1016/j.rinp.2022.105216Search in Google Scholar
[17] Mahmood A, Basir MFMD, Ali U, Mohd Kasihmuddin MS, Mansor M. Numerical solutions of heat transfer for magnetohydrodynamic jefferyhamel flow using spectral Homotopy analysis method. Processes. 2019;7:626.10.3390/pr7090626Search in Google Scholar
[18] Lu F, Qi L, Jiang X, Liu G, Liu Y, Chen B, et al. NNWGridStar: interactive structured mesh generation software for aircrafts. Adv Eng Softw. 2020;145:102803.10.1016/j.advengsoft.2020.102803Search in Google Scholar
[19] Kumar M, Joshi P. A mathematical model and numerical solution of a one dimensional steady state heat conduction problem by using high order immersed interface method on nonuniform mesh. Int J Nonlinear Sci. 2012;14:11–22.Search in Google Scholar
[20] EbrahimiFizik A, Lakzian E, Hashemian A. Numerical investigation of wet inflow in steam turbine cascades using NURBSbased mesh generation method. Int Commun Heat Mass Transf. 2020;118:104812.10.1016/j.icheatmasstransfer.2020.104812Search in Google Scholar
[21] Ali U, Mastoi S, Othman WA, Khater MM, Sohail M. Computation of traveling wave solution for nonlinear variableorder fractional model of modified equal width equation. AIMS Math. 2021;6(9):10055–69.10.3934/math.2021584Search in Google Scholar
[22] Ali U, Kamal R, MohyudDin ST. On nonlinear fractional differential equations. Int J Mod Math Sci. 2012;3:58–73. 10.1016/j.compfluid.2019.04.011.Search in Google Scholar
[23] Miller KS. Partial differential equations in engineering problems. Courier Dover Publications; 2020. https://store.doverpublications.com/0486843297.html.10.1201/97810030668354Search in Google Scholar
[24] Strauss WA. Partial differential equations: An introduction. John Wiley & Sons; 2007. https://www.wiley.com/enus/Partial+Differential+Equations%3A+An+Introduction%2C+2nd+Editionp9781119496694.Search in Google Scholar
[25] Wang H, Yamamoto N. Using a partial differential equation with Google Mobility data to predict COVID19 in Arizona. Math Biosci Eng. 2020;17:4891–4904.10.3934/mbe.2020266Search in Google Scholar PubMed
[26] Alam MK, Memon K, Siddiqui A, Shah S, Farooq M, Ayaz M, et al. Modeling and analysis of high shear viscoelastic Ellis thin liquid film phenomena. Phys Scr. 2021;96:055201.10.1088/14024896/abe4f2Search in Google Scholar
[27] Andreasen J, Huge BN. Finite difference based calibration and simulation. 2010. p. 1697545, Available at SSRN. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1697545.10.2139/ssrn.1697545Search in Google Scholar
[28] Ali U, Sohail M, Abdullah FA. An efficient numerical scheme for variableorder fractional subdiffusion equation. Symmetry. 2020;12:1437.10.3390/sym12091437Search in Google Scholar
[29] Liu Q, Liu F, Turner I, Anh V. Approximation of the Lévy–Feller advection–dispersion process by random walk and finite difference method. J Comput Phys. 2007;222:57–70.10.1016/j.jcp.2006.06.005Search in Google Scholar
[30] Kamrani M. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discret Cont Dyn SystB. 2019;24:5337.10.3934/dcdsb.2019061Search in Google Scholar
[31] Khater M, Ali U, Khan MA, Mousa AA, Attia RA. A new numerical approach for solving 1D fractional diffusionwave equation. J Funct Spaces. 2021;2021:1–7. 10.1155/2021/6638597.Search in Google Scholar
[32] Dinesh TAVVSSPMSS. Potential flow simulation through lagrangian interpolation meshless method coding. J Appl Fluid Mech. 2018;11:7.10.36884/jafm.11.SI.29429Search in Google Scholar
[33] Song C, Ooi ET, Natarajan S. A review of the scaled boundary finite element method for twodimensional linear elastic fracture mechanics. Eng Fract Mech. 2018;187:45–73.10.1016/j.engfracmech.2017.10.016Search in Google Scholar
[34] Bibi M, Nawaz Y, Arif MS, Abbasi JN, Javed U, Nazeer A. A finite difference method and effective modification of gradient descent optimization algorithm for MHD fluid flow over a linearly stretching surface. Computers Mater Continua. 2020;62:657–77.10.32604/cmc.2020.08584Search in Google Scholar
[35] Song P, Karniadakis GE. Fractional magnetohydrodynamics: Algorithms and applications. J Comput Phys. 2019;378:44–62.10.1016/j.jcp.2018.10.047Search in Google Scholar
[36] Ali U, Abdullah FA. Modified implicit difference method for onedimensional fractional wave equation. InAIP Conf Proc. 2019;2184(1):060021, AIP Publishing LLC.10.1063/1.5136453Search in Google Scholar
[37] BarSinai Y, Hoyer S, Hickey J, Brenner MP. Learning datadriven discretizations for partial differential equations. Proc Natl Acad Sci U S A. 2019;116:15344–9.10.1073/pnas.1814058116Search in Google Scholar PubMed PubMed Central
[38] Chen CS, Fan CM, Wen P. The method of approximate particular solutions for solving certain partial differential equations. Numer Methods Partial Differ Equ. 2012;28:506–22.10.1002/num.20631Search in Google Scholar
[39] Xiao Z, He S, Xu G, Chen J, Wu Q. A boundary elementbased automatic domain partitioning approach for semistructured quad mesh generation. Eng Anal Bound Elem. 2020;113:133–44.10.1016/j.enganabound.2020.01.003Search in Google Scholar
[40] Meng H, Lien FS, Yee E, Shen J. Modelling of anisotropic beam for rotating composite wind turbine blade by using finitedifference timedomain (FDTD) method. Renew Energy. 2020;162:2361–79.10.1016/j.renene.2020.10.007Search in Google Scholar
[41] Xue W, Wan P, Li Q, Zhong P, Yu G, Tao T. An online conjugate gradient algorithm for largescale data analysis in machine learning. AIMS Math. 2012;6:1515–37.10.3934/math.2021092Search in Google Scholar
[42] Ahmad H. Variational iteration method with an auxiliary parameter for solving differential equations of the fifth order. Nonlinear Sci Lett A. 2018;9:27–35.10.5899/2018/jnaa00417Search in Google Scholar
[43] Ahmad H. Auxiliary parameter in the variational iteration algorithmII and its optimal determination. Nonlinear Sci Lett A. 2018;9:62–72.Search in Google Scholar
[44] Wen C, Hu QY, Pu BY, Huang YY. Acceleration of an adaptive generalized Arnoldi method for computing PageRank. AIMS Math. 2021;6:893–907.10.3934/math.2021053Search in Google Scholar
[45] Shchepetkin AF, McWilliams JC. The regional oceanic modeling system (ROMS): a splitexplicit, freesurface, topographyfollowingcoordinate oceanic model. Ocean Model. 2005;9:347–404.10.1016/j.ocemod.2004.08.002Search in Google Scholar
[46] Zhang ZH, Liao XL, Shi ZY, Lowry AR, Yu A, Lu RQ, et al. Highprecision downward continuation of potential fields algorithm utilizing adaptive damping coefficient of generalized minimal residuals. Appl Geophys. 2021;17:1–15.10.1007/s117700200858ySearch in Google Scholar
[47] Duan J, Tang H. Entropy stable adaptive moving mesh schemes for 2D and 3D special relativistic hydrodynamics. J Comput Phys. 2020;426:109949.10.1016/j.jcp.2020.109949Search in Google Scholar
[48] Yan Z, Rennie CD, Mohammadian A. Numerical modeling of local scour at a submerged weir with a downstream slope using a coupled movingmesh and maskedelement approach. Int J Sediment Res. 2020;36:279–290.10.1016/j.ijsrc.2020.06.007Search in Google Scholar
[49] Uzunca M, Karasözen B, Küçükseyhan T. Moving mesh discontinuous Galerkin methods for PDEs with traveling waves. Appl Math Comput. 2017;292:9–18.10.1016/j.amc.2016.07.034Search in Google Scholar
[50] Ghosh U. Electromagnetohydrodynamics of nonlinear viscoelastic fluids. J NonNewtonian Fluid Mech. 2020;277:104234.10.1016/j.jnnfm.2020.104234Search in Google Scholar
[51] Liu Z. Algebraic L2decay of weak solutions to the magnetohydrodynamic equations. Nonlinear Anal: Real World Appl. 2019;50:267–89.10.1016/j.nonrwa.2019.05.001Search in Google Scholar
[52] Riaz A, Alolaiyan H, Razaq A. Convective heat transfer and magnetohydrodynamics across a peristaltic channel coated with nonlinear nanofluid. Coatings. 2019;9:816.10.3390/coatings9120816Search in Google Scholar
[53] Shyu SJ. Image encryption by random grids. Pattern Recognit. 2007;40:18–1031.10.1016/j.patcog.2006.02.025Search in Google Scholar
[54] Abbruzzese G, Gómez M, CorderoGracia M. Unstructured 2D grid generation using oversetmesh cutting and singlemesh reconstruction. Aerosp Sci Technol. 2018;78:637–47.10.1016/j.ast.2018.05.004Search in Google Scholar
[55] Areias P, Reinoso J, Camanho P, de Sá JC, Rabczuk T. Effective 2D and 3D crack propagation with local mesh refinement and the screened Poisson equation. Eng Fract Mech. 2018;189:339–60.10.1016/j.engfracmech.2017.11.017Search in Google Scholar
[56] Sohail M, Ali U, Zohra FT, AlKouz W, Chu YM, Thounthong P. Utilization of updated version of heat flux model for the radiative flow of a nonNewtonian material under Joule heating: OHAM application. Open Phys. 2021;19(1):100–10.10.1515/phys20210010Search in Google Scholar
[57] RodaCasanova V, SanchezMarin F. Development of a multiblock procedure for automated generation of twodimensional quadrilateral meshes of gear drives. Mech Mach Theory. 2020;143:103631.10.1016/j.mechmachtheory.2019.103631Search in Google Scholar
[58] Zhang Y, Jia Y. 2D automatic bodyfitted structured mesh generation using advancing extraction method. J Comput Phys. 2018;353:316–35.10.1016/j.jcp.2017.10.018Search in Google Scholar
[59] Khaled AR. Modeling and computation of heat transfer through permeable hollowpin systems. Adv Mech Eng. 2012;4:587165.10.1155/2012/587165Search in Google Scholar
© 2022 Sanaullah Mastoi et al., published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.