Authors:Sanaullah Mastoi,Wan Ainun Mior othman,Umair Ali,Umair Ahmed Rajput,Ghulam Fizza,
Keywords:Partial differential equation,Finite difference method,Polar coordinates,Randomly generated grids,Uniform meshes,fractional-order Legendre functions,
AbstractThere are various methods to solve the physical life problem involving engineering, scientific and biological systems. It is found that numerical methods are approximate solutions. In this way, randomly generated finite difference grids achieve an approximation with fewer iterations. The idea of randomly generated grids in cartesian coordinates and polar form are compared with the exact, iterative method, uniform grids, and approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions. The most ideal and benchmarking method is the finite difference method over randomly generated grids on Cartesian coordinates, polar coordinates used for numerical solutions. This concept motivates the investigation of the effects of the randomly generated meshes. The two-dimensional equation is solved over randomly generated meshes to test randomly generated grids and the implementation. The feasibility of the numerical solution is analyzed by comparing simulation profiles.
I. A. Mahmood, M.F. Md Basir, U. Ali, M.S. Mohd Kasihmuddin, andM. Mansor, Numerical solutions of heat transfer for magneto hydrodynamic jeffery-hamel flow using spectral Homotopy analysis method. Processes 7 (2019) 626.
II. A.-R. Khaled, Modeling and computation of heat transfer through permeable hollow-pin systems. Advances in Mechanical Engineering 4 (2012) 587165.
III. A.E. Abouelregal, M.V. Moustapha, T.A. Nofal, S. Rashid, andH. Ahmad, Generalized thermoelasticity based on higher-order memory-dependent derivative with time delay. Results in Physics 20 (2021) 103705.
IV. C.S.E.T.O.S. Natarajan, A review of the scaled boundary finite element method for two-dimensional linear elastic fracture mechanics. Engineering Fracture Mechanics 187 (2018) 43.
V. C. Song, E.T. Ooi, andS. Natarajan, A review of the scaled boundary finite element method for two-dimensional linear elastic fracture mechanics. Engineering Fracture Mechanics 187 (2018) 45-73.
VI. D.S. Lubinsky, and I.E. Pritsker, Variance of real zeros of random orthogonal polynomials. Journal of Mathematical Analysis and Applications 498 (2021) 124954.
VII E. Samaniego, C. Anitescu, S. Goswami, V.M. Nguyen-Thanh, H. Guo, K. Hamdia, X. Zhuang, and T. Rabczuk, An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering 362 (2020) 112790.
VIII. G.B. Folland, Introduction to partial differential equations, Princeton university press, 2020.
IX H. Meng, F.-S. Lien, E. Yee, andJ. Shen, Modelling of anisotropic beam for rotating composite wind turbine blade by using finite-difference time-domain (FDTD) method. Renewable Energy 162 (2020) 2361-2379.
X. H. Ahmad, Auxiliary parameter in the variational iteration algorithm-II and its optimal determination. Nonlinear Sci. Lett. A 9 (2018) 62-72.
XI. H. Ahmad, Variational iteration method with an auxiliary parameter for solving differential equations of the fifth order. Nonlinear Sci. Lett. A 9 (2018) 27-35.
XII. H. Ahmad, A.R. Seadawy, T.A. Khan, andP. Thounthong, Analytic approximate solutions for some nonlinear Parabolic dynamical wave equations. Journal of Taibah University for Science 14 (2020) 346-358.
XIII. H. Wang, and N. Yamamoto, Using a partial differential equation with Google Mobility data to predict COVID-19 in Arizona. Mathematical Biosciences and Engineering 17 (2020).
XIV. J.W. Thomas, Numerical partial differential equations: finite difference methods, Springer Science & Business Media, 2013.
XV. J.H. He, and Y.O. El‐Dib, The reducing rank method to solve third‐order Duffing equation with the homotopy perturbation. Numerical Methods for Partial Differential Equations 37 (2021) 1800-1808.
XVI. J. Andreasen, and B.N. Huge, Finite difference based calibration and simulation. Available at SSRN 1697545 (2010).
XVII. K.S. Miller, Partial differential equations in engineering problems, Courier Dover Publications, 2020.
XVIII. M. Bibi, Y. Nawaz, M.S. Arif, J.N. Abbasi, U. Javed, andA. Nazeer, A Finite Difference Method and Effective Modification of Gradient Descent Optimization Algorithm for MHD Fluid Flow Over a Linearly Stretching Surface. Computers, Materials & Continua 62 (2020) 657-677.
XIX. Q. Liu, F. Liu, I. Turner, andV. Anh, Approximation of the Lévy–Feller advection–dispersion process by random walk and finite difference method. Journal of Computational Physics 222 (2007) 57-70.
XX. Rafiuddin, Noushima Humera. G., : NUMERICAL SOLUTION OF UNSTEADY, : TWO DIMENSIONAL HYDROMAGNETICS FLOW WITH HEAT AND MASS TRANSFER OF CASSON FLUID. J. Mech. Cont.& Math. Sci., Vol.-15, No.-9, September (2020) pp 17-30. DOI: 10.26782/jmcms.2020.09.00002
XI. S. Mastoi, A.B. Mugheri, N.B. Kalhoro, andA.S. Buller, Numerical solution of Partial differential equations(PDE’s) for nonlinear Local Fractional PDE’s and Randomly generated grids. International Journal of Disaster Recovery and Business Continuity 11 (2020) 2429-2436.
XXII. S. Mastoi, N.B. Kalhoro, A.B.M. Mugheri, U.A. Rajput, R.B. Mastoi, N. Mastoi, and W.A.M. Othman, Numerical solution of Partial differential equation using finite random grids. International Journal of Advanced Research in Engineering and Technology (IJARET) (2021).
XXIII. S. Yao, W. Gu, C. Zhu, S. Zhou, Z. Wu, andZ. Zhang, A Novel Cross Iteration Method for Dynamic Energy Flow Calculation of the Hot-water Heating Network in the Integrated Energy System, 2020 IEEE 4th Conference on Energy Internet and Energy System Integration (EI2), IEEE, pp. 23-28.
XXIV. S. Mastoi, and W.A.M. Othman, A Finite difference method using randomly generated grids as non-uniform meshes to solve the partial differential equation. International Journal of Disaster Recovery and Business Continuity 11 (2020) 1766-1778.
XXV. S. Mastoi, N.B. Kalhoro, A.B. Mugheri, A.S. Buller, U.A. Rajput, R.B. Mastoi, andN. Mastoi, A Statistical analysis for Mathematics & Statistics in Engineering Technologies (Random Sampling). International Journal of Management 12 (2021) 416-421.
XXVI. S. Mastoi, A.B.M. Mugheri, N.B. Kalhoro, U.A. Rajput, R.B. Mastoi, N. Mastoi, andW.A.M. Othman, Finite difference algorithm on Finite random grids. International Journal of Advanced Research in Engineering and Technology (IJARET) (2021).
XXVII. S. Mastoi, W.A.M. Othman, andK. Nallasamy., Numerical Solution of Second order Fractional PDE’s by using Finite difference Method over randomly generated grids. International Journal of Advance Science and Technology 29 (2020) 373-381.
XXVIII. S. Mastoi, W.A.M. Othman, andK. Nallasamy., Randomly generated grids and Laplace Transform for Partial differential equation. International Journal Disaster Recovery and Business continuity 11 (2020) 1694-1702.
XXIX. S.J. Shyu, Image encryption by random grids. Pattern Recognition 40 (2007) 18.
XXX. S.-T. Saeed, M.-B. Riaz, D. Baleanu, A. Akgül, andS.-M. Husnine, Exact Analysis of Second Grade Fluid with Generalized Boundary Conditions. Intelligent Automation & Soft Computing 28 (2021) 547–559.
XXXI. T. Bonnafont, R. Douvenot, andA. Chabory, A local split‐step wavelet method for the long range propagation simulation in 2D. Radio Science 56 (2021) e2020RS007114.
XXXII. T.A.V.V.S.S.P.M.S.S. Dinesh., Potential Flow Simulation through Lagrangian Interpolation Meshless Method Coding. Journal of Applied Fluid Mechanics 11 (2018) 7.
XXXIII. U. Ali, M. Sohail, andF.A. Abdullah, An efficient numerical scheme for variable-order fractional sub-diffusion equation. Symmetry 12 (2020) 1437.
XXXIV. U. Rashid, H. Liang, H. Ahmad, M. Abbas, A. Iqbal, andY. Hamed, Study of (Ag and TiO2)/water nanoparticles shape effect on heat transfer and hybrid nanofluid flow toward stretching shrinking horizontal cylinder. Results in Physics 21 (2021) 103812.
XXXV. W. Cao, W. Huang, andR.D. Russell, Anr-adaptive finite element method based upon moving mesh PDEs. Journal of Computational physics 149 (1999) 221-244.
XXXVI. Y. Sun, W. Sun, andH. Zheng, Domain decomposition method for the fully-mixed Stokes–Darcy coupled problem. Computer Methods in Applied Mechanics and Engineering 374 (2021) 113578.
XXXVII. Y. Bar-Sinai, S. Hoyer, J. Hickey, andM.P. Brenner, Learning data-driven discretizations for partial differential equations. Proc Natl Acad Sci U S A 116 (2019) 15344-15349.
XXXVIII. Y. Gu, L. Wang, W. Chen, C. Zhang, andX. He, Application of the meshless generalized finite difference method to inverse heat source problems. International Journal of Heat and Mass Transfer 108 (2017) 721-729.