Solving Double Integration With The Help of Monte Carlo Simulation: A Python Approach

Priyanshi Mishra¹ , Ayush Sharma² , Dhananjay R. Mishra³ , Pankaj Dumka⁴

1. Department of Computer Science Engineering, Jaypee University of Engineering and Technology, Guna-473226, Madhya Pradesh, India.
2. Department of Computer Science Engineering, Jaypee University of Engineering and Technology, Guna-473226, Madhya Pradesh, India.
3. Department of Mechanical Engineering, Jaypee University of Engineering and Technology, Guna-473226, Madhya Pradesh, India Department of Mechanical Engineering, Jaypee University of Engineering and Technology, Guna-473226, Madhya Pradesh, India.

Section:Research Paper, Product Type: Journal-Paper
Vol.9 , Issue.3 , pp.6-10, Mar-2023

Online published on Mar 31, 2023

Copyright © Priyanshi Mishra, Ayush Sharma, Dhananjay R. Mishra, Pankaj Dumka . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

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Abstract :
In this article, a model of the double integration process has been attempted using Monte Carlo simulation. Both non-rectangular and rectangular domains are dealt separately. Python programming has been used to automate the entire random number-based integration process. Three example problems are used to test the newly created cods, and the results achieved are consistent with those reported in the literature.

Key-Words / Index Term :
Double integration; Monte Carlo simulation; Monte Carlo integration; Python programming; Integration; Stochastic Integration

References :
[1] J. D. Faires and R. L. Burden, Numerical methods. Thomson, 2003.
[2] G. Beer, B. Marussig, and C. Duenser, “Numerical integration,” Lect. Notes Appl. Comput. Mech., vol. 90, pp. 129–149, 2020, doi: 10.1007/978-3-030-23339-6_7.
[3] S. Mordechai, Applications of Monte Carlo Method in Science and Engineering. InTech, 2012. doi: 10.5772/1954.
[4] R. L. Harrison, “Introduction to Monte Carlo simulation,” in AIP Conference Proceedings, 2009, vol. 1204, no. 1, pp. 17–21. doi: 10.1063/1.3295638.
[5] C. Doloi, J. Gogoi, and G. Pradhani, “Research Paper A Cascade System Reliability Model for Rama Distributed Random Variables,” Int. J. Sci. Res. Math. Stat. Sci., vol. 10, no. 1, pp. 1–9, 2023.
[6] Nikita and S. Malik, “Generalized Logarithmic Ratio and Product-Type Estimators in Simple Random Sampling ( SRS ),” Int. J. Sci. Res. Math. Stat. Sci., vol. 9, no. 6, pp. 56–60, 2022.
[7] J. F. Epperson, An introduction to numerical methods and analysis. John Wiley & Sons, 2021. doi: 10.1002/9781119604754.
[8] P. Dumka, R. Dumka, and D. R. Mishra, Numerical Methods Using Python. BlueRose, 2022.
[9] B. Fornberg, “Improving the accuracy of the trapezoidal rule,” SIAM Rev., vol. 63, no. 1, pp. 167–180, 2021, doi: 10.1137/18M1229353.
[10] A. F. Abdulhameed and Q. A. Memon, “An improved Trapezoidal rule for numerical integration,” in Journal of Physics: Conference Series, 2021, vol. 2090, no. 1, p. 12104. doi: 10.1088/1742-6596/2090/1/012104.
[11] K. Binder, D. Heermann, L. Roelofs, A. J. Mallinckrodt, and S. McKay, “Monte Carlo Simulation in Statistical Physics,” Comput. Phys., vol. 7, no. 2, p. 156, 1993, doi: 10.1063/1.4823159.
[12] R. L. Harrison, “Introduction to Monte Carlo simulation,” in AIP Conference Proceedings, 2009, vol. 1204, pp. 17–21. doi: 10.1063/1.3295638.
[13] M. N. Akhtar, M. H. Durad, and A. Ahmed, “Solving Initial Value Ordinary Differential Equations By Monte Carlo Method,” Proceeding IAM, pp. 149–174, 2015.
[14] Wei Zhong and Zhou Tian, “Solving initial value problem of ordinary differential equations by Monte Carlo method,” in Multimedia Technology (ICMT), IEEE, 2011, pp. 2577–2579. doi: 10.1109/icmt.2011.6002604.
[15] S. K. R., “Python -The Fastest Growing Programming Language,” Int. Res. J. Eng. Technol., vol. 4, no. 12, pp. 354–357, 2017.
[16] K. Gajula, V. Sharma, B. Sharma, D. R. Mishra, and P. Dumka, “Modelling of Energy in Transit Using Python,” Int. J. Innov. Sci. Res. Technol., vol. 7, no. 8, pp. 1152–1156, 2022.
[17] P. Dumka, A. Deo, K. Gajula, V. Sharma, R. Chauhan, and D. R. Mishra, “Load and Load Duration Curves Using Python,” Int. J. All Res. Educ. Sci. Methods, vol. 10, no. 8, pp. 2127–2134, 2022.
[18] P. Dumka and D. R. Mishra, “Understanding the TDMA / Thomas algorithm and its Implementation in Python,” Int. J. All Res. Educ. Sci. Methods, vol. 10, no. 10, pp. 998–1002, 2022.
[19] P. Dumka, P. S. Pawar, A. Sauda, G. Shukla, and D. R. Mishra, “Application of He’s homotopy and perturbation method to solve heat transfer equations: A python approach,” Adv. Eng. Softw., vol. 170, no. May, p. 103160, 2022, doi: 10.1016/j.advengsoft.2022.103160.
[20] J. Ranjani, A. Sheela, and K. Pandi Meena, “Combination of NumPy, SciPy and Matplotlib/Pylab-A good alternative methodology to MATLAB-A Comparative analysis,” in Proceedings of 1st International Conference on Innovations in Information and Communication Technology, ICIICT 2019, 2019, pp. 1–5. doi: 10.1109/ICIICT1.2019.8741475.
[21] S. Van Der Walt, S. C. Colbert, and G. Varoquaux, “The NumPy array: A structure for efficient numerical computation,” Comput. Sci. Eng., vol. 13, no. 2, pp. 22–30, 2011, doi: 10.1109/MCSE.2011.37.
[22] P. Dumka, R. Chauhan, A. Singh, G. Singh, and D. Mishra, “Implementation of Buckingham ’ s Pi theorem using Python,” Adv. Eng. Softw., vol. 173, no. July, p. 103232, 2022, doi: 10.1016/j.advengsoft.2022.103232.
[23] P. Dumka, K. Rana, S. Pratap, S. Tomar, P. S. Pawar, and D. R. Mishra, “Modelling air standard thermodynamic cycles using python,” Adv. Eng. Softw., vol. 172, no. July, p. 103186, 2022, doi: 10.1016/j.advengsoft.2022.103186.
[24] P. S. Pawar, D. R. Mishra, and P. Dumka, “Solving First Order Ordinary Differential Equations using Least Square Method?: A comparative study,” Int. J. Innov. Sci. Res. Technol., vol. 7, no. 3, pp. 857–864, 2022.