International Journal of Mathematical Sciences and Computing (IJMSC)
IJMSC Vol. 9, No. 3, 8 Aug. 2023
Cover page and Table of Contents: PDF (size: 636KB)
Author(s)
Priyanshi Mishra
1
Pramiti Tewari
1
Dhananjay R. Mishra
2
Pankaj Dumka
2,*
Index Terms
Integration, Definite Integral, Monte Carlo Simulation, Python Programming
Abstract
In this short article an attempt has been made to model Monte Carlo simulation to solve integration problems. The Monte Carlo method employs random sampling and the theory of big numbers to generate values that are very close to the integral's true solution. Python programming has been used to implement the developed algorithm for integration. The developed Python functions are tested with the help of six different integration examples which are difficult to solve analytically. It has been observed that that the Monte Carlo simulation has given results which are in good agreement with the exact analytical results.
Cite This Paper
Priyanshi Mishra, Pramiti Tewari, Dhananjay R. Mishra, Pankaj Dumka, "Integration based on Monte Carlo Simulation", International Journal of Mathematical Sciences and Computing(IJMSC), Vol.9, No.3, pp. 58-65, 2023. DOI:10.5815/ijmsc.2023.03.05
Reference
[1]M.N. Hounkonnou, J.D.B. Kyemba, R (p, q)-calculus: differentiation and integration, SUT J. Math. 49 (2013) 145–167.
[2]P.W. Thompson, G. Harel, Ideas foundational to calculus learning and their links to students’ difficulties, ZDM - Math. Educ. 53 (2021) 507–519. https://doi.org/10.1007/s11858-021-01270-1.
[3]W. Miranker, E. Isaacson, H.B. Keller, Analysis of Numerical Methods, Courier Corporation, 1967. https://doi.org/10.2307/2003280.
[4]J.D. Faires, R.L. Burden, Numerical methods., Thomson, 2003.
[5]P. Dumka, R. Dumka, D.R. Mishra, Numerical Methods Using Python, BlueRose, 2022.
[6]M. Fippel, Basics of monte carlo simulations, in: Monte Carlo Tech. Radiat. Ther., CRC Press, 2016: pp. 17–28. https://doi.org/10.1587/bplus.2008.6_11.
[7]R.M. Feldman, C. Valdez-Flores, R.M. Feldman, C. Valdez-Flores, Basics of Monte Carlo Simulation, Appl. Probab. Stoch. Process. (2010) 45–72.
[8]B. V Gnedenko, Theory of probability, Routledge, 2006. https://doi.org/10.2307/3028701.
[9]P. Dattalo, Strategies to Approximate Random Sampling and Assignment, Oxford University Press, 2010. https://doi.org/10.1093/acprof:oso/9780195378351.001.0001.
[10]R.L. Harrison, Introduction to Monte Carlo simulation, in: AIP Conf. Proc., 2009: pp. 17–21. https://doi.org/10.1063/1.3295638.
[11]P.L. Bonate, A brief introduction to Monte Carlo simulation, Clin. Pharmacokinet. 40 (2001) 15–22. https://doi.org/10.2165/00003088-200140010-00002.
[12]R.L. Harrison, Introduction to Monte Carlo simulation, in: AIP Conf. Proc., 2009: pp. 17–21. https://doi.org/10.1063/1.3295638.
[13]K. Binder, D. Heermann, L. Roelofs, A.J. Mallinckrodt, S. McKay, Monte Carlo Simulation in Statistical Physics, Comput. Phys. 7 (1993) 156. https://doi.org/10.1063/1.4823159.
[14]Y.H. Kwak, L. Ingall, Exploring monte carlo simulation applications for project management, IEEE Eng. Manag. Rev. 37 (2009) 83–83. https://doi.org/10.1109/emr.2009.5235458.
[15]G.A. Bird, Monte-Carlo Simulation in an Engineering Context, Rarefied Gas Dyn. Parts I II. 74 (1981) 239–255. https://doi.org/10.2514/5.9781600865480.0239.0255.
[16]S. Mordechai, Applications of Monte Carlo Method in Science and Engineering, InTech, 2012. https://doi.org/10.5772/1954.
[17]E. Zio, E. Zio, Monte Carlo simulation: The method, Springer, 2013. https://doi.org/10.1007/978-1-4471-4588-2_3.
[18]J.C. Walter, G.T. Barkema, An introduction to Monte Carlo methods, Phys. A Stat. Mech. Its Appl. 418 (2015) 78–87. https://doi.org/10.1016/j.physa.2014.06.014.
[19]P. Brandimarte, Handbook in Monte Carlo Simulation: Applications in Financial Engineering, Risk Management, and Economics, John Wiley \& Sons, 2014. https://doi.org/10.1002/9781118593264.
[20]P. Mishra, A. Sharma, D.R. Mishra, P. Dumka, Solving Double Integration With The Help of Monte Carlo Simulation : A Python Approach, Int. J. Sci. Res. Multidiscip. Stud. 9 (2023) 6–10.
[21]Y.C. Huei, Benefits and introduction to python programming for freshmore students using inexpensive robots, in: Proc. IEEE Int. Conf. Teaching, Assess. Learn. Eng. Learn. Futur. Now, TALE 2014, 2015: pp. 12–17. https://doi.org/10.1109/TALE.2014.7062611.
[22]T.A. Review, C.E. Issn, Comparative Study of Wind Pressure Variations on Rectangular Buildings Using Python Programming, 11 (2022) 15–24.
[23]P. Dumka, N. Samaiya, S. Gandhi, D.R. Mishra, Modelling of Hardy Cross Method for Pipe Networks, 10 (2023) 1–8.
[24]P. Dumka, D.R. Mishra, Understanding the TDMA / Thomas algorithm and its Implementation in Python, Int. J. All Res. Educ. Sci. Methods. 10 (2022) 998–1002.
[25]P. Dumka, K. Rana, S. Pratap, S. Tomar, P.S. Pawar, D.R. Mishra, Modelling air standard thermodynamic cycles using python, Adv. Eng. Softw. 172 (2022) 103186. https://doi.org/10.1016/j.advengsoft.2022.103186.
[26]P. Dumka, R. Chauhan, A. Singh, G. Singh, D. Mishra, Implementation of Buckingham ’ s Pi theorem using Python, Adv. Eng. Softw. 173 (2022) 103232. https://doi.org/10.1016/j.advengsoft.2022.103232.