Implementation of Numerical Methods for Solving Differential Equations using Python

Aayushi Sahgal

doi:10.36676/jrps.2023-v14i4-019

Authors

Aayushi Sahgal

DOI:

https://doi.org/10.36676/jrps.2023-v14i4-019

Keywords:

mathematical, engineering, computer science, practitioners

Abstract

Numerical analysis is a branch of mathematics that deals with the development, analysis, and implementation of numerical algorithms for solving mathematical problems. In particular, it focuses on finding approximate solutions to problems that cannot be solved analytically, often using computers and other numerical methods. The numerical analysis involves the use of various mathematical techniques, including linear algebra, calculus, optimization, and statistics, to develop numerical algorithms that can solve a wide range of problems in fields such as engineering, physics, finance, and computer science. These problems may involve the calculation of derivatives and integrals, the solution of differential equations, the optimization of functions, and the approximation of functions. The goal of numerical analysis is to provide accurate and efficient solutions to mathematical problems that are difficult or impossible to solve using traditional analytical methods. This field has become increasingly important in modern science and engineering, as it enables researchers and practitioners to simulate complex systems, design new products, and solve challenging mathematical problems.

References

Avellaneda, M., & Paras, A. (1996). Dynamic hedging portfolios for derivative securities in the presence of large transaction costs. Applied Mathematical Finance, 3(1), 21-52. DOI: https://doi.org/10.1080/13504869600000002

Bhide, A., et al. (2021). Finite Element Analysis of a Composite Material. International Journal of Engineering and Advanced Technology, 10(1), 115-118.

Bridson, R., & Müller-Fischer, M. (2007). Fluid simulation for computer graphics. AK Peters/CRC Press. DOI: https://doi.org/10.1145/1281500.1281681

Brigo, D., & Mercurio, F. (2001). Interest rate models: Theory and practice. Springer Science & Business Media. DOI: https://doi.org/10.1007/978-3-662-04553-4

Cazacu, O., et al. (2017). Numerical analysis of fluid flow in a microchannel. Proceedings of the International Conference on Applied Mathematics and Computational Methods, 48-52.

Chen, Y., Li, J., & Li, J. (2020). Simulation of the electric field in a graphene-based supercapacitor using the trapezoidal rule. Journal of Physics and Chemistry of Solids, 136, 109119.

Farin, G. (1990). Curves and surfaces for computer-aided geometric design. Academic Press. DOI: https://doi.org/10.1016/B978-0-12-460515-2.50020-2

Ghaffari, S., Ziaei-Rad, S., & Yang, K. (2019). Mechanical properties of high-entropy alloys: a molecular dynamics study. Journal of Materials Science, 54(9), 7024-7036.

Glassner, A. (1995). An introduction to ray tracing. Academic Press.

Gogtay, N. J., Ranade, S. S., & Thatte, U. M. (2020). Use of least squares regression analysis in medical research. Journal of Postgraduate Medicine, 66(2), 67-71. 11. Goldstine, H. H. (2012). A History of Numerical Analysis from the 16th through the 19th Century (Vol. 2). Springer Science & Business Media.

Günther, M., Keeton, C. R., & Suyu, S. H. (2020). Gravitational lens identification with deep learning: exploring the limitations of current data sets. Astronomy & Astrophysics, 633, A139.

Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. The review of financial studies, 6(2), 327-343. DOI: https://doi.org/10.1093/rfs/6.2.327

Higham, N. J. (2002). Handbook of writing for the mathematical sciences. SIAM.

Hinton, D. L., & Campbell, J. M. (2003). Finite element modeling in aerospace engineering: a numerical tool for solving complex problems. Progress in Aerospace Sciences, 39(5), 393-430.

Kearns, M., Nevmyvaka, Y., & Pfeffer, J. (2009). Machine learning for market microstructure and high-frequency trading. IEEE Intelligent Systems, 24(5), 80-90.

Kumar, A., et al. (2019). Structural Analysis of Wind Turbine Blade Using Finite Element Analysis. International Journal of Engineering and Advanced Technology, 8(5), 1562-1566.

Merton, R. C. (1974). On the pricing of corporate debt: The risk structure of interest rates. The Journal of Finance, 29(2), 449-470. DOI: https://doi.org/10.1111/j.1540-6261.1974.tb03058.x

Paudel, R., Jha, M., & Pokharel, R. K. (2018). Design optimization of microstrip patch antenna using particle swarm optimization. Journal of Electrical and Electronics Engineering, 6(2), 43-49.

Prakash, S., et al. (2021). Optimization of Injection Molding Process Parameters Using Taguchi Method. International Journal of Engineering and Advanced Technology, 10(1), 121-125.

Witkin, A., & Baraff, D. (1991). Physically based modeling: Principles and practice. In Proceedings of the 18th annual conference on Computer graphics and interactive techniques (pp. 235-242).