Fourier series are a powerful tool in applied mathematics; indeed, their importance is twofold since Fourier series are used to represent both periodic real functions as well as solutions admitted by linear partial differential equations with assigned initial and boundary conditions. The idea inspiring the introduction of Fourier series is to approximate a regular periodic function, of period T, via a linear superposition of trigonometric functions of the same period T; thus, Fourier polynomials are constructed. They play, in the case of regular periodic real functions, a role analogue to that one of Taylor polynomials when smooth real functions are considered. In this thesis we will study function approximation by FS method. We will make an attempt to approximate square wave function, line function by FS, and line function by Fourier exponential and trigonometric polynomial. DFT will also be used to approximate function values from data set. We compare the accuracy and the error of Fourier approximation with the actual function and we find that the approximate function is very close to the actual function. We also study the solution of 1D heat equation and Laplace equation by Fourier series method. We compare the solution of heat equation obtained by Fourier series with BTCS. We also compare the solution of Laplace equation obtained by Fourier series with Jacobi iterative method. MATLAB codes for each scheme are presented in appendix and results of running the codes give the numerical solution and graphical solution.
| Published in | Science Journal of Applied Mathematics and Statistics (Volume 10, Issue 4) |
| DOI | 10.11648/j.sjams.20221004.12 |
| Page(s) | 57-84 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Fourier Series, Sine Wave, Discrete Fourier Transform, Heat Equation and Laplace Equation
| [1] | Arthur, L. S. 2005. An Introduction to Fourier analysis, Partial Differential Equations and Fourier Transforms. Naval Postgraduate School, Monterey, California. |
| [2] | Attia, J. 1999. Electronics and circuit analysis using MATLAB. CRC Press LLC. Boca Raton, London. |
| [3] | Auslander, L., Grunbaum, F. A, (1989). The Fourier transform and the discrete Fourier transform. Inverse Problems, 5: 149-164. |
| [4] | Boggesse, A and F. Narcowich, (2009). A first course in wavelets with Fourier analysis. Prentice Hall, USA. |
| [5] | Christopher, J. Z, (2004). An Introduction to Numerical Analysis for Electrical and compute Engineers. University of Albert, Canada. |
| [6] | Codruta Chis, F. Cret. (2005). Scientifically Researches. Agro alimentary Processes and Technologies, Volume XI, No. 2: 481-484. |
| [7] | Colletz, L. (1999). The numerical treatment of differential equation. 3rd edition. Berlin, Springer-Verlag. |
| [8] | Curtis, F. G. and Patrice, O. W. (1994). Applied numerical analysis, 5th edition. Addison-Wesley Publishing Company. |
| [9] | Elali, T. (2005). Discrete system and digital processing with MATLAB. CRC Press. Boca Raton, London. |
| [10] | Gerald, W. R. (2011). Finite-Difference Approximations to the Heat Equation. Portland State University, Portland, Oregon. |
| [11] | Gottlieb, D., Shu, C-W. (1997). On the Gibbs phenomenon and its resolution. SIAReview 39: 644-668. |
| [12] | Hsu, H. 1995. Schaom’s outlines of theory and problems of signal systems. McGraw-Hill campanies, Inc. New York city, USA. |
| [13] | Jeffrey, A. (2002). Advanced engineering mathematics. Harcourt or Academic press. |
| [14] | Karris, S. (2004). Signal and system with MATLAB applications. 2nd edition. Orchard publications, USA. |
| [15] | Mandal, M and M. Asif. (2007). Continuous and discrete time signals. Cambridge Universit Press. Cambridge. |
| [16] | Mark, S. G. 2002. Partial differential equations, analytical and numerical methods. Society for industrial and applied mathematics, Philadelphia, USA. |
| [17] | Mussoko, V. (2005). Biomedical signal processing and image processing. Ph.D. Thesis. Prague. |
| [18] | Rocchesso, D. (2003). Internal Combusion engine of signal monitoring. Department of Aerospace and mechanical engineering University of Notre Dame, Australia. |
| [19] | Stoer, J., Bulirsch, R. (1980). Introduction to numerical analysis. Springer, Heidelberg. |
| [20] | Walter, A. S. (2007). Partial differential equations an introduction. 2nd edition. Lauri Rosatone, Brown University. |
| [21] | Wiley, P. (20040. Communication on pure and applied mathematics. Vol. LVII I: 0001-0015. |
| [22] | Wong, v. (2006). Signal and system of lecture notes. The university of Adelaide. Australia. |
| [23] | Yang, W. Y. (2005). Applied Numerical Methods using MATLAB. New Jersey, John Wiley and sons. |
APA Style
Desta Sodano Sheiso. (2022). Approximation of Functions Using Fourier Series and Its Application to the Solution of Partial Differential Equations. Science Journal of Applied Mathematics and Statistics, 10(4), 57-84. https://doi.org/10.11648/j.sjams.20221004.12
ACS Style
Desta Sodano Sheiso. Approximation of Functions Using Fourier Series and Its Application to the Solution of Partial Differential Equations. Sci. J. Appl. Math. Stat. 2022, 10(4), 57-84. doi: 10.11648/j.sjams.20221004.12
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.sjams.20221004.12,
author = {Desta Sodano Sheiso},
title = {Approximation of Functions Using Fourier Series and Its Application to the Solution of Partial Differential Equations},
journal = {Science Journal of Applied Mathematics and Statistics},
volume = {10},
number = {4},
pages = {57-84},
doi = {10.11648/j.sjams.20221004.12},
url = {https://doi.org/10.11648/j.sjams.20221004.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20221004.12},
abstract = {Fourier series are a powerful tool in applied mathematics; indeed, their importance is twofold since Fourier series are used to represent both periodic real functions as well as solutions admitted by linear partial differential equations with assigned initial and boundary conditions. The idea inspiring the introduction of Fourier series is to approximate a regular periodic function, of period T, via a linear superposition of trigonometric functions of the same period T; thus, Fourier polynomials are constructed. They play, in the case of regular periodic real functions, a role analogue to that one of Taylor polynomials when smooth real functions are considered. In this thesis we will study function approximation by FS method. We will make an attempt to approximate square wave function, line function by FS, and line function by Fourier exponential and trigonometric polynomial. DFT will also be used to approximate function values from data set. We compare the accuracy and the error of Fourier approximation with the actual function and we find that the approximate function is very close to the actual function. We also study the solution of 1D heat equation and Laplace equation by Fourier series method. We compare the solution of heat equation obtained by Fourier series with BTCS. We also compare the solution of Laplace equation obtained by Fourier series with Jacobi iterative method. MATLAB codes for each scheme are presented in appendix and results of running the codes give the numerical solution and graphical solution.},
year = {2022}
}
TY - JOUR T1 - Approximation of Functions Using Fourier Series and Its Application to the Solution of Partial Differential Equations AU - Desta Sodano Sheiso Y1 - 2022/09/19 PY - 2022 N1 - https://doi.org/10.11648/j.sjams.20221004.12 DO - 10.11648/j.sjams.20221004.12 T2 - Science Journal of Applied Mathematics and Statistics JF - Science Journal of Applied Mathematics and Statistics JO - Science Journal of Applied Mathematics and Statistics SP - 57 EP - 84 PB - Science Publishing Group SN - 2376-9513 UR - https://doi.org/10.11648/j.sjams.20221004.12 AB - Fourier series are a powerful tool in applied mathematics; indeed, their importance is twofold since Fourier series are used to represent both periodic real functions as well as solutions admitted by linear partial differential equations with assigned initial and boundary conditions. The idea inspiring the introduction of Fourier series is to approximate a regular periodic function, of period T, via a linear superposition of trigonometric functions of the same period T; thus, Fourier polynomials are constructed. They play, in the case of regular periodic real functions, a role analogue to that one of Taylor polynomials when smooth real functions are considered. In this thesis we will study function approximation by FS method. We will make an attempt to approximate square wave function, line function by FS, and line function by Fourier exponential and trigonometric polynomial. DFT will also be used to approximate function values from data set. We compare the accuracy and the error of Fourier approximation with the actual function and we find that the approximate function is very close to the actual function. We also study the solution of 1D heat equation and Laplace equation by Fourier series method. We compare the solution of heat equation obtained by Fourier series with BTCS. We also compare the solution of Laplace equation obtained by Fourier series with Jacobi iterative method. MATLAB codes for each scheme are presented in appendix and results of running the codes give the numerical solution and graphical solution. VL - 10 IS - 4 ER -
Email: