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Diferansiyel Denklemler İçin Matlab Kullanarak Nümerik Çözümler

Year 2025, Volume: 41 Issue: 2, 616 - 630, 30.08.2025

Bengü Çına

Abstract

Diferansiyel denklemlerin fizik, mühendislik, biyoloji ve ekonomi gibi disiplinlerdeki bir dizi olgunun modellenmesindeki rolü büyük önem taşımaktadır. Birçok diferansiyel denklem analitik olarak çözülebilmesine rağmen, diğerleri bu konuda bir zorluk teşkil etmektedir. MATLAB, bu karmaşık denklemlerin çözümünü kolaylaştırmak için kullanılabilir. Bu çalışmanın amacı, analitik çözümü olmayan veya çözümü karmaşık olan diferansiyel denklemlerin MATLAB kullanılarak sayısal çözümlerini elde etmek ve bu çözümlerin grafiklerini analiz etmektir. Bu sayede denklemlerin dinamik davranışları ve çözüm aralıkları hakkında daha derin bir anlayış kazanmayı hedefliyoruz. Bu amaçla çoğunlukla ode45 fonksiyonu ve Runge-Kutta yöntemi kullanılacaktır.

Keywords

Ode45 Runge-Kutta Yöntemi Dormand-Prince (4 5) Matlab.

References

  • Yang, W. Y., Cao, W., Kim, J., Park, K. W., Park, H. H., Joung, J., ... & Im, T. 2020. Applied numerical methods using MATLAB, John Wiley & Sons.
  • Corless, R. M., Nicolas, F. 2013. A graduate introduction to numerical methods, AMC (2013) 10: 12.
  • Denis, B. 2020. An overview of numerical and analytical methods for solving ordinary differential equations, arXiv preprint arXiv: 2012.07558.
  • Yüzbasi, S., Gök, E., Sezer, M. 2016. A numerical method for solving systems of higher order linear functional differential equations, Open Physics, 14(1) (2016) 15-25.
  • Saqib, M., et al. 2024. Dynamical Behavior of Nonlinear Coupled Reaction-Diffusion Model: A Numerical Study Utilizing ADI and Staggered Grid Finite Volume Method in Matlab, IEEE Access.
  • Gopal, D., et al. 2021. Numerical analysis of higher order chemical reaction on electrically MHD nanofluid under influence of viscous dissipation, Alexandria Engineering Journal, 60.1 (2021) 1861-1871.
  • Shah, K., Fahd, J., Thabet, A. 2020. Stable numerical results to a class of time-space fractional partial differential equations via spectral method, Journal of Advanced Research, 25 (2020) 39-48.
  • Ong, B. W., Spiteri, R. J. 2020. Deferred correction methods for ordinary differential equations, Journal of Scientific Computing, 83(3) (2020) 60.
  • Christlieb, A., Ong, B., Qiu, J.M. 2009. Comments on high-order integrators embedded within integral deferred correction methods, Commun. Appl. Math. Comput. Sci., 4 (2009) 27–56.
  • Christlieb, A., Ong, B., Qiu, J.M., Integral deferred correction methods constructed with high order Runge–Kutta integrators, Math. Comput., 79(270) (2010) 761–783.
  • Christlieb, A.J., Macdonald, C.B., Ong, B.W. 2010. Parallel high-order integrators, SIAM J. Sci. Comput., 32(2) (2010) 818–835.
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  • Hansen, A.C., Strain, J. 2011. On the order of deferred correction, Appl. Numer. Math., 61(8) (2011) 961–973.
  • Fox, L. 1947. Some improvements in the use of relaxation methods for the solution of ordinary and partial differential equations. Proc. R. Soc. Lond. Ser. A, 190 (1947) 31–59.
  • Pereyra, V. 1966. On improving an approximate solution of a functionalequation by deferred corrections, Numerische Mathematik, 8 (1966) 376-391.
  • Pereyra, V. 1968. Iterated deferred corrections for nonlinear boundary value problems, Numerische Mathematik, 11 (1968) 111– 125.
  • Kress, W., Gustafsson, B. 2002. Deferred correction methods for initial boundary value problems, Proceedings of the 5th International Conference on Spectral and High Order Methods (ICOSAHOM-01) (Uppsala), vol. 17 (2002) 241–251.
  • Rangan, A.V. 2003. Adaptive solvers for partial differential and differential-algebraic equations, Ph.D. thesis, University of California, Berkeley.
  • Huang, J., Jia, J., Minion, M. 2006. Accelerating the convergence of spectral deferred correction methods, J. Comput. Phys., 214(2) (2006) 633–656.
  • Chu, K.W., Spence, A. 1981. Deferred correction for the integral equation eigenvalue problem, The ANZIAM Journal, 22(4) (1981) 474–487.
  • Reuter, B., et al. 2021. FESTUNG 1.0: Overview, usage, and example applications of the MATLAB/GNU Octave toolbox for discontinuous Galerkin methods, Computers & Mathematics with Applications, 81 (2021) 3-41.
  • Shampine, L. F. 2018. Numerical solution of ordinary differential equations, Routledge.
  • Cooper, J. M. 2012. Introduction to partial differential equations with MATLAB. Springer Science & Business Media.
  • Coleman, M. P., Bukshtynov, V. 2024 An introduction to partial differential equations with MATLAB, CRC Press.
  • Zheng, L., Zhang, X. 2017. Modeling and analysis of modern fluid problems. Academic Press.
  • Dormand, J. R., & Prince, P. J. 1980. A family of embedded Runge-Kutta formulae, Journal of Computational and Applied Mathematics, 6(1), 19–26.
  • Hale, J. K. 2006. Functional differential equations. In: Analytic Theory of Differential Equations: The Proceedings of the Conference at Western Michigan University, Kalamazoo, from 30 April to 2 May 1970. Berlin, Heidelberg: Springer Berlin Heidelberg, 2006. p. 9-22.
  • Murray, J. D. 2007. Mathematical biology: I. An introduction (Vol. 17), Springer Science & Business Media, (2007).
  • Zaya, N. E., Hassan, L. H., & Bilgil, H. 2018. Mathematical Modeling for Prediction of Heating and Air-Conditioning Energies of Multistory Buildings in Duhok City, Acad. J. Nawroz Univ, 7, 153-167.
  • https://doi.org/10.38016/jista.1447980

Numerical Solutions for Differential Equations Using Matlab

Year 2025, Volume: 41 Issue: 2, 616 - 630, 30.08.2025

Bengü Çına

Abstract

The role of differential equations in modelling a range of phenomena across disciplines, including physics, engineering, biology and economics, is of great significance. Despite many differential equations can be solved analytically, others present a challenge in this regard. MATLAB can be employed to facilitate the resolution of these intricate equations. The aim of this study is to obtain numerical solutions of differential equations that have no analytical solutions or whose solutions are complex using MATLAB and to analyse the graphs of these solutions. In this way, we aim to gain a deeper understanding of the dynamic behaviour of the equations and their solution ranges. For this purpose, the ode45 function and the Runge-Kutta method will be mostly used.

Keywords

Ode45 Runge-Kutta method Dormand-Prince (4 5) Method Matlab.

References

  • Yang, W. Y., Cao, W., Kim, J., Park, K. W., Park, H. H., Joung, J., ... & Im, T. 2020. Applied numerical methods using MATLAB, John Wiley & Sons.
  • Corless, R. M., Nicolas, F. 2013. A graduate introduction to numerical methods, AMC (2013) 10: 12.
  • Denis, B. 2020. An overview of numerical and analytical methods for solving ordinary differential equations, arXiv preprint arXiv: 2012.07558.
  • Yüzbasi, S., Gök, E., Sezer, M. 2016. A numerical method for solving systems of higher order linear functional differential equations, Open Physics, 14(1) (2016) 15-25.
  • Saqib, M., et al. 2024. Dynamical Behavior of Nonlinear Coupled Reaction-Diffusion Model: A Numerical Study Utilizing ADI and Staggered Grid Finite Volume Method in Matlab, IEEE Access.
  • Gopal, D., et al. 2021. Numerical analysis of higher order chemical reaction on electrically MHD nanofluid under influence of viscous dissipation, Alexandria Engineering Journal, 60.1 (2021) 1861-1871.
  • Shah, K., Fahd, J., Thabet, A. 2020. Stable numerical results to a class of time-space fractional partial differential equations via spectral method, Journal of Advanced Research, 25 (2020) 39-48.
  • Ong, B. W., Spiteri, R. J. 2020. Deferred correction methods for ordinary differential equations, Journal of Scientific Computing, 83(3) (2020) 60.
  • Christlieb, A., Ong, B., Qiu, J.M. 2009. Comments on high-order integrators embedded within integral deferred correction methods, Commun. Appl. Math. Comput. Sci., 4 (2009) 27–56.
  • Christlieb, A., Ong, B., Qiu, J.M., Integral deferred correction methods constructed with high order Runge–Kutta integrators, Math. Comput., 79(270) (2010) 761–783.
  • Christlieb, A.J., Macdonald, C.B., Ong, B.W. 2010. Parallel high-order integrators, SIAM J. Sci. Comput., 32(2) (2010) 818–835.
  • Dutt, A., Greengard, L., Rokhlin, V. 2000. Spectral deferred correction methods for ordinary differential equations, BIT Numerical Mathematics, 40(2) (2000) 241–266.
  • Fox, L., Goodwin, E.T. 1949. Some new methods for the numerical integration of ordinary differential equations, Proc. Camb. Philos. Soc., 45 (1949) 373–388.
  • Hansen, A.C., Strain, J. 2011. On the order of deferred correction, Appl. Numer. Math., 61(8) (2011) 961–973.
  • Fox, L. 1947. Some improvements in the use of relaxation methods for the solution of ordinary and partial differential equations. Proc. R. Soc. Lond. Ser. A, 190 (1947) 31–59.
  • Pereyra, V. 1966. On improving an approximate solution of a functionalequation by deferred corrections, Numerische Mathematik, 8 (1966) 376-391.
  • Pereyra, V. 1968. Iterated deferred corrections for nonlinear boundary value problems, Numerische Mathematik, 11 (1968) 111– 125.
  • Kress, W., Gustafsson, B. 2002. Deferred correction methods for initial boundary value problems, Proceedings of the 5th International Conference on Spectral and High Order Methods (ICOSAHOM-01) (Uppsala), vol. 17 (2002) 241–251.
  • Rangan, A.V. 2003. Adaptive solvers for partial differential and differential-algebraic equations, Ph.D. thesis, University of California, Berkeley.
  • Huang, J., Jia, J., Minion, M. 2006. Accelerating the convergence of spectral deferred correction methods, J. Comput. Phys., 214(2) (2006) 633–656.
  • Chu, K.W., Spence, A. 1981. Deferred correction for the integral equation eigenvalue problem, The ANZIAM Journal, 22(4) (1981) 474–487.
  • Reuter, B., et al. 2021. FESTUNG 1.0: Overview, usage, and example applications of the MATLAB/GNU Octave toolbox for discontinuous Galerkin methods, Computers & Mathematics with Applications, 81 (2021) 3-41.
  • Shampine, L. F. 2018. Numerical solution of ordinary differential equations, Routledge.
  • Cooper, J. M. 2012. Introduction to partial differential equations with MATLAB. Springer Science & Business Media.
  • Coleman, M. P., Bukshtynov, V. 2024 An introduction to partial differential equations with MATLAB, CRC Press.
  • Zheng, L., Zhang, X. 2017. Modeling and analysis of modern fluid problems. Academic Press.
  • Dormand, J. R., & Prince, P. J. 1980. A family of embedded Runge-Kutta formulae, Journal of Computational and Applied Mathematics, 6(1), 19–26.
  • Hale, J. K. 2006. Functional differential equations. In: Analytic Theory of Differential Equations: The Proceedings of the Conference at Western Michigan University, Kalamazoo, from 30 April to 2 May 1970. Berlin, Heidelberg: Springer Berlin Heidelberg, 2006. p. 9-22.
  • Murray, J. D. 2007. Mathematical biology: I. An introduction (Vol. 17), Springer Science & Business Media, (2007).
  • Zaya, N. E., Hassan, L. H., & Bilgil, H. 2018. Mathematical Modeling for Prediction of Heating and Air-Conditioning Energies of Multistory Buildings in Duhok City, Acad. J. Nawroz Univ, 7, 153-167.
  • https://doi.org/10.38016/jista.1447980
There are 31 citations in total.

Details

Primary Language English
Subjects Ordinary Differential Equations, Difference Equations and Dynamical Systems, Pure Mathematics (Other)
Journal Section Research Article
Authors

Bengü Çına 0000-0003-1294-0983

Submission Date April 10, 2025
Acceptance Date June 4, 2025
Publication Date August 30, 2025
Published in Issue Year 2025 Volume: 41 Issue: 2

Cite

APA Çına, B. (2025). Numerical Solutions for Differential Equations Using Matlab. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi, 41(2), 616-630.
AMA Çına B. Numerical Solutions for Differential Equations Using Matlab. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi. August 2025;41(2):616-630.
Chicago Çına, Bengü. “Numerical Solutions for Differential Equations Using Matlab”. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi 41, no. 2 (August 2025): 616-30.
EndNote Çına B (August 1, 2025) Numerical Solutions for Differential Equations Using Matlab. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi 41 2 616–630.
IEEE B. Çına, “Numerical Solutions for Differential Equations Using Matlab”, Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi, vol. 41, no. 2, pp. 616–630, 2025.
ISNAD Çına, Bengü. “Numerical Solutions for Differential Equations Using Matlab”. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi 41/2 (August2025), 616-630.
JAMA Çına B. Numerical Solutions for Differential Equations Using Matlab. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi. 2025;41:616–630.
MLA Çına, Bengü. “Numerical Solutions for Differential Equations Using Matlab”. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi, vol. 41, no. 2, 2025, pp. 616-30.
Vancouver Çına B. Numerical Solutions for Differential Equations Using Matlab. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi. 2025;41(2):616-30.

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