# An Approximate Optimization Method for Solving Stiff Ordinary Differential Equations With Combinational Mutation Strategy of Differential Evolution Algorithm

### Abstract

This paper examines the implementation of simple combination mutation of differential evolution algorithm for solving stiff ordinary differential equations. We use the weighted residual method with a series expansion to approximate the solutions of stiff ordinary differential equations. We solve the problems from an ordinary stiff differential equation for linear and nonlinear problems. Then, we also implement our method for solving stiff systems of ordinary differential equations. We find that our algorithm can approximate the exact solution of a stiff ordinary differential equation with the smallest error for each length of series that we have chosen. Thus, this approximation method, by using the optimization method of simple combination differential evolution, can be a good tool for solving stiff ordinary differential equations.

### References

Adee, S., and Atabo, V. Improved two-point block backward differentiation formulae for solving first order stiff initial value problems of ordinary differential equations. NIGERIAN ANNALS OF PURE AND APPLIED SCIENCES 3, 2 (2020), 200–209.

Ahmad, R., Yaacob, N., and Mohd Murid, A.-H. Explicit methods in solving stiff ordinary differential equations. International Journal of Computer Mathematics 81, 11 (2004), 1407–1415.

Atay, M. T., and Kilic, O. The semianalytical solutions for stiff systems of ordinary differential equations by using variational iteration method and modified variational iteration method with comparison to exact solutions. Mathematical Problems in Engineering 2013 (2013).

Babaei, M. A general approach to approximate solutions of nonlinear differential equations using particle swarm optimization. Applied Soft Computing 13, 7 (2013), 3354–3365.

Burden, R. L., Faires, J. D., and Burden, A. M. Numerical analysis. Cengage learning, 2015.

Cao, H., Kang, L., Chen, Y., and Yu, J. Evolutionary modeling of systems of ordinary differential equations with genetic programming. Genetic Programming and Evolvable Machines 1, 4 (2000), 309–337.

Chapra, S. C., Canale, R. P., et al. Numerical methods for engineers, vol. 1221. Mcgraw-hill New York, 2011.

Chaquet, J. M., and Carmona, E. J. Solving differential equations with fourier series and evolution strategies. Applied Soft Computing 12, 9 (2012), 3051–3062.

El-Zahar, E. R., Tenreiro Machado, J., and Ebaid, A. A new generalized taylor-like explicit method for stiff ordinary differential equations. Mathematics 7, 12 (2019), 1154.

Febrianti, W., Sidarto, K. A., and Sumarti, N. Solving some ordinary differential equations numerically using differential evolution algorithm with a simple adaptive mutation scheme. In AIP Conference Proceedings (2021), vol. 2329, AIP Publishing LLC, p. 040003.

Febrianti, W., Sidarto, K. A., and Sumarti, N. Solving systems of ordinary differential equations using differential evolution algorithm with the best base vector of mutation scheme. In AIP Conference Proceedings (2021), vol. 2423, AIP Publishing LLC, p. 020006.

Gear, C. W. Numerical initial value problems in ordinary differential equations. Prentice-Hall series in automatic computation (1971). [13] Ibrahim, Z. B., Mohd Noor, N., and Othman, K. I. Fixed coefficient a (α) stable block backward differentiation formulas for stiff ordinary differential equations. Symmetry 11, 7 (2019), 846.

Ibrahim, Z. B., and Nasarudin, A. A. A class of hybrid multistep block methods with a–stability for the numerical solution of stiff ordinary differential equations. Mathematics 8, 6 (2020), 914.

Jana Aksah, S., Ibrahim, Z. B., and Mohd Zawawi, I. S. Stability analysis of singly diagonally implicit block backward differentiation formulas for stiff ordinary differential equations. Mathematics 7, 2 (2019), 211.

Musa, H., Suleiman, M., and Senu, N. Fully implicit 3-point block extended backward differentiation formula for stiff initial value problems. Applied Mathematical Sciences 6, 85 (2012), 4211–4228.

Nasarudin, A. A., Ibrahim, Z. B., and Rosali, H. On the integration of stiff odes using block backward differentiation formulas of order six. Symmetry 12, 6 (2020), 952.

Panagant, N., and Bureerat, S. Solving partial differential equations using a new differential evolution algorithm. Mathematical Problems in Engineering 2014 (2014).

Price, K., Storn, R. M., and Lampinen, J. A. Differential evolution: a practical approach to global optimization. Springer Science & Business Media, 2006.

Raymond, D., Donald, J., Michael, A., and Ajileye, G. A self-starting five-step eight-order block method for stiff ordinary differential equations. Journal of Advances in Mathematics and Computer Science 26, 4 (2018), 1–9.

Reich, C. Simulation of imprecise ordinary differential equations using evolutionary algorithms. In Proceedings of the 2000 ACM symposium on Applied computing-Volume 1 (2000), pp. 428–432.

Ukpebor, L., and Omole, E. Three-step optimized block backward differentiation formulae (tobbdf) for solving stiff ordinary differential equations. African Journal of Mathematics and Computer Science Research 13, 1 (2020), 51–57.

*MENDEL*. 28, 2 (Dec. 2022), 54-61. DOI:https://doi.org/10.13164/mendel.2022.2.054.

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