Optimum solutions of partial differential equation with initial condition using optimization techniques

Muhammad Bilal, Shakoor Muhammad, Nekmat Ullah, Fazal Hanan, Subahan Ullah


This paper proposes a new minimization technique for the solutions of partial differential equation with initial conditions. The proposed procedure is used to minimize the obtained solutions through any numerical technique. For
the minimization process, Non-linear Nelder-Mead Simplex algorithm and genetic algorithm are used as optimization techniques. The designed partial differential equation has been calculated as an error function for the minimization process. Both Non-linear Nelder-Mead Simplex and genetic algorithm guarantees the minimization of nonlinear partial differential equation with initial conditions. The resultant technique has a global validity for the solutions minimization of partial differential equations. Non-linear Nelder-Mead simplex showed better performance than genetic algorithm when tested on numerical instances.

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Lagarias, J.C., Reeds, J.A., Wright, M.H. and Wright, P.E., 1998. Convergence properties of the Nelder-Mead simplex method in low dimensions. SIAM Journal on optimization, 9(1), pp.112-147.

McKinnon, K.I., 1998. Convergence of the Nelder–Mead Simplex Method to a Non stationary Point. SIAM Journal on Optimization, 9(1), pp.148-158.

Price, C.J., Coope, I.D. and Byatt, D., 2002. A convergent variant of the Nelder–Mead algorithm. Journal of optimization theory and applications, 113(1), pp.5-19.

Nelder, J.A. and Mead, R., 1965. A Simplex method for function minimization. The computer journal, 7(4), pp.308313.

Wright, M.H., 2010. Nelder, Mead, and the other simplex method. Documenta Mathematica, 7, pp.271-276.

Han, L. and Neumann, M., 2006. Effect of dimensionality on the Nelder–Mead simplex method. Optimization Methods and Software, 21(1), pp.1-16.

Gao, F. and Han, L., 2012. Implementing the Nelder-Mead simplex algorithm with adaptive parameters. Computational Optimization and Applications, 51(1), pp.259-277.

Lagarias, J.C., Poonen, B. and Wright, M.H., 2012. Convergence of the Restricted Nelder– Mead Algorithm in Two Dimensions. SIAM Journal on Optimization, 22(2), pp.501-532.

Chen, D.H., Saleem, Z. and Grace, D.W., 1986. A new simplex procedure for function minimization. International Journal of modeling and simulation, 6(3), pp.81-85.

Pashaie, M., Sadeghi, M. and Jafarian, A., Arti1cial Neural Networks with Nelder-Mead Optimization Method for solving nonlinear integral equations.

Jafarian, A., Nia, S.A.M., Golmankhaneh, A.K. and Balenu, D., 2013. Numerical solution of linear integral equations

system using the Bernstein collection method. Advances in Difference Equations, 2013(1), p.123.

Pham, N., 2012. Improved Nelder Mead’s simplex method and applications (Doctoral dissertation).

Ouria, A. and Tou1gh, M.M., 2009. Application of Nelder-Mead simplex method for uncon1ned seepage problems.

Applied Mathematical Modelling, 33(9), pp.3589-3598.

Muhammad, S., Coelho, V.N., Guimarães, F.G. and Takahashi, R.H., 2016. An infeasibility certi1cate for nonlinear programming based on Pareto criticality condition. Operations Research Letters, 44(3), pp.302-306.

Ali, R., Muhammad, S. and Takahashi, R.H., 2021. Decision making viva genetic algorithm for the utilization of leftovers. International Journal of Intelligent Systems, 36(4), pp.1746-1769.

Muhammad, S., Ali, R., Abdullah, S. and Okyere, S., 2022. A New Approach to Decision-Making Problem under Complex Pythagorean Fuzzy Information. Complexity, 2022.

Tahir, M., Sardaraz, M., Mehmood, Z. and Muhammad, S., 2021. CryptoGA: a cryptosystem based on genetic algorithm for cloud data security. Cluster Computing, 24(2), pp.739-752.

Gul, H., Alrabaiah, H., Ali, S., Shah, K. and Muhammad, S., 2020. Computation of solution to fractional order partial reaction diffusion equations. Journal of Advanced Research, 25, pp.31-38.

DOI: http://dx.doi.org/10.21015/vtm.v10i2.1170


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