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Meta-heuristic algorithms for solving nonlinear differential equations based on multivariate Bernstein polynomials

  • Mathematical methods in data science
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Abstract

This paper concerns with solving nonlinear boundary value problems with Dirichlet boundary conditions by a novel approximation technique based on Bernstein polynomials and meta-heuristic algorithms. A trial solution is expressed as a weighted sum of Bernstein polynomials of degree \(n\). Some weights of this trial solution are determined by enforcing exact satisfaction of the Dirichlet boundary conditions. The remaining weights are determined such that they minimize the sum of squares of the residuals of the differential equation computed at arbitrary set of interior points in the domain of the problem. Error bounds for the approximate solutions for ordinary and partial differential equations are derived. The resulting optimization problem is solved using meta-heuristic algorithms, namely particle swarm optimization (PSO), genetic algorithm (GA), and a hybrid PSO–GA algorithm. The accuracy of the proposed approach is demonstrated by solving some nonlinear boundary value problems including Bratu problems in one- and two-dimensional spaces.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt

    Rasha M. Abo-bakr

  2. Department of Physics and Engineering Mathematics, Faculty of Engineering, P.O. Box 44519, Zagazig, Egypt

    N. A. Mohamed & S. A. Mohamed

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  1. Rasha M. Abo-bakr
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  2. N. A. Mohamed
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  3. S. A. Mohamed
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Correspondence to N. A. Mohamed.

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Abo-bakr, R.M., Mohamed, N.A. & Mohamed, S.A. Meta-heuristic algorithms for solving nonlinear differential equations based on multivariate Bernstein polynomials. Soft Comput 26, 605–619 (2022). https://doi.org/10.1007/s00500-021-06535-1

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