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Biologically inspired computing framework for solving two-point boundary value problems using differential evolution

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Abstract

In the present study, a design of biologically inspired computing framework is presented for solving second-order two-point boundary value problems (BVPs) by differential evolution (DE) algorithm employing finite difference-based cost function. The DE has been implemented to minimize the combined residue from all nodes in a least square sense. The proposed methodology has been evaluated using five numerical examples in linear and nonlinear regime of BVPs in order to demonstrate the process and check the efficacy of the implementation. The assessment and validation of the DE algorithm have been carried out by comparing the DE-computed results with exact solution as well as with the corresponding data obtained using continuous genetic algorithms. These benchmark comparisons clearly establish DE as a competitive solver in this domain in terms of computational competence and precision.

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Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Pakistan Institute of Engineering and Applied Sciences (PIEAS), Nilore, Islamabad, 45650, Pakistan

    Muhammad Faisal Fateh

  2. Department of Computer and Information Sciences, Pakistan Institute of Engineering and Applied Sciences (PIEAS), Nilore, Islamabad, 45650, Pakistan

    Aneela Zameer

  3. Department of Physics and Applied Mathematics, Pakistan Institute of Engineering and Applied Sciences (PIEAS), Nilore, Islamabad, 45650, Pakistan

    Nasir M. Mirza & Sikander M. Mirza

  4. Department of Electrical Engineering, COMSATs Institute of Information Technology, Attock Campus, Attock, Pakistan

    Muhammad Asif Zahoor Raja

Authors
  1. Muhammad Faisal Fateh
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  2. Aneela Zameer
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  3. Nasir M. Mirza
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  4. Sikander M. Mirza
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  5. Muhammad Asif Zahoor Raja
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Corresponding author

Correspondence to Muhammad Asif Zahoor Raja.

Appendix

Appendix

Complexity study of the proposed scheme is given in this section as follows.

Before giving the results, firstly, we introduce the function set of the proposed scheme:

  • Unifrnd(): MATLAB function for uniform random number generation.

  • Diff_evol: User-defined differential evolution minimizer for objective function fitval().

  • Mutcb: Mutation function with current to best strategy.

  • Cross1: Exponential crossover function of differential evolution.

  • Fitval: User-defined objective function to be minimized.

  • Cond: Returns boundary condition for each case.

  • InitPOP: Initialization function to create population for DE.

Results of computational complexity (on average) for solving a two-point boundary value problem by proposed methodology are given in Table 12.

Table 12 Complexity analysis of differential evolution algorithm for solution of a two-point boundary value problem
Full size table

The total and self time given in above table is defined as:

  • Self time: it is the time spent in a function excluding the time spent in its child functions.

  • Total time: it is the time spent in a function including the time spent in its child functions.

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Fateh, M.F., Zameer, A., Mirza, N.M. et al. Biologically inspired computing framework for solving two-point boundary value problems using differential evolution. Neural Comput & Applic 28, 2165–2179 (2017). https://doi.org/10.1007/s00521-016-2185-z

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