Differential evolution based computation intelligence solver for elliptic partial differential equations

Muhammad Faisal FATEH; Aneela ZAMEER; Sikander M. MIRZA; Nasir M. MIRZA; Muhammad Saeed ASLAM; Muhammad Asif Zahoor RAJA

doi:10.1631/FITEE.1900221

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Differential evolution based computation intelligence solver for elliptic partial differential equations

Regular Papers | Updated：2022-05-19

- Differential evolution based computation intelligence solver for elliptic partial differential equations
- 基于差分进化的椭圆型偏微分方程计算智能求解器
- Frontiers of Information Technology & Electronic Engineering Vol. 20, Issue 10, Pages: 1445-1456(2019)
- Affiliations：
  
  Department of Electrical Engineering, Pakistan Institute of Engineering and Applied Sciences (PIEAS), Islamabad 45650, Pakistan
  Department of Computer and Information Sciences, Pakistan Institute of Engineering and Applied Sciences (PIEAS), Islamabad 45650, Pakistan
  Department of Physics and Applied Mathematics, Pakistan Institute of Engineering and Applied Sciences (PIEAS), Islamabad 45650, Pakistan
  School of Electrical and Electronic Engineering, University of Adelaide, Adelaide 5005, Australia
  Department of Electrical and Computer Engineering, COMSATS University Islamabad, Attock Campus, Attock 43600, Pakistan
- Author bio：
  
  Muhammad Saeed ASLAM, E-mail: muhammad.aslam@adelaide.edu.au
- Funds：
- DOI：10.1631/FITEE.1900221
  CLC： O175
- Received：01 May 2019，
  
  Revised：2019-10-;8，
  
  Published：2019-10
- Accepted：
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Muhammad Faisal FATEH, Aneela ZAMEER, Sikander M. MIRZA, et al. Differential evolution based computation intelligence solver for elliptic partial differential equations[J]. Frontiers of Information Technology & Electronic Engineering, 2019, 20(10): 1445-1456.
DOI：

Muhammad Faisal FATEH, Aneela ZAMEER, Sikander M. MIRZA, et al. Differential evolution based computation intelligence solver for elliptic partial differential equations[J]. Frontiers of Information Technology & Electronic Engineering, 2019, 20(10): 1445-1456. DOI： 10.1631/FITEE.1900221.

摘要

介绍了一种基于差分进化的方法，用以解决具有狄里克莱和/或诺依曼边界条件的椭圆型偏微分方程。通过最小化群体间的节点偏差，解决方案在整个内部节点的有界域上演化。用对应系统的有限差分近似代替椭圆型偏微分方程，得到节点留数的表达式。将全局留数声明为节点留数的均方根值，并将其作为代价函数。利用标准微分进化方法将椭圆型偏微分方程转化为全局留数的极小化问题求解。同时考虑线性与非线性椭圆偏微分方程的一系列基准问题，验证了该算法的有效性。为证明该算法的鲁棒性，对不同差分进化算子和参数进行灵敏度分析。将基于差分进化的计算节点值与用精确解析表达式得到的对应数据进行比较，比较结果显示了该方法的精确度和收敛性。

Abstract

A differential evolution based methodology is introduced for the solution of elliptic partial differential equations (PDEs) with Dirichlet and/or Neumann boundary conditions. The solutions evolve over bounded domains throughout the interior nodes by minimization of nodal deviations among the population. The elliptic PDEs are replaced by the corresponding system of finite difference approximation

yielding an expression for nodal residues. The global residue is declared as the root-mean-square value of the nodal residues and taken as the cost function. The standard differential evolution is then used for the solution of elliptic PDEs by conversion to a minimization problem of the global residue. A set of benchmark problems consisting of both linear and nonlinear elliptic PDEs has been considered for validation

proving the effectiveness of the proposed algorithm. To demonstrate its robustness

sensitivity analysis has been carried out for various differential evolution operators and parameters. Comparison of the differential evolution based computed nodal values with the corresponding data obtained using the exact analytical expressions shows the accuracy and convergence of the proposed methodology.

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