Study of Hybrid Evolutionary Computation Techniques for Solving Large Set of Linear Equations and Partial Differential Equations

Abstract

Hybrid Algorithms for solving set of linear equations are hybridization of evolutionary techniques and classical methods for solving set of linear equations. The classical iterative methods for solving set of linear equations are slow in terms of convergence and can be made faster by introducing relaxation factor ω (0 <ω< 2). The process in very sensitive to the relaxation factor and the estimation of its optimum value is very difficult. Adaptation and selection mechanism of evolutionary computations serves the purpose of finding the optimum value of the relaxation factor and then the solution come out. The four Hybrid Evolutionary Algorithms (JBUA, GSBUA, JBTVA and GSBTVA) were in front of us. Thorough study of the Uniform Adaptive Hybrid Evolutionary Algorithms JBUA and GSBUA showed that the crossover operation present in them are needless and thus we have proposed two modified Algorithms MJBUA and MGSBUA. We have tested the proposed MGSBUA separately for solving partial differential equations (especially in case of Laplace's equation). The solution of the discretized form is compared with the analytical one and the same set is also solved by the Gauss-Scidel method. It is found that our proposed method is faster and better accuracy can be achieved. We also have solved a sample Poisson's equation using our proposed algorithm. It is found that MJBUA and MGSBUA hybrid algorithms are faster and memory effective than their original counterparts.