The particle swarm optimization against the Runge's phenomenon: Application to the generalized integral quadrature method

Abstract —In the present work, we introduce the particle swarmoptimization called (PSO in short) to avoid the Runge’s phenomenonoccurring in many numerical problems. This new approach is testedwith some numerical examples including the generalized integralquadrature method in order to solve the Volterra’s integral equations. Keywords —Integral equation, particle swarm optimization,Runge’s phenomenon. I. I NTRODUCTION I N recent years, much attention has been devoted to theinvestigation of new mathematical models and numericalapproaches to evaluate the solutions of the EDP and theintegral equations. Excellent surveys which contain both nu-merical and theoretical researches are given in [1-20].The primary argument for the interest of the type of thisproblem comes naturally from its wide applications almostin any branches of science and engineering described bysystems of ODEs and PDEs [21-31] which in some situationsthe solutions present the Runge’s phenomenon in the edgesof the interval. This situation can be avoided by a specificutilization of the algorithm PSO. The PSO algorithm is a paral-lel evolutionary computation technique proposed by Kennedyand Eberhart in 1995. The PSO has nowadays gained greatimportance in computer optimization.The latest numerical approach to date is the generalizedintegral quadrature method introduced by Zerarka and Soukeur[32]. It was first applied to one-dimensional Volterra integralin the linear and nonlinear cases, where the solution is notcompletely reproduced in the domain in which strong oscil-lations can arise. This method studies the situation in whichthe unknown function is identified as the Lagrange polynomial[33] and the interpolating points of the Tchebychev type areused.New calculations are performed for the construction of thesolution by a suitable choice of the interpolating points usingthe particle swarm optimization (PSO) in order to avoid theRunge’s phenomenon [34]. Our main purpose is to show howthe Runge’s phenomenon can be completely removed fromthe solution of interest. We examine two specific examples in

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