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In this study, a new numerical method based on the combination of High Exploration Particle Swarm Optimization (HEPSO) and Moving Least Squares (MLS) is introduced to solve nonlinear porous media equations. The MLS scheme is employed to describe an appropriate discretized function, and the penalty method is implemented to convert the constrained problem into an unconstrained one via satisfying the initial conditions. The identified objective function is minimized by the HEPSO to find the approximated nodal values for the nonlinear porous media equation. In order to illustrate the effectiveness of the HEPSO, the optimization trajectories are compared with those of a Standard Particle Swarm Optimization (SPSO) algorithm. Moreover, comparisons are made between the exact solution and the introduced strategy to expose the accuracy, effectiveness and simplicity of the proposed method.

NUMERICAL SOLUTIONS OF THE NONLINEAR POROUS MEDIA EQUATION BASED ON HIGH EXPLORATION PARTICLE SWARM OPTIMIZATION AND MOVING LEAST SQUARES