A general approach to approximate solutions of nonlinear differential equations using particle swarm optimization

A general algorithm is presented to approximately solve a great variety of linear and nonlinear ordinary differential equations (ODEs) independent of their form, order, and given conditions. The ODEs are formulated as optimization problem. Some basic fundamentals from different areas of mathematics are coupled with each other to effectively cope with the propounded problem. The Fourier series expansion, calculus of variation, and particle swarm optimization (PSO) are employed in the formulation of the problem. Both boundary value problems (BVPs) and initial value problems (IVPs) are treated in the same way. Boundary and initial conditions are both modeled as constraints of the optimization problem. The constraints are imposed through the penalty function strategy. The penalty function in cooperation with weighted-residual functional constitutes fitness function which is central concept in evolutionary algorithms. The robust metaheuristic optimization technique of the PSO is employed to find the solution of the extended variational problem. Finally, illustrative examples demonstrate practicality and efficiency of the presented algorithm as well as its wide operational domain.

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► A general algorithm is presented to approximately solve a great variety of ordinary differential equations (ODEs).
► The algorithm can be applied to linear and nonlinear ODEs independent of their form, order, and given conditions.
► The ODEs are formulated as optimization problem.
► Fourier series expansions, calculus of variation, and the particle swarm optimization are employed in problem formulation.
► Illustrative examples demonstrate practicality and efficiency of the algorithm as well as its wide operational domain.