Numerical solution of q-dynamic equations

Abstract

The variational iteration method (VIM) was used to find approximate numerical solutions of classical and fractional dynamical system equations. To the best of our knowledge, no work on the numerical treatment of q-nonlinear dynamic systems NLDSs is done in the literature. This motivated us to study the numerical solutions of this problem. In this paper, the VIM is extended to find the numerical solutions of q-NLDSs. The proof of the convergence theorem and the error bound analysis are presented. Exact and numerical solutions, by using the extended VIM, of the q-logistic and Lotka–Volterra equations are found. And the comparison shows an excellent matching between the exact and numerical solutions. Approximate numerical solutions of the NLDS of predator–prey with and without self (or cross) difference between two patches are found. In the case of Lotka–Volterra equation, it cannot be solved exactly. Numerical solutions are obtained and a good accuracy is found via evaluating the residual error function. The results show an excellent error tolerance after few iteration steps.