Abstract
The aim of the research, we extended the one-dimensional to multi-dimensional, we applied the homotopy perturbation and analysis methods to solve Volterra integral equations and to obtain approximate analytical solutions of systems of the second kind multi-dimensional Volterra integral equations. We proved the convergence of the homotopy analysis method (HAM). The HAM solutions contained an auxiliary parameter that provides a convenient way of controlling the convergence region of series solutions. It is shown that the solutions obtained by the homotopy-perturbation method (HPM) are only special cases of the HAM solutions. Several examples are given to illustrate the efficiency and implementation of the method. The results indicate that this method is efficient for the linear and non - linear models with the dissipative terms.
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