Series Solution of the System of Integro-Differential Equations

The homotopy analysis method [1, 2], is developed to search the accurate asymptotic solutions of nonlinear problems. This technique has been successfully applied to many nonlinear problems such as the viscous flows of non-Newtonian fluids [3, 4], the Korteweg-de Vries-type equations [5, 6], nonlinear heat transfer [7, 8], finance problems [9, 10], Riemann problems related to nonlinear shallow water equations [11], projectile motion [12], Glauertjet flow [13], nonlinear water waves [14], groundwater flows [15], Burgers-Huxley equation [16], time-dependent Emden-Fowler type equations [17], differential-difference equation [18], Laplace equation with Dirichlet and Neumann boundary conditions [19], thermal-hydraulic networks [20], and recently for the Fitzhugh-Nagumo equation [21], and so on. On the other hand, one of the interesting topics among researchers is solving integro-differential equations. In fact, integro-differential equations arise in many physical processes, such as glass-forming process [22], nanohydrodynamics [23], drop wise condensation [24], and wind ripple in the desert [25]. There are various numerical and analytical methods to solve such problems, but each method limits to a special class of integro-differential equations. El-Sayed et al. applied the decomposition method to solve high-order linear Volterra-Fredholm integrodifferential equations [26]. In [27], the variational iteration method was applied to solve the system of linear integro-differential equations. Also, Biazar et al.