The proposed work focuses on study of homotopy based approximation techniques for evaluating partial differential equations (PDEs) with continuous and piece-wise initial conditions. The well proposed homotopy analysis method (HAM) is taken into consideration. The approximate method based on a zero-order deformation equations in topology contains the auxiliary operator for mapping an initial estimation to the unknown solution and ensure rapid convergence of the given series approximate solution. Implementation of residual error through algebraic equations depicts that the proposed zero-order deformation equation essentially increase the rate of the convergence region and series solution and concede greater freedom in the choosing of convergence operators than the traditional approximate methods. As a starting point of approach, we applied HAM to linear system of PDEs in first order form with prescribed conditions including presence and absence of governing parameters. The convergence of system of PDEs is analysed through auxiliary parameter value. The derived series solution obtained from HAM is compared with exact solution in order to confirm accuracy and effectiveness. The HAM shows rapid convergence than the usual numerical and approximate approach and confirm accuracy in comparison with exact solution in limiting case.
Kumar et al. (Tue,) studied this question.