New semi-analytical approach and its convergence analysis for a classical hyperbolic fragmentation model: A homotopy perturbation method

Abstract

In the past few decades, the homotopy perturbation method (HPM) has been used intensively for finding analytical and semi-analytical solutions for linear and nonlinear differential equations. This method is well-known for its flexibility and efficiency to solve complicated physical problems. Moreover, HPM offers physical insight into the underlying dynamics of the system, as well as on the evolution of solutions. In this work, HPM is employed to obtain semi-analytic solutions for the hyperbolic particle-fragmentation equation. A recursive scheme is derived for finding the approximate solution. Mathematical formulation of the scheme is further studied by conducting a detailed convergence analysis of the method. Different physics-embedded test problems are solved using this recursive scheme and their series solutions are obtained. The proposed method is validated by comparing the approximated solutions with the exact results. For two test problems, new series solutions are presented in closed form. New analytical solutions are obtained for the particle number density c(x,t), with breakage kernels ℱ(x,y)=1,x+y, and initial condition c(0,x)=exp(−x)x. Moreover, the analytical solutions for ℱ(x,y)=1x+y corresponding to initial conditions c(0,x)=exp(−x) and exp(−x)x are derived for the first time in literature. Due to the non-availability of the analytical solutions of the particle density functions, the results are verified against the finite volume scheme (Saha and Bück, 2021). It is observed that the proposed method predicts the particle density as well as significant integral moments with high accuracy by considering only a fewer number of terms in the series solution.