An iterative scheme for nonlinear collision-induced breakage equation and convergence analysis

Abstract

The particulate process (Population balance equation (PBE)) has significant applications in milling processes, astrophysics, and the formation of raindrops. A novel PBE is presented, where particle collisions result in one particle fragmenting into multiple pieces (two or more) due to the impact of elastic collisions. This article aspires to offer a semi-analytical solution of a nonlinear collision-induced breakage equation (CBE) using the Temimi and Ansari method (TAM). Firstly, we describe the contraction mapping theorem for the local existence of the solution to CBE. Then, the convergence analysis of the TAM iterative solution is exhibited under some physical assumptions on the collision kernels. In addition to this, the maximum error bound is calculated for the finite term truncated solution. In order to show the accuracy and efficiency of the proposed method, we have numerically simulated the finite-term approximate density functions and moments with the available analytical results at various time stages considering several numerical examples. In all numerical cases, TAM yields closed-form solutions for the zeroth and first moments. Furthermore, it is noted that the TAM consumes less computing time despite producing results with precision comparable to the Homotopy Perturbation method [1]. Finally, it has been shown that the proposed method provides the first-order convergence rate.