Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-linear collision-induced breakage equation: approximate solution and error estimation

Sanjiv Kumar Bariwal, Rajesh Kumar

Abstract

This article aims to provide approximate solutions for the non-linear collision-induced breakage equation using two different semi-analytical schemes, i.e., variational iteration method (VIM) and optimized decomposition method (ODM). The study also includes the detailed convergence analysis and error estimation for ODM in the case of product collisional ($K(ε,ρ)=ερ$) and breakage ($b(ε,ρ,σ)=\frac{2}ρ$) kernels with an exponential decay initial condition. By contrasting estimated concentration function and moments with exact solutions, the novelty of the suggested approaches is presented considering three numerical examples. Interestingly, in one case, VIM provides a closed-form solution, however, finite term series solutions obtained via both schemes supply a great approximation for the concentration function and moments.

Non-linear collision-induced breakage equation: approximate solution and error estimation

Abstract

This article aims to provide approximate solutions for the non-linear collision-induced breakage equation using two different semi-analytical schemes, i.e., variational iteration method (VIM) and optimized decomposition method (ODM). The study also includes the detailed convergence analysis and error estimation for ODM in the case of product collisional () and breakage () kernels with an exponential decay initial condition. By contrasting estimated concentration function and moments with exact solutions, the novelty of the suggested approaches is presented considering three numerical examples. Interestingly, in one case, VIM provides a closed-form solution, however, finite term series solutions obtained via both schemes supply a great approximation for the concentration function and moments.
Paper Structure (11 sections, 4 theorems, 69 equations, 11 figures, 3 tables)

This paper contains 11 sections, 4 theorems, 69 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Literature review and motivation
  3. Semi-analytical Approach
  4. Variational iteration method
  5. VIM for CBE
  6. Optimized decomposition method
  7. ODM for CBE
  8. Convergence Analysis
  9. Numerical Discussion
  10. Conclusions
  11. Acknowledgments

Key Result

Theorem 3.1

Let the operator $A[f],$ mentioned in (vim4), be defined on a Hilbert space $D$ to $D$. The series solution $f(\varsigma,\epsilon)=\sum_{k=0}^{\infty}f_k(\varsigma,\epsilon)$ converges, if $i.e., \|f_{k+1}(\varsigma,\epsilon)\| \leq \alpha \|f_{k}(\varsigma,\epsilon)\|$, where $0< \alpha < 1$ and $\forall k \in \{0\} \cup \mathbb{N}$.

Figures (11)

  • Figure 4.1(A): Series solutions of VIM, ODM and exact solution
  • Figure 4.1(B): Concentration plots at time $\varsigma$= 0.6
  • Figure 4.1(C): Concentration plots at time $\varsigma$= 1
  • Figure 4.1(D): Absolute error between exact solution and $\varphi_{10}$ as well as $\psi_{10}$
  • Figure 4.1(E): Moments comparision: VIM, ODM and Exact
  • ...and 6 more figures

Theorems & Definitions (11)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Example 4.1
  • Example 4.2
  • ...and 1 more