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Non-linear collision-induced breakage equation: finite volume and semi-analytical methods

Sanjiv Kumar Bariwal, Saddam Hussain, Rajesh Kumar

Abstract

The non-linear collision-induced breakage equation has significant applications in particulate processes. Two semi-analytical techniques, namely homotopy analysis method (HAM) and accelerated homotopy perturbation method (AHPM) are investigated along with the well-known finite volume method (FVM) to comprehend the dynamical behavior of the non-linear system, i.e., the concentration function, the total number and the total mass of the particles in the system. The theoretical convergence analyses of the series solutions of HAM and AHPM are discussed. In addition, the error estimations of the truncated solutions of both methods equip the maximum absolute error bound. To justify the applicability and accuracy of these methods, numerical simulations are compared with the findings of FVM and analytical solutions considering three physical problems.

Non-linear collision-induced breakage equation: finite volume and semi-analytical methods

Abstract

The non-linear collision-induced breakage equation has significant applications in particulate processes. Two semi-analytical techniques, namely homotopy analysis method (HAM) and accelerated homotopy perturbation method (AHPM) are investigated along with the well-known finite volume method (FVM) to comprehend the dynamical behavior of the non-linear system, i.e., the concentration function, the total number and the total mass of the particles in the system. The theoretical convergence analyses of the series solutions of HAM and AHPM are discussed. In addition, the error estimations of the truncated solutions of both methods equip the maximum absolute error bound. To justify the applicability and accuracy of these methods, numerical simulations are compared with the findings of FVM and analytical solutions considering three physical problems.
Paper Structure (9 sections, 3 theorems, 78 equations, 7 figures, 1 table)

This paper contains 9 sections, 3 theorems, 78 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Literature review and motivation
  3. Finite Volume Method
  4. Semi-analytical Methods (SAM)
  5. Homotopy analysis method
  6. Accelerated homotopy perturbation method
  7. Convergence Analysis
  8. Numerical Results and Discussion
  9. Conclusions

Key Result

Theorem 4.1

Assume that the non-linear operator ${\mathcal{S}}$ is defined in (operator2). If the following hypotheses; hold, then the operator ${\mathcal{S}}$ has contractive nature, i.e., $\|{\mathcal{S}}f-{\mathcal{S}}f^{\star}\|\leq \xi \|f_-f^{\star}\|, \forall \, (f,f^{\star}) \in \mathbb{Y}_{r,s}^{+}(t) \times \mathbb{Y}_{r,s}^{+}(t),$ where $\xi = 2t\mathcal{K}_1(\mu+1)L_0<1$ with $\mu=\max\{\bar{N}

Figures (7)

  • Figure 1: Log-log plots of concentration functions at time 1
  • Figure 2: Absolute error plots
  • Figure 3: Moments comparison at time 1: FVM, HAM, AHPM and Exact
  • Figure 4: Concentration and consecutive terms error plots
  • Figure 5: Moments comparison at time 1: FVM, HAM, AHPM and Exact
  • ...and 2 more figures

Theorems & Definitions (10)

  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Remark 4.3
  • Theorem 4.4
  • proof
  • Example 5.1
  • Example 5.2
  • Example 5.3