A global collocation quasi–linearization approach for solving coalescence population balance models

Taruni Alekhya Sarvasuddi; Prashanth Maroju; Sattam Alharbi; Ramandeep Behl; Taruni Alekhya Sarvasuddi; Prashanth Maroju; Sattam Alharbi; Ramandeep Behl

doi:10.3934/math.20261112

AIMS Mathematics

2026, Volume 11, Issue 9: 27826-27866. doi: 10.3934/math.20261112

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Research article

A global collocation quasi–linearization approach for solving coalescence population balance models

1.
Department of Mathematics, School of Advanced Sciences, VIT-AP University, Amaravati-522237, Andhra Pradesh, India
2.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
3.
Mathematical Modelling and Applied Computation Research Group (MMAC), Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received: 15 June 2026 Revised: 06 August 2026 Accepted: 11 August 2026 Published: 01 September 2026
MSC : 45G10, 45K05, 65J15, 65M70, 65R20

Population balance models (PBMs) are fundamental to understanding the dynamics of particulate systems in chemical, environmental, and biological engineering. This study presents the global collocation quasi-linearization method (GCQLM), an efficient and robust approach to solving nonlinear coalescence population balance models (CPBMs). The proposed framework combines the rapid convergence of the quasi-linearization method (QLM) with the high spatial accuracy of global spectral collocation. Consequently, the nonlinear CPBM is transformed into a sequence of linear subproblems solved iteratively on a finite domain mapped with Chebyshev-Gauss-Lobatto nodes. The associated bilinear integral terms are computed through Clenshaw–Curtis quadrature to ensure numerical accuracy and structural stability.
Numerical experiments conducted across benchmark kernels (including product, constant, sum, and Pulvermacher kernels) demonstrate that the GCQLM consistently outperforms conventional approaches, such as the variational iteration method (VIM), modified VIM (MVIM), homotopy analysis method (HAM), optimal VIM (OVIM), and finite volume schemes (FVS) ^[1]. Rigorous error norm analysis confirms that the method achieves rapid iterative convergence, exponential decay of spatial errors, and first-order temporal accuracy. Furthermore, the GCQLM produces extremely low $ L_\infty $ error norms, reaching as low as $ 1.13 \times 10^{-12} $ for the product aggregation kernel, while maintaining excellent computational efficiency across various kernel complexities.
Moreover, the parameter-free formulation of the GCQLM entirely eliminates the necessity for auxiliary convergence controls, thereby significantly enhancing algorithm robustness and precision. Comprehensive tabular and graphical assessments validate the method's stability under highly nonlinear and stiff kinetic conditions, establishing it as a highly reliable tool for modeling complex particle aggregation processes.
Citation: Taruni Alekhya Sarvasuddi, Prashanth Maroju, Sattam Alharbi, Ramandeep Behl. A global collocation quasi–linearization approach for solving coalescence population balance models[J]. AIMS Mathematics, 2026, 11(9): 27826-27866. doi: 10.3934/math.20261112

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Abstract

Population balance models (PBMs) are fundamental to understanding the dynamics of particulate systems in chemical, environmental, and biological engineering. This study presents the global collocation quasi-linearization method (GCQLM), an efficient and robust approach to solving nonlinear coalescence population balance models (CPBMs). The proposed framework combines the rapid convergence of the quasi-linearization method (QLM) with the high spatial accuracy of global spectral collocation. Consequently, the nonlinear CPBM is transformed into a sequence of linear subproblems solved iteratively on a finite domain mapped with Chebyshev-Gauss-Lobatto nodes. The associated bilinear integral terms are computed through Clenshaw–Curtis quadrature to ensure numerical accuracy and structural stability.

Numerical experiments conducted across benchmark kernels (including product, constant, sum, and Pulvermacher kernels) demonstrate that the GCQLM consistently outperforms conventional approaches, such as the variational iteration method (VIM), modified VIM (MVIM), homotopy analysis method (HAM), optimal VIM (OVIM), and finite volume schemes (FVS) ^[1]. Rigorous error norm analysis confirms that the method achieves rapid iterative convergence, exponential decay of spatial errors, and first-order temporal accuracy. Furthermore, the GCQLM produces extremely low $ L_\infty $ error norms, reaching as low as $ 1.13 \times 10^{-12} $ for the product aggregation kernel, while maintaining excellent computational efficiency across various kernel complexities.

Moreover, the parameter-free formulation of the GCQLM entirely eliminates the necessity for auxiliary convergence controls, thereby significantly enhancing algorithm robustness and precision. Comprehensive tabular and graphical assessments validate the method's stability under highly nonlinear and stiff kinetic conditions, establishing it as a highly reliable tool for modeling complex particle aggregation processes.

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