Department of Mathematics

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Author: search.filters.author.Kumar, RajeshSubject: search.filters.subject.Semi-analytical techniques

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    Semi-analytical methods for solving non-linear differential equations: A review
    (Elsevier, 2024-03) Kumar, Rajesh
    This article develops a new semi-analytical technique based on the homotopy analysis approach for solving linear or non-linear differential equations and the results are compared to the well-known approaches such as the Adomian decomposition method (ADM), homotopy perturbation method (HPM), homotopy analysis method (HAM), and optimized decomposition method (ODM). We discuss the decomposition of the non-linear operator to expedite the HAM solution's convergence to its precise values by using the convergence control parameter. The theoretical convergence analysis and the error estimates are studied. Numerical illustrations show that our proposed scheme improves the accuracy of the non-linear problems discussed in the recently published articles [30] and [31] to an excellent extent and also indicate rapid convergence.
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    Elzaki projected differential transform method for multi-dimensional aggregation and combined aggregation-breakage equations
    (Elsevier, 2024-01) Kumar, Rajesh
    Numerous real-world fields, including planetary science, bio-pharmaceutical, chemical study, food processing industry, and many more are profoundly impacted by population balance equations. Model complexity limits the analytical investigations to a few aggregation-breakage parameters, although various numerical and semi-analytical schemes are available. This article proposes a new semi-analytical approach, the Elzaki integral transform as a pre-treatment to reinforce domain decomposition for better accuracy and convergence, in conjunction with the projected differential transform method for finding the closed form or approximated series solutions for non-linear aggregation, aggregation-breakage, and multi-dimensional aggregation equations. The technique’s key benefit over traditional numerical methods is its ability to solve linear or non-linear differential equations directly without discretization or linearization. Theoretical convergence analysis and error estimates of series solutions for both one and two dimensional aggregation models are of particular interest. Finally, several numerical examples of aggregation, aggregation-breakage, and two dimensional aggregation equations are taken to validate the accuracy of the proposed method by comparing numerical simulations with exact solutions. Interestingly, we have obtained closed form solutions for the pure aggregation equation when considering constant and product aggregation kernels. Additionally, error tables and graphs help to highlight the method’s innovative nature
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    An analytical treatment to spatially inhomogeneous population balance model
    (Elsevier, 2024-09) Kumar, Rajesh
    In modern liquid–liquid contact components, there is an increasing use of droplet population balance models. These components include differential and completely mixed contractors. These models aim to explain the complex hydrodynamic processes occurring in the dispersion phase. The hydrodynamics of these interacting dispersions include droplet breaking, coalescence, axial dispersion, and both entry and exit events. The resulting equations for population balance are represented as integro-partial differential equations, which rarely have analytical solutions, especially when spatial dependency is apparent. Consequently, the pursuit predominantly lies in seeking numerical solutions to resolve these complex equations. In this study, we have devised analytical solutions for inhomogeneous breakage and coagulation by employing the population balance equation (PBEs) applicable to both batch and continuous flow systems. The innovative approaches for solving PBEs in these systems leverage the Adomian decomposition method (ADM) and the homotopy analysis method (HAM). These semi-analytical methodologies effectively tackle the significant challenges related to numerical discretization and stability, which have often plagued previous solutions of the homogeneous PBEs. Our findings across all test examples demonstrate that the approximated particle size distributions utilizing these two methods converge to the analytical solutions continuously.
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    Collisional breakage population balance equation: An analytical approach
    (Elsevier, 2025-01) Kumar, Rajesh
    This work presents a unique semi-analytical approach based on the homotopy analysis method (HAM), called accelerated HAM, recently proposed in (Hussain et al., “Semi-analytical methods for solving non-linear differential equations: A review.”, JMAA, 2023), to solve the collisional breakage population balance model, which is an integro-partial differential equation. We compare our findings with those obtained using the Adomian decomposition method, a well-known technique for solving various forms of differential equations. By decomposing the non-linear operator, we investigate how to utilize the convergence control parameter to expedite the convergence of the HAM solution towards its precise value in accelerated HAM. The other objective of the article is to examine the theoretical convergence analysis of the two proposed methods. Additionally, we conduct theoretical research on the error estimates for both the techniques. To validate our schemes, several numerical examples are considered and the numerical simulations demonstrate that the suggested techniques provide accurate estimates for the solution and moments of the collisional breakage equation.
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    Analytical and numerical treatments of a coagulation population balance model
    (Elsevier, 2025-03) Kumar, Rajesh
    The Redner–Ben-Avraham–Kahng (RBK) coagulation model, initially proposed as a discrete framework for investigating cluster growth kinetics, has recently been reformulated to encompass a continuous representation. While the existence, uniqueness, and long-term dynamics of solutions for the continuous model have been examined, both analytical and numerical solutions have yet to be thoroughly addressed. This study undertakes a comprehensive investigation of the continuous RBK coagulation model utilizing both numerical and semi-analytical methodologies, specifically the Finite Volume Method (FVM) and the Homotopy Perturbation Method (HPM). Analytical expressions for the number density function are derived for a variety of coagulation kernels, including constant, sum, product, and bilinear kernels, based on exponential and gamma initial distributions. The efficacy of the HPM is rigorously assessed through an extensive convergence analysis, which encompasses the order of convergence and error estimates pertinent to the series solutions. Furthermore, the outcomes obtained from HPM are validated against those derived from the established FVM, thereby demonstrating the reliability and effectiveness of HPM in addressing the continuous RBK model.