Department of Mathematics

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Author: search.filters.author.Kumar, RajeshSubject: search.filters.subject.Convergence analysis

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    Mathematical study of BLUES function method for KdV Burgers’ and BBM-Burgers’ equations
    (Elsevier, 2025-10) Kumar, Rajesh
    The Korteweg–De Vries (KdV) Burgers’ and Benjamin–Bona–Mohoney (BBM) Burgers’ equations are crucial in understanding wave dynamics, heat transfer, and plasma waves. It is essential to solve these models over a long time domain to study how energy will transmit and dissipate, or whether waves will remain coherent or disperse due to dissipation effects. Researchers study various semi-analytical and numerical methods to solve these models. However, numerical methods come with the drawback of discretizing the domain, which leads to some errors in the solutions. In a recent paper (Berx and Indekeu, 2021), the authors introduced a new semi-analytical technique, namely the beyond linear use of the superposition (BLUES) function method for partial differential equations, and showed that the proposed method provides better accuracy compared to existing methods. Therefore, the purpose of this article is to describe the BLUES function method for the KdV and BBM Burgers’ equations. The absence of assumptions, convergence control parameters, linearization, and discretization demonstrates the method’s superiority over conventional numerical and semi-analytical techniques. The article mainly focuses on the stability and convergence analysis of the method. Additionally, the numerical validation of the results includes two instances of KdV-Burgers equations and two instances of BBM-Burgers equations. The efficacy and precision of the suggested methodology are illustrated through the utilization of graphical representations and tabular data.
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    An improved version of homotopy perturbation method for multi-dimensional burgers' equations
    (Wilmington Scientific Publisher, 2024) Kumar, Rajesh
    The accelerated homotopy perturbation Elzaki transform method (AHPETM), which is based on the homotopy perturbation method (HPM), is used in this article to solve the Burgers equation and system of Burgers equations. AHPETM presents the Elzaki integral transform as a pre-treatment in combination with the decomposition of nonlinear variables to speed up the convergence of the HPM solution to its precise values. When the suggested method's findings are compared to HPM's, the results show a considerable improvement. Theoretical convergence analysis and error estimations are also crucial in this work. Multiple numerical examples of 1D, 2D, and 3D Burgers equations, as well as systems of 1D and 2D Burgers equations, are examined to confirm the method's accuracy. Interestingly, the proposed approach offers the closed-form results to most of the problems, which are essentially the exact solutions.
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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.