Department of Mathematics

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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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    Non-linear collision-induced breakage equation: finite volume and semi-analytical methods
    (Springer, 2024-07) Kumar, Rajesh
    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 numerical scheme finite volume method (FVM) to comprehend the dynamical behavior of the non-linear system, i.e., the concentration function, the total number, total mass and energy dissipation of the particles in the system. These semi-analytical methods provide approximate analytical solutions by truncating the infinite series form. The theoretical convergence analyses of the series solutions of HAM and AHPM are discussed under some assumptions on the collisional kernels. In addition, the error estimations of the truncated solutions of both methods equip the maximum absolute error bound. Moreover, HAM simulations are computationally costly compared to AHPM because of an additional auxiliary parameter. To justify the applicability and accuracy of these series methods, approximated solutions are compared with the findings of FVM and analytical solutions considering three physical problems.