Department of Mathematics
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Item A Novel Optimized Decomposition Method for Solving Smoluchowski’s Aggregation Equation. Journal of Computational and Applied Mathematics(Elsevier, 2023-02) Kumar, RajeshThe Smoluchowski’s aggregation equation has applications in the field of bio-pharmaceuticals (Zidar et al., 2018 [1]), financial sector (Pushkin et al., 2004 [2]), aerosol science (Shen et al., 2020 [3]) and many others. Several analytical, numerical and semi-analytical approaches have been devised to calculate the solutions of this equation. Semi-analytical methods are commonly employed since they do not require discretization of the space variable. The article deals with the introduction of a novel semi-analytical technique called the optimized decomposition method (ODM) (see Odibat (2020)) to compute solutions of this relevant integro-partial differential equation. The series solution computed using ODM is shown to converge to the exact solution. The theoretical results are validated using numerical examples for scientifically relevant aggregation kernels for which the exact solutions are available. Additionally, the ODM approximated results are compared with the solutions obtained using the Adomian decomposition method (ADM) in Singh et al., (2015). The novel method is shown to be superior to ADM for the examples considered and thus establishes as an improved and efficient method for solving the Smoluchowski’s equation.Item Comparison of variational iteration and Adomian decomposition methods to solve growth, aggregation and aggregation-breakage equations(Elsevier, 2023-03) Kumar, RajeshIn this work, semi-analytical approaches such as the Adomian decomposition method (ADM), and variational iteration method (VIM) are examined to solve the aggregation, aggregation-breakage and pure growth equations in series forms. The analytical and truncated series solutions are compared for the number density and various moments. The solutions produced using ADM and VIM are mathematically equal in the pure growth case and provide closed-form solutions for constant growth rate. Additionally, Optimal variational iteration method (OVIM) is implemented to solve the growth and aggregation equations, which reduces the error compared to ADM and VIM to some extent but increases the computational cost. Furthermore, in this work, we provide the ADM and VIM formulations for the coupled aggregation-breakage model. Various test cases of each problem are taken to justify the efficiency and accuracy of the series approximated methods. These observations are shown numerically by comparing the finite term series solutions with the exact solutions of number density and moments.Item An efficient semi-analytical technique to solve multi-dimensional Burgers’ equation(Springer, 2023-12) Kumar, RajeshThe work of this paper is motivated by the recently published article (Zeidan et al., Math Methods Appl Sci 43(5):2171–2188, 2020) in which the authors have discussed the Adomian decomposition method (ADM) to solve one dimensional Burgers’ equation in viscous and inviscid forms. Here, we propose an effective and efficient semi-analytical method named variational iteration method (VIM) (He, Int J Non-linear Mech 34(4):699–708, 1999) to solve the Burgers’ equations considered in Zeidan et al. (Math Methods Appl Sci 43(5):2171–2188, 2020). The novelty of the proposed scheme over ADM is proven by comparing the truncated series solutions and presented in the form of graphs and error tables. In addition to this, VIM is extended to solve 2D, 3D, and systems of Burgers’ equations. Thanks to the scheme, closed-form solutions are obtained in most of the cases. The convergence analysis is also investigated for all the test problems.