BITS Faculty Publications
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Item Mathematical study of BLUES function method for KdV Burgers’ and BBM-Burgers’ equations(Elsevier, 2025-10) Kumar, RajeshThe 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.Item An iterative scheme for nonlinear collision-induced breakage equation and convergence analysis(Elsevier, 2025-07) Kumar, RajeshThe particulate process (Population balance equation (PBE)) has significant applications in milling processes, astrophysics, and the formation of raindrops. A novel PBE is presented, where particle collisions result in one particle fragmenting into multiple pieces (two or more) due to the impact of elastic collisions. This article aspires to offer a semi-analytical solution of a nonlinear collision-induced breakage equation (CBE) using the Temimi and Ansari method (TAM). Firstly, we describe the contraction mapping theorem for the local existence of the solution to CBE. Then, the convergence analysis of the TAM iterative solution is exhibited under some physical assumptions on the collision kernels. In addition to this, the maximum error bound is calculated for the finite term truncated solution. In order to show the accuracy and efficiency of the proposed method, we have numerically simulated the finite-term approximate density functions and moments with the available analytical results at various time stages considering several numerical examples. In all numerical cases, TAM yields closed-form solutions for the zeroth and first moments. Furthermore, it is noted that the TAM consumes less computing time despite producing results with precision comparable to the Homotopy Perturbation method [1]. Finally, it has been shown that the proposed method provides the first-order convergence rate.Item Weak convergence analysis for non-linear collisional induced breakage equation with singular kernel(2024-12) Kumar, RajeshThe phenomenon of collisional breakage in particulate processes has garnered significant interest due to its wide-ranging applications in fields such as milling, astrophysics, and disk formation. This study investigates the analysis of the pure collisional breakage equation (CBE), characterized by its nonlinear nature with presence of locally bounded collision kernels and singular breakage kernels. Employing a finite volume scheme (FVS), we discretize the continuous equation and investigate the weak convergence of the approximated solution of the conservative scheme towards the continuous solution of CBE. A weight function is introduced to ensure the conservation of the scheme. The non-negativity of the approximated solutions is also shown with the assistance of the mathematical induction approach. Our approach relies on the weak compactness argument, complemented by introducing a stable condition on the time step.Item Convergence and error estimation of weighted finite volume scheme for coagulation-fragmentation equation(Wiley, 2022-12) Kumar, RajeshThis article is dedicated to analyze a finite volume scheme for solving coagulation and multiple fragmentation equation. The rates of coagulation and fragmentation are chosen locally bounded and unbounded (singularity near the origin), respectively. It is shown that using weak compactness method, the numerically approximated solution tends to the weak solution of the continuous problem under a stability condition on the time step for non-uniform mesh. Further, considering a uniform mesh, first order error approximation is demonstrated when kernels are in space. The accuracy of the scheme is also authenticated numerically for several test problems.Item Weak solution and global qualitative behaviour of a prion proliferation model in the presence of chaperone(ACM Digital Library, 2022-08) Kumar, RajeshIn this article, a prion proliferation system in the presence of a chaperone, which involves two ODEs and an integro-partial differential equation, is studied. The existence of weak solution results obtained in Laurençot and Walker (J. Evol. Equ. 7:241–264, 2007) is extended by incorporating chaperone. Further, we study the uniqueness of solution under the sufficient conditions proposed in Laurençot and Walker (J. Evol. Equ. 7:241–264, 2007). In addition, the qualitative global results of disease and disease-free equilibrium points are proved analytically. The effect of the chaperone on prion population is also presented numerically.Item Analysis of a prion proliferation model with polymer coagulation in the presence of chaperone(Wiley, 2023-03) Kumar, RajeshIn the present work, a mathematical model which consists of a nonlinear partial integro-differential equation coupled with two ordinary differential equations (ODEs) is analyzed. This model describes the relation between infectious, noninfectious prion proteins, and chaperone. The well-posedness of the system is proved for bounded kernels by using evolution operator theory in the state space . The existence of a global weak solution for unbounded kernels is also discussed by a weak compactness argument. In addition, we investigated the stability analysis results theoretically and effect of chaperone on prion proliferation numerically.Item Homotopy perturbation and adomian decomposition methods for condensing coagulation and Lifshitz-Slyzov models(Springer, 2023-03) Kumar, RajeshThis article presents analytic approximate solutions of the condensing coagulation model (CCM) and Lifshitz-Slyzov equation (LSE) using two different semi-analytical schemes, namely the homotopy perturbation and Adomian decomposition methods. It is shown mathematically that the series solutions obtained using these techniques converge to the same set of solutions and therefore, justified the method’s reliability. Interestingly, for the CCM, scheme provides closed form solutions for the constant and product kernels. However, finite term approximated solutions are given for sum and Ruckenstein kernels which are physically relevant. Proceeding further, the truncated series solutions are discussed for LSE with constant kernel. To see the novelty of our proposed methods, numerical findings for number density and zeroth moment are compared to the exact solutions with given initial conditions and the errors between the approximated results are shown graphically.Item Optimized decomposition method for solving multi-dimensional Burgers’ equation(Elsevier, 2023-06) Kumar, RajeshThe objectives of this article are to deal with computing the series solutions of 1D dimensionless Burgers’ equation using the optimized decomposition method (ODM) and the extension of ODM to the system of PDEs which aids in dealing with multi-dimensional Burgers’ equation. Several examples of the inviscid and viscous 1D Burgers’ equations are considered to demonstrate the implementation of the scheme. In this case, it is shown that ODM provides better estimates than the existing Adomian decomposition method (ADM). Owing to the advantage of ODM over ADM, the extension of ODM is used to calculate the semi-analytical approximate solutions of the dimensionless 2D and 3D Burgers’ equations. In most cases, it is observed that the series solution gives the closed-form solution. Moreover, in all the examples, the finite term approximate solutions obtained by the proposed method are shown to provide good accuracy with the exact solutions. The theoretical convergence results are also established to showcase the efficacy of our techniqueItem Theoretical analysis of a discrete population balance model with sum kernel(EMS Press, 2023-05) Kumar, RajeshThe Oort–Hulst–Safronov equation is a relevant population balance model. Its discrete form, developed by Pavel Dubovski, is the main focus of our analysis. The existence and density conservation are established for non-negative symmetric coagulation rates satisfying Vi,j⩽i+j, ∀i,j∈N. Differentiability of the solutions is investigated for kernels with Vi,j⩽iα+jα where 0⩽α⩽1 with initial conditions with bounded (1+α)-th moments. The article ends with a uniqueness result under an additional assumption on the coagulation kernel and the boundedness of the second momentItem 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.