Department of Mathematics

Permanent URI for this collectionhttp://localhost:4000/handle/123456789/1920

Browse

Now showing 1 - 20 of 44

Analysis of a prion proliferation model with polymer coagulation in the presence of chaperone
(Wiley, 2023-03) Kumar, Rajesh
In 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.
An analytic approach for nonlinear collisional fragmentation model arising in bubble column
(AIP, 2024-10) Kumar, Rajesh
The phenomenon of coagulation and breakage of particles plays a pivotal role in diverse fields. It aids in tracking the development of aerosols and granules in the pharmaceutical sector, coagulation or breakage of droplets in chemical engineering, understanding blood clotting mechanisms in biology, and facilitating cheese production through the action of enzymes within the dairy industry. A significant portion of research in this direction concentrates on coagulation or linear breakage processes. In the case of linear case, bubble particles break down due to inherent stresses or specific conditions of the breakage event. However, in many practical situations, particle division is primarily due to forces exerted during collisions between particles, necessitating an approach that accounts for nonlinear collisional breakage. Despite its critical role in a wide array of engineering and physical operations, the study of this nonlinear fragmentation phenomenon has not been extensively pursued. This article introduces an innovative semi-analytical method that leverages the beyond linear use of equation superposition function to address the nonlinear integro-partial differential model of collisional breakage population balance. This approach is versatile, allowing for the resolution of both linear/nonlinear equations while sidestepping the complexities associated with discretization of domain. To assess the precision of this method, we conduct a thorough convergence analysis. This process utilizes the principle of contractive mapping in the Banach space, a globally recognized strategy for verifying convergence. We explore a variety of kernel parameters associated with collisional kernels, alongside breakage and initial distribution functions, to derive novel iterative solutions. Comparing our findings with those obtained through the finite volume method regarding number density functions and their integral moments, we demonstrate the reliability and accuracy of our approach. The consistency and correctness of our method are further validated by depicting the errors between the exact and approximated solutions in graphical and tabular formats.
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.
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.
Asymptotic behavior of solution of Whitham–Broer–Kaup type equations with negative dispersion
(De Gruyter, 2021-10) Kumar, Rajesh
In this work, we discuss the long time behavior of solutions of the Whitham–Broer–Kaup system with Lipschitz nonlinearity and negative dispersion term. We prove the global well-posedness when α+β2<0 as well as the convergence to 0 of small solutions at rate O(t−1/2) .
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.
Comparison of variational iteration and Adomian decomposition methods to solve growth, aggregation and aggregation-breakage equations
(Elsevier, 2023-03) Kumar, Rajesh
In 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.
Convergence analysis of a finite volume scheme for solving non-linear aggregation-breakage population balance equations
(ARXIV, 2014) Kumar, Rajesh
This paper presents stability and convergence analysis of a finite volume scheme (FVS) for solving aggregation, breakage and the combined processes by showing Lipschitz continuity of the numerical fluxes. It is shown that the FVS is second order convergent independently of the meshes for pure breakage problem while for pure aggregation and coupled equations, it shows second order convergent on uniform and non-uniform smooth meshes. Furthermore, it gives only first order convergence on non-uniform grids. The mathematical results of convergence analysis are also demonstrated numerically for several test problems.
Convergence and error estimation of weighted finite volume scheme for coagulation-fragmentation equation
(Wiley, 2022-12) Kumar, Rajesh
This 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.
Disintegration process of surface stabilized sol-gel TiO2 nanoparticles by population balances
(Elsevier, 2009-12) Kumar, Rajesh
Titanium oxide () is one of the most useful oxide materials, because of its widespread applications in photocatalysis, solar energy conversion, sensors and optoelectronics. The control of particle size and monodispersity of nanoparticles is a challenging task in processing. The control and prediction of these dynamics are based on the process conditions and the nature of chemicals. In this work, we investigate the effect on the surface stabilization with different surfactants and temperature. The steric stabilization of the polymer and various functional groups of dispersants are also considered. Narrow distributed spherical titania particles in the size range 10–100 nm are produced in a sol–gel synthesis from titanium tetra-isopropoxide. The influence of various precursor concentrations and different surfactants on the particle size distribution is investigated. The population balance model for disintegration leads to a system of integro-partial differential equations which is numerically solved by the cell average technique. The experimental results are also compared with the simulation using two different disintegration kernels.
An efficient semi-analytical technique to solve multi-dimensional Burgers’ equation
(Springer, 2023-12) Kumar, Rajesh
The 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.
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
Existence and Uniqueness of Mass Conserving Solutions to Safronov-Dubovski Coagulation Equation for Product Kernel
(ARXIV, 2022-05) Kumar, Rajesh
The article presents the existence and mass conservation of solution for the discrete Safronov-Dubovski coagulation equation for the product coalescence coefficients ϕ such that ϕi,j≤ij ∀ i,j∈N. Both conservative and non-conservative truncated systems are used to analyse the infinite system of ODEs. In the conservative case, Helly's selection theorem is used to prove the global existence while for the non-conservative part, we make use of the refined version of De la Vallée-Poussin theorem to establish the existence. Further, it is shown that these solutions conserve density. Finally, the solutions are shown to be unique when the kernel ϕi,j≤min{iη,jη} where η∈[0,2].
Finite volume scheme for multiple fragmentation Equations
(ISCI, 2012) Kumar, Rajesh
In this paper we study a finite volume approximation for the conservative formulation of multiple fragmentation models. We investigate the convergence of the numerical solutions towards a weak solution of the continuous problem by considering locally bounded kernels. The proof is based on the Dunford-Pettis theorem by using the weak L1 compactness method. The analysis of the method allows us to prove the convergence of the discretized approximated solution towards a weak solution to the continuous problem in a weighted L1 space.
Global classical and weak solutions of the prion proliferation model in the presence of chaperone in a banach space
(AIMS Press, 2022-08) Kumar, Rajesh
The present work is based on the coupling of prion proliferation system together with chaperone which consists of two ODEs and a partial integro-differential equation. The existence and uniqueness of a positive global classical solution of the system is proved for the bounded degradation rates by the idea of evolution system theory in the state space Moreover, the global weak solutions for unbounded degradation rates are discussed by weak compactness technique.
Homotopy perturbation and adomian decomposition methods for condensing coagulation and Lifshitz-Slyzov models
(Springer, 2023-03) Kumar, Rajesh
This 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.
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.
Investigation on agglomeration kinetics of acetaminophen using fluidized bed wet granulation
(Wiley, 2020-01) Kumar, Rajesh
Acetaminophen is a well-known medicine frequently used as analgesic in fever treatment. In pharmaceutical formulation procedure, batch fluidized bed wet granulation is a bottleneck process for the continuous processing of acetaminophen from powder to solid dosage form. To meet the market demand and reduce operating costs, fluid bed wet granulation needs process intensification by converting batch to continuous process. For the scale-up and batch to continuous conversion procedure, investigation on acetaminophen agglomeration kinetics is necessary. Therefore, this work investigates agglomeration kinetics of acetaminophen through batch fluidized bed wet granulation experiments, and the kinetic parameters are estimated using inverse modeling. Experiments are conducted on a 5-L capacity pilot scale batch fluidized bed granulator. The effects of various process parameters, namely, binder concentration, spray rate, atomization pressure, and batch size, on particle size distribution are investigated. A 1-D population balance model with Equi-Kinetic Energy kernel for agglomeration is simulated to compare with the experimental data. The mean particle diameter increased when binder spray rate and binder concentration are increased and that the mean particle diameter decreased with increase in the atomization pressure and batch size. Experiments data comparison with the model can be used for process intensification with reasonable accuracy.
An iterative scheme for nonlinear collision-induced breakage equation and convergence analysis
(Elsevier, 2025-07) Kumar, Rajesh
The 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.
Laplace transform-based approximation methods for solving pure aggregation and breakage equations
(Wiley, 2023-06) Kumar, Rajesh
The varied applications of the aggregation and breakage equations in several fields of science have attracted many researchers to explore accurate novel methods to calculate their solutions. Due to the complexity of these models, the exact solutions are computable only for a few cases of aggregation and breakage kernel parameters. So, to obtain solutions for physically relevant kernels, various numerical and semi-analytical approaches have been explored. It is observed in the literature that the numerical methods are accurate, but they require some unrealistic assumptions. This has led to the development of semi-analytical methods that need fewer parameters and are bereft of discretization of the variables. The researchers explore accurate and less time-consuming methods to solve such equations. So, the objective of this article is the introduction of novel and accurate semi-analytical techniques to solve the pure aggregation and breakage equations. We have used the Laplace optimized decomposition method (LODM) to calculate the series solution for the aggregation equation and the Laplace Adomian decomposition method (LADM) to solve pure breakage equation. The novelty of this work is that it deals with the theoretical convergence of the LADM and LODM solutions toward the exact solutions. In addition to this, several numerical test cases are presented to validate our theoretical findings. For the aggregation equation, LODM results are compared with the solutions obtained via well-developed finite volume technique. The methods are found to be highly accurate to solve these partial integro-differential equations.