Department of Mathematics
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Item Adomian decomposition and homotopy perturbation method for the solution of time fractional partial integro-differential equations(Springer, 2021-07) Santra, SudarshanThis article deals with two different methods to solve a time fractional partial integro-differential equation. The fractional derivatives are defined here in Caputo sense. The model problem is solved using the Adomian decomposition method and homotopy perturbation method. Moreover, this paper proves the convergence analysis of the solution based on the present methods. Numerical evidences are illustrated in support of the theoretical analysis.Item Numerical treatment of multi-term time fractional nonlinear KdV equations with weakly singular solutions(Taylor & Francis, 2021-12) Santra, SudarshanThe main aim of this work is to construct an efficient recursive numerical technique for solving multi-term time fractional nonlinear KdV equation. The fractional derivatives are defined in Caputo sense. A modified Laplace decomposition method is introduced to approximate the solution. The Adomian polynomials play an important role to execute such a recursive process. In addition, the mathematical importance and some applications of KdV equation are discussed. The approximate solution obtained by the proposed method can be expressed in the form of an infinite convergent series. The experimental evidences demonstrate the effectiveness of the proposed method.Item Hybridizable discontinuous galerkin method for strongly damped wave problem(Springer, 2025-01) Yadav, SangitaWe introduce and analyze a hybridizable discontinuous Galerkin (HDG) approach for the strongly damped linear wave equation. In our investigation, we derive a priori error estimates to demonstrate the optimal convergence of the approximations for both the solution and its gradient. Further, with the help of the dual problem, we present a post-processed solution and analyze its convergence rate, which is of order for , where k is the degree of the polynomial. We also propose a fully discrete scheme, which is of . To validate our theoretical findings, we perform numerical experiments.Item A priori error estimates for sobolev equation using HDG method(Springer, 2025-08) Yadav, SangitaA hybridizable discontinuous Galerkin (HDG) method is introduced and analyzed to solve the Sobolev equation. The analysis includes the derivation of a priori error estimates, demonstrating that the approximations for both the flux and displacement exhibit convergence at a rate of order where h represents the mesh size and k is the polynomial degree. Additionally, the solution is further improved by applying a post-processing technique, and it has been demonstrated that, for , the post-processed solution converges at an enhanced rate of order . A fully discrete scheme is also proposed, achieving second-order accuracy in time; numerical results are needed to validate the theoretical results.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 Legendre wavelet modified petrov–galerkin method in two-dimensional moving boundary problem(De Gruyter, 2017-12) Yadav, SangitaIn this study, we developed the two-dimensional Legendre wavelet modified Petrov–Galerkin method for solving the two-dimensional moving boundary problem arising during melting of solid whose one surface is kept under most generalised boundary condition, and other two surfaces are insulated. The particular cases when surface subjected to the boundary condition of first, second and third kinds are discussed in detail. For validity of the present method, we have plotted graphs between residual (obtained from the original differential equation and its associated boundary conditions) and x-axis and found the effect of an error on moving layer thickness and y coordinate, respectively. Furthermore, we proved the convergence analysis of present method. The effect of parameters (Predvoditelev number, Kirpichev number, Biot number) on the moving layer thickness is discussed in detail. The whole analysis is presented in a dimensionless form.Item An improved version of homotopy perturbation method for multi-dimensional burgers' equations(Wilmington Scientific Publisher, 2024) Kumar, RajeshThe 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.Item Collisional breakage population balance equation: An analytical approach(Elsevier, 2025-01) Kumar, RajeshThis 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.