Vijay Kumar Kukreja
@sliet.ac.in
Professor, Department of Mathematics
Sant Longowal Institute of Engineering & Technology, Longowal
RESEARCH, TEACHING, or OTHER INTERESTS
Mathematics, Computational Mathematics, Numerical Analysis, Modeling and Simulation
Scopus Publications
Scholar Citations
Scholar h-index
Scholar i10-index
Scopus Publications
ARCHNA KUMARI and VIJAY KUMAR KUKREJA
Cambridge University Press (CUP)
Abstract This study aims to formulate a highly accurate numerical method, specifically a seventh-order Hermite technique with an error term of sixth order, to solve the Fisher and Burgers–Fisher equations. This technique employs a combination of orthogonal collocation on the finite element method and hepta Hermite basis functions. By ensuring continuity of the dependent variable and its first three derivatives across the entire solution domain, it achieves a remarkable level of accuracy and smoothness. The space discretization is handled through the application of hepta Hermite polynomials, while the time discretization is managed by the Crank–Nicholson scheme. The stability and convergence analysis of the scheme are discussed in detail. To validate the accuracy of the proposed technique, three examples are taken. The results obtained from these examples are thoroughly analysed and compared against the exact solutions and reliable data from the existing literature. It is established that the proposed technique is easy to implement and gives better results as compared with existing ones.
Shallu and V. K. Kukreja
Informa UK Limited
Archna Kumari, Sudhir Kumar, and Vijay Kumar Kukreja
Springer Science and Business Media LLC
Archna Kumari and Vijay Kumar Kukreja
Elsevier BV
Archna Kumari and Vijay K. Kukreja
With progress on both the theoretical and the computational fronts, the use of Hermite interpolation for mathematical modeling has become an established tool in applied science. This article aims to provide an overview of the most widely used Hermite interpolating polynomials and their implementation in various algorithms to solve different types of differential equations, which have important applications in different areas of science and engineering. The Hermite interpolating polynomials, their generalization, properties, and applications are provided in this article.
Archna Kumari and Vijay Kumar Kukreja
Elsevier BV
Shallu and V. K. Kukreja
Informa UK Limited
In this work, a generalized regularized long-wave equation is solved using the optimal cubic B-spline collocation method for space discretization. Two different approaches are followed for the time-domain discretization. One is the implicit Crank–Nicolson scheme and the other is strong explicit stability preserving Runge–Kutta method of four stages and third-order. Also, the stability analysis of the techniques is carried out and it is shown that the implicit technique is unconditional stable, whereas the explicit technique is conditionally stable. These methods are applied to three problems involving a single solitary wave, the interaction of two solitary waves, and the evolution of solitons via the Maxwellian initial condition. These equations possess three invariants of motion that are mass, momentum, and energy. The value of these invariants is calculated, which is found to remain preserved for a long time. To demonstrate the robustness of both the techniques, the and error norms are calculated.
Archna Kumari and V. K. Kukreja
Springer Science and Business Media LLC
Archna Kumari, Shallu, and V. K. Kukreja
Springer Science and Business Media LLC
Shallu and Vijay Kumar Kukreja
Springer Science and Business Media LLC
AbstractIn the proposed work, an improvised collocation technique with cubic B-spline as basis functions is applied to obtain the numerical solution of non-linear generalized Burgers’–Huxley equation, which has application in the soliton theory. In this technique, posteriori corrections are made to the cubic B-spline interpolant and its higher-order derivatives, which leads to enhancement in the order of convergence in the spatial domain. The temporal domain is discretized using a weighted finite difference scheme, to obtain the solution at each time level and the spatial domain is discretized using the improvised cubic B-spline collocation method. The stability analysis is carried out using the von Neumann scheme and the technique is found to be unconditionally stable. The theoretical proof of the order of convergence is discussed in detail using Green’s function approach. The $$L_{2}$$ L 2 , $$L_{\\infty }$$ L ∞ , and absolute error norms are calculated as well as compared with the results available in the literature, which shows the improvement and efficacy of the proposed technique over the existing ones.
Shallu and V.K. Kukreja
Elsevier BV
Archna Kumari and V. K. Kukreja
Springer Science and Business Media LLC
Ravneet Kaur and V. K. Kukreja
AIP Publishing
Shallu and Vijay Kumar Kukreja
Wiley
In the present work, the numerical solution of the Kuramoto–Sivashinsky (KS) equation is obtained using an improvised quintic B‐spline extrapolated collocation technique. This equation helps in the study of wave production in dissipative medium, investigates the hydrodynamic uncertainty in laminar flames, describes the turbulence of diverse physical processes, and so on. In this work, splines are used due to their higher smoothness property and the sparse nature of the matrices corresponding to the B‐splines. The improvised B‐splines are formed by forcing the quintic B‐spline interpolant to satisfy the interpolatory and some special end conditions. These basis functions are used in the collocation method for space integration and a weighted finite difference scheme is used for temporal domain integration. The stability of the technique is analyzed using the von Neumann scheme, and it is found to be unconditionally stable. The proposed method is found to be sixth‐order convergent in space and second‐order convergent in the time direction. The theoretical order of convergence is matching perfectly with the numerical one. The efficiency of this scheme is demonstrated by applying it to a number of examples. The L2, L∞, and global relative errors are calculated and compared with the previous work, especially with those where quintic B‐splines are used as basis functions. Also, the behavior of some KS equations is discussed for which the exact solution is not available. The aim of the paper is to show that such an improvised technique can be implemented to solve partial differential equations like the KS equation.
Archna Kumari, Shallu Shallu, and V. K. Kukreja
Springer International Publishing
Archna Kumari and Vijay Kumar Kukreja
Elsevier BV
Shelly Arora, Rajiv Jain, and V.K. Kukreja
Elsevier BV
Ravneet Kaur, Shallu, V.K. Kukreja, Nabendra Parumasur, and Pravin Singh
Elsevier BV
Archna Kumari and V.K. Kukreja
Informa UK Limited
In this paper, the singularly perturbed generalized Hodgkin–Huxley equation is solved by the septic Hermite collocation method (SHCM). In this method, septic Hermite interpolating polynomials are used to approximate the trial function because of their special properties such as continuity of the function and the continuity of its tangent at the grid points. The Crank–Nicolson scheme is applied for time discretization and the septic Hermite interpolating polynomials are used for space discretization. The Von-Neumann stability analysis is applied and the algorithm is found to be unconditionally stable. The efficiency of the numerical technique is demonstrated by solving some test examples and comparing the output with the literature data. The analysis shows that the present scheme is easy to implement and gives better results in contrast to the earlier ones.
Shallu and V.K. Kukreja
Elsevier BV
Satinder Pal Kaur, Ajay Kumar Mittal, V. K. Kukreja, Archna Kaundal, N. Parumasur, and P. Singh
Springer Science and Business Media LLC
Shallu and Vijay Kumar Kukreja
Springer Science and Business Media LLC
Shallu, Archna Kumari, and Vijay Kumar Kukreja
Springer Science and Business Media LLC