VIJAY KUMAR PATEL
@iitbhu.ac.in
Indian Institute of Technology Banaras Hindu University, Varanasi, India
Scopus Publications
Scholar Citations
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Scopus Publications
Vijay Kumar Patel and Dhirendra Bahuguna
Elsevier BV
Vijay Kumar Patel, Somveer Singh, and Vineet Kumar Singh
Wiley
Vinita Devi, Rahul Kumar Maurya, Vijay Kumar Patel, and Vineet Kumar Singh
Springer Science and Business Media LLC
The current study is presented to develop two approaches and methodologies to find the numerical solution of linear and non-linear initial value problems such as Lane–Emden type equation, Riccati’s equation and Bessel’s equation of order zero based on approximation. The function approximations (scheme-I and scheme-II) are presented to find the numerical solutions of linear and non-linear initial value problems by using Gauss Legendre roots as node points and random node points in the domain [0, 1]. In the scheme-I, the roots of Legendre polynomial are used as node points for Lagrange polynomials and in scheme-II, we have taken random node points in the domain [0, 1] and orthogonalize the resulting Lagrange polynomials using Gram–Schmidt orthogonalization process. Firstly, we have introduced the function approximations by using generating interpolating scaling functions (ISF) and orthonormal Lagrangian basis functions (OLBF) over the space $$L^{2}[0,1]$$L2[0,1] then we have constructed the operational matrices of integration and product operational matrices based on newly designed approximations namely ISF and OLBF. These operational matrices convert given linear and non-linear initial value problems into the associated system of algebraic equations. Finally, we have established error bounds (Lemmas 1, 2) of both scheme-I and scheme-II including the function approximations. The efficiency of the proposed schemes has been confirmed with several test examples including numerical stability. So, the schemes are simple, efficient and produces very accurate numerical results in considerably small number of basis functions and hence reduces computational effort.
Vijay Kumar Patel, Somveer Singh, Vineet Kumar Singh, and Emran Tohidi
Springer Science and Business Media LLC
In this article, we have study a 2D Legendre and Chebyshev wavelets collocation scheme for solving a class of linear and nonlinear weakly singular Volterra partial integro-differential equations (PIDEs). The scheme is based on wavelets collocation for PIDEs with uniquely designed matrices over the Hilbert space defined on the domain $$\\left( [0, 1]\\times [0, 1]\\right) $$[0,1]×[0,1]. Using piecewise approximation associated with 2D Legendre wavelet, 2D Chebyshev wavelet and its operational matrices, the considered PIDEs will be reduced into the corresponding system of linear and nonlinear algebraic equations. The corresponding linear and nonlinear system of equations solved by collocation scheme and well-known Newton–Raphson scheme at collocation points respectively. In addition, the convergence and error analysis of the numerical scheme is provided under several mild conditions. The numerical results are correlated with the exact solutions and the execution of the proposed scheme is determined by estimating the maximum absolute errors, $$l_{2}\\hbox {-}norm$$l2-norm errors and $$l_{\\infty }\\hbox {-}norm$$l∞-norm errors. The numerical result shows that the scheme is simply applicable, efficient, powerful and very precisely at small number of basis function. The main important applications of the proposed wavelets collocation scheme is that it can be applied on linear as well as nonlinear problems and can be applied on higher order partial differential equations too.
Somveer Singh, Vijay Kumar Patel, and Vineet Kumar Singh
Wiley
Somveer Singh, Vijay Kumar Patel, Vineet Kumar Singh, and Emran Tohidi
Elsevier BV
Abstract The present article is devoted to develop a new approach and methodology to find the approximate solution of second order two-dimensional telegraph equations with the Dirichlet boundary conditions. We first transform the telegraph equations into equivalent partial integro-differential equations (PIDEs) which contain both initial and boundary conditions and therefore can be solved numerically in a more appropriate manner. Operational matrices of integration and differentiation of Bernoulli polynomials together with the completeness of these polynomials are used to reduce the PIDEs into the associated algebraic generalized Sylvester equations which can be solved by an efficient Krylov subspace iterative (i.e., BICGSTAB) method. The efficiency of the proposed method has been confirmed with several test examples and it is clear that the results are acceptable and found to be in good agreement with exact solutions. We have compared the numerical results of the proposed method with radial basis function method and differential quadrature method. Also, the method is simple, efficient and produces very accurate numerical results in considerably small number of basis functions and hence reduces computational effort. Moreover, the technique is easy to apply for multidimensional problems.
Somveer Singh, Vijay Kumar Patel, and Vineet Kumar Singh
Elsevier BV
In this work, we developed an efficient computational method based on Legendre and Chebyshev wavelets to find an approximate solution of one dimensional hyperbolic partial differential equations (HPDEs) with the given initial conditions. The operational matrices of integration for Legendre and Chebyshev wavelets are derived and utilized to transform the given PDE into the linear system of equations by combining collocation method. Convergence analysis and error estimation associated to the presented idea are also investigated under several mild conditions. Numerical experiments confirm that the proposed method has good accuracy and efficiency. Moreover, the use of Legendre and Chebyshev wavelets are found to be accurate, simple and fast.
Vijay Kumar Patel, Somveer Singh, and Vineet Kumar Singh
Wiley
In this paper, a numerical solution of fractional partial differential equations (FPDEs) for electromagnetic waves in dielectric media will be discussed. For the solution of FPDEs, we developed a numerical collocation method using an algorithm based on two-dimensional shifted Legendre polynomials approximation, which is proposed for electromagnetic waves in dielectric media. By implementing the partial Riemann–Liouville fractional derivative operators, two-dimensional shifted Legendre polynomials approximation and its operational matrix along with collocation method are used to convert FPDEs first into weakly singular fractional partial integro-differential equations and then converted weakly singular fractional partial integro-differential equations into system of algebraic equation. Some results concerning the convergence analysis and error analysis are obtained. Illustrative examples are included to demonstrate the validity and applicability of the technique. Copyright © 2017 John Wiley & Sons, Ltd.
Vijay Kumar Patel, Somveer Singh, and Vineet Kumar Singh
Elsevier BV
In this article, we deal with a numerical wavelet collocation method (NWCM) using a technique based on two-dimensional wavelets (TDWs) approximation proposed for the fractional partial differential equations (FPDEs) for electromagnetic waves in dielectric media (EWDM). By implementing the RiemannLiouville fractional derivative, TDWs approximation and its operational matrix along with collocation method are utilized to reduce FPDEs firstly into weakly singular fractional partial integro-differential equations (FPIDEs) and then reduced weakly singular FPIDEs into system of algebraic equation. Using Legendre wavelet approximation (LWA) and Chebyshev wavelet approximation (CWA), we investigated the convergence analysis and error analysis of the proposed problem. Finally, some examples are included for demonstrating the efficiency of the proposed method via LWA and CWA respectively.
Somveer Singh, Vijay Kumar Patel, Vineet Kumar Singh, and Emran Tohidi
Elsevier BV
In this paper, we propose and analyze an efficient matrix method based on shifted Legendre polynomials for the solution of non-linear volterra singular partial integro-differential equations(PIDEs). The operational matrices of integration, differentiation and product are used to reduce the solution of volterra singular PIDEs to the system of non-linear algebraic equations. Some useful results concerning the convergence and error estimates associated to the suggested scheme are presented. illustrative examples are provided to show the effectiveness and accuracy of proposed numerical method.
Somveer Singh, Vijay Kumar Patel, and Vineet Kumar Singh
Elsevier BV
In this paper, an effective numerical method is introduced for the treatment of Volterra singular partial integro-differential equations. They are based on the operational and almost operational matrix of integration and differentiation of 2D shifted Legendre polynomials. The methods convert the singular partial integro-differential equation in to a system of algebraic equations. Convergence analysis and error estimates are derived for the proposed method. Illustrative examples are included to demonstrate the validity and applicability of the technique.