Vineet K. Singh

AN ANALYTIC ALGORITHM OF LANE-EMDEN TYPE EQUATIONS ARISING IN ASTROPHYSICS USING MODIFIED HOMOTOPY ANALYSIS METHOD

Abstract Lane–Emden type equation models many phenomena in mathematical physics and astrophysics. It is a nonlinear differential equation which describes the equilibrium density distribution in self-gravitating sphere of polytropic isothermal gas, has a singularity at the origin, and is of fundamental importance in the field of stellar structure, radiative cooling, modeling of clusters of galaxies. An efficient analytic algorithm is provided for Lane–Emden type equations using modified homotopy analysis method, which is different from other analytic techniques as it itself provides us with a convenient way to adjust convergence regions even without Pade technique. Some examples are given to show its validity.

A stable algorithm for Hankel transforms using hybrid of Block-pulse and Legendre polynomials

Abstract A new numerical method, based on hybrid of Block-pulse and Legendre polynomials for numerical evaluation of Hankel transform is proposed in this paper. Hybrid of Block-pulse and Legendre polynomials are used as a basis to expand a part of the integrand, r f ( r ) , appearing in the Hankel transform integral. Thus transforming the integral into a Fourier–Bessel series. Truncating the series, an efficient algorithm is obtained for the numerical evaluations of the Hankel transforms of order ν > − 1 . The method is quite accurate and stable, as illustrated by given numerical examples with varying degree of random noise terms e θ i added to the data function f ( r ) , where θ i is a uniform random variable with values in [ − 1 , 1 ] . Finally, an application of the proposed method is given in solving the heat equation in an infinite cylinder with a radiation condition.

Efficient algorithms to solve singular integral equations of Abel type

In the present paper, we obtain the approximate solution of Abels integral equation by using the following powerful, efficient but simple methods: (i) Hes homotopy perturbation method (HPM), (ii) Modified homotopy perturbation method (MHPM), (iii) Adomian decomposition method (ADM) and (iv) Modified Adomian decomposition method (MADM). The validity and applicability of these techniques are illustrated through various particular cases which demonstrate their efficiency and simplicity in solving these types of integral equations compared with the other existing methods.

Numerical evaluation of the Hankel transform by using linear Legendre multi-wavelets

Abstract An efficient algorithm for evaluating the Hankel transform F n ( p ) of order n of a function f ( r ) is given. As the continuous Legendre multi-wavelets forms an orthonormal basis for L 2 ( R ) ; we expand the part r f ( r ) of the integrand in its wavelet series reducing the Hankel transform integral as a series of Bessel functions multiplied by the wavelet coefficients of the input function. Numerical examples are given to illustrate the efficiency of the proposed method.

Efficient algorithms to compute Hankel transforms using wavelets

Abstract The aim of the paper is to propose two efficient algorithms for the numerical evaluation of Hankel transform of order ν , ν > − 1 using Legendre and rationalized Haar (RH) wavelets. The philosophy behind the algorithms is to replace the part x f ( x ) of the integrand by its wavelet decomposition obtained by using Legendre wavelets for the first algorithm and RH wavelets for the second one, thus representing F ν ( y ) as a Fourier–Bessel series with coefficients depending strongly on the input function x f ( x ) in both the cases. Numerical evaluations of test functions with known analytical Hankel transforms illustrate the proposed algorithms.

Operational matrix approach for the solution of partial integro-differential equation

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.

An improved method for computing Hankel transform

Abstract The purpose of this paper is to compute the Hankel transform Fn(y) of order n of a function f(x) and its inverse transform using rationalized Haar wavelets. The integrand x f ( x ) is replaced by its wavelet decomposition. Thus representing Fn(y) as a Fourier–Bessel series with coefficients depending strongly on the local behavior of the function x f ( x ) , thereby getting an efficient algorithm for their numerical evaluation. Numerical evaluations of test functions with known analytical Hankel transforms illustrate the proposed algorithm.

Numerical solution of nonlinear weakly singular partial integro-differential equation via operational matrices

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.

A stable algorithm for numerical evaluation of Hankel transforms using Haar wavelets

The purpose of the paper is to propose a stable algorithm for the numerical evaluation of the Hankel transform Fn(y) of order n of a function f(x) using Haar wavelets. The integrand

Numerical solution of system of Volterra integral equations using Bernstein polynomials

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