CLOSED NEWTON COTES QUADRATURE RULES WITH DERIATIVES

In this research paper, a new family of numerical integration of closed newton cotes is introduced which uses the mean of arithmetic and geometric means at derivative value for the Evaluation of Definite Integral. These quadrature methods are shown to be more efficient than the existing quadrature rules. The error terms are obtained by using the concept of precision. Finally, the accuracy of proposed method is verified with numerical examples and the results are compared with existing methods numerically and graphically. Keywords – Numerical Integration, Closed Newton-cotes formula, Definite integral, Arithmetic mean, Geometric mean, Numerical examples. DOI: 10.7176/MTM/9-5-06 Publication date: May 31st 201

An efficient high-order Nystr\"om scheme for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

This text proposes a fast, rapidly convergent Nystr\"{o}m method for the solution of the Lippmann-Schwinger integral equation that mathematically models the scattering of time-harmonic acoustic waves by inhomogeneous obstacles, while allowing the material properties to jump across the interface. The method works with overlapping coordinate charts as a description of the given scatterer. In particular, it employs "partitions of unity" to simplify the implementation of high-order quadratures along with suitable changes of parametric variables to analytically resolve the singularities present in the integral operator to achieve desired accuracies in approximations. To deal with the discontinuous material interface in a high-order manner, a specialized quadrature is used in the boundary region. The approach further utilizes an FFT based strategy that uses equivalent source approximations to accelerate the evaluation of large number of interactions that arise in the approximation of the volumetric integral operator and thus achieves a reduced computational complexity of $O(N \log N)$ for an $N$-point discretization. A detailed discussion on the solution methodology along with a variety of numerical experiments to exemplify its performance in terms of both speed and accuracy are presented in this paper

Arithmetic Mean Derivative-Based Quartet Midpoint Rule

A definite integral that is difficult to solve analytically can be calculated using the numerical integration methods. The midpoint rule is a prominent rule for approximating definite integrals. This article discusses a version of the quartet midpoint rule that includes the derivative of the arithmetic mean . The proposed rule increases precision over the previous rules. Furthermore, the error term is obtained by using the concept of precision between quadrature and exact values. Finally, the proposed rule is more effective than the present rule, according to numerical simulation results

Studies in numerical quadrature

Various types of quadrature formulae for oscillatory integrals are studied with a view to improving the accuracy of existing techniques. Concentration is directed towards the production of practical algorithms which facilitate the efficient evaluation of integrals of this type arising in applications. [Continues.

Modified Trapezoidal Rule Based Different Averages for Numerical Integration

In this paper, A Modified Trapezoidal Rule is presented for the evaluation of numerical integration; the proposed method appears to be efficient modification of trapezoidal rule which is the composition of arithmetic mean of subintervals while the proposed method uses arithmetic mean and Heronian mean at the subintervals. The accuracy of this rule is higher than the original trapezoidal rule .the comparison between the modified trapezoidal rule and original trapezoidal rule is made by using numerical experiments on the basis of local truncation Error. Keywords: Modified trapezoidal rule, Arithmetic mean, Heronian mean, Numerical examples, Accuracy, Numerical Integration.

Efficient solution of parabolic equations by Krylov approximation methods

Numerical techniques for solving parabolic equations by the method of lines is addressed. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus, the resulting approximation consists of applying an evolution operator of a very small dimension to a known vector which is, in turn, computed accurately by exploiting well-known rational approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications, and as a result the algorithm can easily be parallelized and vectorized. Some relevant approximation and stability issues are discussed. We present some numerical experiments with the method and compare its performance with a few explicit and implicit algorithms