Numerical method for evaluation triple integrals by using midpoint's rule

In this paper, we derive method to find the values of the triple integrals numerically its integrands continuous but have singularity in partial derivatives in the region of the integrals by using Midpoint's rule on the three x,y and dimensions z , and  how to findthe general form of the errors (correction terms) and we will improve the results by using Romberg acceleraion[3],[6] from correction terms that we found it when the number of (l)subintervals that divided interval integral on the exterior dimension z equal to twice the number of subintervals(n) on the interior dimension x and the number of subintervals (m) on the middle dimension y ,that is mean ( h3=1/2 h1 , h1 = h2 ) when h1 means the distances between the ordinates on the x- axis, h2 means the distances between the ordinates on the y- axis and h3 means the distances between the ordinates on the z-axis and we denote to this method by  Mid3 (h1,h2,h3 ) and we can depend on it to calculate the triple integrals because it gave high accuracy in results by few subintervals

Numerical quadrature methods for integrals of singular periodic functions and their application to singular and weakly singular integral equations

High accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Such periodic equations are used in the solution of planar elliptic boundary value problems, elasticity, potential theory, conformal mapping, boundary element methods, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples

A comparison of the error function and the tanh transformation as progressive rules for double and triple singular integrals

AbstractTransformations of the form x = tanh(g(t)) for g(t) = n and for g(t) = c sinh(t), as well as transformations of the error function type are employed on double and triple singular integrals. Extensions of the one-dimensional approach allow variable limit multiple integrals to be attempted successfully. Some shortcomings of the double exponential or DE rule are exposed in the comparisons. Finally, comparison is made with the method of good lattice points (Sugihara, 1987)

Some quadrature methods for general and singular integrals in one and two dimensions

In this thesis numerical integration in one and two dimensions is considered. In chapter two transformation methods are considered primarily for singular integrals and methods of computing the transformations themselves are derived. The well-known transformation based on the IMT rule and error function are extended to non-standard functions. The implementation of these rules and their performances are demonstrated. These transformations are then extended to two-dimensions and are used to develop accurate rules for integrating singular integrals. In addition to this, a polynomial transformation with the aim of the reduction in the number of function evaluations is also considered and the resultant product rule is applied to two-dimensional non-singular integrals. Finally, the use of monomials in the construction of integration rules for non-singular two-dimensional integrals is considered and some rules developed. In all these situations the rules developed are tested and compared with existing methods. The results show that the new rules compare favourably with existing ones

Mixed method for the product integral on the infinite interval

In this note, quadrature formula is constructed for product integral on the infinite interval I(f) = ∫ w(x)f(x)dx, where w(x) is a weight function and f(x) is a smooth decaying function for x > N (large enough) and piecewise discontinuous function of the first kind on the interval a ≤ x ≤ N. For the approximate method we have reduced infinite interval x [a, ∞) into the interval t[0,1] and used the mixed method: Cubic Newton’s divided difference formula on [0, t3) and Romberg method on [t3,1] with equal step size, ti = t0+ih,i=0, …,n, h=1/n, where t0 = 0,tn=1. Error term is obtained for mixed method on different classes of functions. Finally, numerical examples are presented to validate the method presented

Relative and Logarthmic of AI-Tememe Acceleration Methods for Improving the Values of Integrations Numerically of Second Kind

Abstract: The aims of this study are to introduce acceleration methods that called relative and algorithmic acceleration methods, which we generally call Al-Tememe's acceleration methods of the second kind discovered by (Ali Hassan Mohammed). It is very useful to improve the numerical results of continuous integrands in which the main error is of the 4th order, and related to accuracy, the number of used partial intervals and how fast to get results especially to accelerate the results got by Simpson's method. Also, it is possible to utilize it in improving the results of differential equations numerically of the main error of the forth order

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.

Triangular Acceleration Methods of Second Kind for Improving the Values of Integrals Numerically

Abstract: The aims of this study are to introduce acceleration methods that are called triangular acceleration methods, which come within the series of several acceleration methods that generally known as Al-Tememe's acceleration methods of the second kind which are discovered by (Ali Hassan Mohammed). These methods are useful in improving the results of determining numerical integrals of continuous integrands where the main error is of the forth order with respect to accuracy, partial intervals and the fasting of calculating the results specifically to accelerate results come out by Simpson's method. Also, it is possible to make use of it to improve the results of solving differential equations numerically of the main error of the forth orde