Parry Romberg syndrome(PRS) is a rare acquired poorly understood neurocutaneous syndrome of unknown etiology characterized by slow progressive atrophic changes commonly affecting one half of the face. The exact incidence and etiology towards the syndrome remains unclear. Apart from the multifactorial etiology proposed, the possible primary cause is mainly attributed to the cerebral disturbance ...

We consider a Gaussian type quadrature rule for some classes of integrands involving highly oscillatory functions of the form f (x) = f 1 (x) sin ζ x + f 2 (x) cos ζ x, where f 1 (x) and f 2 (x) are smooth, ζ ∈ R. We find weights σ ν and nodes x ν , ν = 1, 2,. .. , n, in a quadrature formula of the form 1 −1 f (x) dx ≈ n ν=1 σ ν f (x ν) such that it is exact for all polynomials f 1 (x) and f 2 ...

A straightforward 3-point quadrature formula of closed type is derived that improves on Simpson’s rule. Just using the additional information of the integrand’s derivative at the two endpoints we show the error is sixth order in grid spacing. Various error bounds for the quadrature formula are obtained to quantify more precisely the errors. Applications in numerical integration are given. With ...

Smolyak’s sparse grid construction is commonly used in a setting involving quadrature of a function of a multidimensional argument over a product region. However, the method can be applied in a straightforward way to the interpolation problem as well. In this discussion, we outline a procedure that begins with a family of interpolants defined on a family of nested tensor product grids, and demo...

This paper presents a Gaussian quadrature method for the evaluation of the triple integral ( , , ) T I f x y z d xd yd z = ∫∫∫ , where ) , , ( z y x f is an analytic function in , , x y z and T refers to the standard tetrahedral region:{( , , ) 0 , , 1, 1} x y z x y z x y z ≤ ≤ + + ≤ in three space( , , ). x y z Mathematical transformation from ( , , ) x y z space to ( , , ) u v w space maps th...

Parry Romberg syndrome (PRS), also referred to as progressive hemifacial atrophy, facial hemiatrophy, or idiopathic was first described by C and M Romberg.1 It manifests in the two decades morphologically normal-born individuals. commonly affects one more dermatomes trigeminal nerve territory. has an early onset of ophthalmic neurological involvement a variable maxillo-facial cardiac involvemen...

We discuss the theoretical convergence and numerical evaluation of simultaneous interpolation quadrature formulas which are exact for rational functions. Basically, the problem consists in integrating a single function with respect to different measures by using a common set of quadrature nodes. Given a multi-index n, the nodes of the integration rule are the zeros of the multiorthogonal Hermit...

In a previous paper, McLean and Thomée (to appear), we studied three numerical methods for the discretization in time of a fractional order evolution equation, in a Banach space framework. Each of the methods applied a quadrature rule to a contour integral representation of the solution in the complex plane, where for each quadrature point an elliptic boundary value problem had to be solved to ...

The importance of singular and hypersingular integral transforms, coming from their many applications, justifies some interest in their numerical approximation. The literature about the numerical evaluation of such integrals on bounded intervals is wide and quite satisfactory; instead only few papers deal with the numerical evaluation of such integral transforms on half-infinite intervals or on...