Quadrature Formulas Research Articles

Clinical utility of the maximum area of detrusor tracing measurement in the urodynamic studies in pediatric population diagnosed with neuropathic bladder: A pilot study

IntroductionNeuropathic bladder (NB) is a lower urinary tract dysfunction (LUTD) that may lead to kidney failure. Urodynamic study is the gold standard for its diagnosis and follow-up. A classic prognostic parameter to predict the risk of upper urinary tract damage (UUTD) is the detrusor leak point pressure (DLPP) ≥ 40 cmH2O, although this factor alone may lack prognostic value. The area under the curve (AUC) of the detrusor pressure tracing has been recently described as a prognostic factor. AimTo analyze the clinical utility of the AUC of the detrusor pressure tracing in the pediatric population (<15 years). MethodsRetrospective study of urodynamic recordings in pediatric population with NB from 2011 to 2020. The following variables were evaluated: detrusor leak point pressure ≥ 40 cmH2O, voiding dysfunction, bladder compliance and overactivity. The sample is classified according to clinical-radiological findings at the time of the UD. AUC is calculated using Newton-Cotes rules (trapezium, Simpson’s rule) and its indexes are calculated according to bladder capacity (TI: Trapezium Index, SI: Simpson Index). Statistical significance: P < .05. ResultsFifty-five recordings are analyzed. Unfavorable clinical condition was identified in 41.8% (n = 23) patients. Voiding dysfunction, low compliance and both indexes were associated with an unfavorable clinical condition (P < .05). ROC analysis showed the following AUC for the new indexes: TI (0.736, P = 0.0006), SI (0.755, P = .0001) with a cut-off value of 10,69 and 8 cmH2O·s/cc, respectively. We did not find differences in the diagnostic performance between them (P > .05). ConclusionsThe analyzed indexes are useful in the diagnosis of patients with NB and unfavorable clinical condition.

Research Spotlights

Research Spotlights

New inequalities for strongly exponentially generalized functions with applications

The aim of this paper is to introduce a new class of functions called strongly exponentially generalized (m, ν1, ν2, g1, g2). Some new integral inequalities of trapezium-type for strongly exponentially generalized (m, ν1, ν2, g1, g2) functions with modulus c via Riemann-Liouville fractional integral are established. Also, some new estimates with respect to trapezium-type integral inequalities for strongly exponentially generalized (m, ν1, ν2, g1, g2) functions with modulus c via general fractional integrals are obtained. We show that the strongly exponentially generalized (m, ν1, ν2, g1, g2) functions with modulus c includes several other classes of functions. At the end, some new error estimates for trapezoidal quadrature formula are provided as well. This results may stimulate further research in different areas of pure and applied sciences.

Modified HAM for solving linear system of Fredholm-Volterra Integral Equations

This paper considers systems of linear Fredholm-Volterra integral equations using a modified homotopy analysis method (MHAM) and the Gauss-Legendre quadrature formula (GLQF) to find approximate solutions. Standard homotopy analysis method (HAM), MHAM, and optimal homotopy asymptotic method (OHAM) are compared for the same number of iterations. It is noted from the chosen examples that MHAM with GLQF is comparable with standard HAM and OHAM. In all cases, MHAM with GLQF approaches exact solutions, where residual rapidly converges to zero when the number of iterations and quadrature nodes increases. The HAM developed in this paper is better than the HAM developed by Shidfar &amp; Molabahrami in "Solving a system of integral equations by an analytic method".

Polar-Analytic Functions: Old and New Results, Applications

Here we review the notion of polar analyticity introduced in a previous paper and successfully applied in Mellin analysis and for quadrature formulae over the positive real axis. This approach provides a simple way of describing functions which are analytic on a part of the Riemann surface of the logarithm. New results are also obtained.

On numerical solution of Fredholm and Hammerstein integral equations via Nyström method and Gaussian quadrature rules for splines

Nyström method is a standard numerical technique to solve Fredholm integral equations of the second kind where the integration of the kernel is approximated using a quadrature formula. Traditionally, the quadrature rule used is the classical polynomial Gauss quadrature. Motivated by the observation that a given function can be better approximated by a spline function of a lower degree than a single polynomial piece of a higher degree, in this work, we investigate the use of Gaussian rules for splines in the Nyström method. We show that, for continuous kernels, the approximate solution of linear Fredholm integral equations computed using spline Gaussian quadrature rules converges to the exact solution for m→∞, m being the number of quadrature points. Our numerical results also show that, when fixing the same number of quadrature points, the approximation is more accurate using spline Gaussian rules than using the classical polynomial Gauss rules. We also investigate the non-linear case, considering Hammerstein integral equations, and present some numerical tests.

Zeros of quasi-paraorthogonal polynomials and positive quadrature

In this paper we illustrate that paraorthogonality on the unit circle T is the counterpart to orthogonality on R when we are interested in the spectral properties. We characterize quasi-paraorthogonal polynomials on the unit circle as the analogues of the quasi-orthogonal polynomials on R. We analyse the possibilities of preselecting some of its zeros, in order to build positive quadrature formulas with prefixed nodes and maximal domain of validity. These quadrature formulas on the unit circle are illustrated numerically.

New fractional identities, associated novel fractional inequalities with applications to means and error estimations for quadrature formulas

In this paper, the authors derive some new generalizations of fractional trapezium-like inequalities using the class of harmonic convex functions. Moreover, three new fractional integral identities are given, and on using them as auxiliary results some interesting integral inequalities are found. Finally, in order to show the efficiency of our main results, some applications to special means for different positive real numbers and error estimations for quadrature formulas are obtained.

Symplectic P-stable additive Runge—Kutta methods

&lt;p style='text-indent:20px;'&gt;Classical symplectic partitioned Runge–Kutta methods can be obtained from a variational formulation where all the terms in the discrete Lagrangian are treated with the same quadrature formula. We construct a family of symplectic methods allowing the use of different quadrature formulas (primary and secondary) for different terms of the Lagrangian. In particular, we study a family of methods using Lobatto quadrature (with corresponding Lobatto IIIA-B symplectic pair) as a primary method and Gauss–Legendre quadrature as a secondary method. The methods have the same implicitness as the underlying Lobatto IIIA-B pair, and, in addition, they are &lt;i&gt;P-stable&lt;/i&gt;, therefore suitable for application to highly oscillatory problems.&lt;/p&gt;

Generalized Averaged Gaussian Formulas for Certain Weight Functions

In this paper we analyze the generalized averaged Gaussian quadrature formulas and the simplest truncated variant for one of them for some weight functions on the interval [0, 1] considered by Milovanović in [10]. We shall investigate internality of these formulas for the equivalents of the Jacobi polynomials on this interval and, in some special cases, show the existence of the Gauss-Kronrod quadrature formula. We also include some examples showing the corresponding error estimates for some non-classical orthogonal polynomials.

Regularization of the procedure for inverting the Laplace transform using quadrature formulas

The problem of inversion of the integral Laplace transform, which belongs to the class of ill-posed problems, is considered. Integral equations are reduced to ill-conditioned systems of linear algebraic equations (SLAE), in which the unknowns are either the expansion coefficients in a series in terms of shifted Legendre polynomials, or the approximate values of the desired original at a number of points. The first step of reduction to SLAE is to apply quadrature formulas that provide the minimum values of the condition number of SLAE. Regularization methods are used to obtain a reliable solution of the system. A common strategy is to use the Tikhonov stabilizer or its modifications. A variant of the regularization method for systems with oscillatory-type matrices is presented, which significantly reduces the conditionality of the problem in comparison with the classical Tikhonov scheme. A method is given for actually constructing special quadratures leading to problems with oscillation matrices.

Error bounds for Gaussian quadrature formulae with Legendre weight function for analytic integrands

In this paper we are concerned with a method for the numerical evaluation of the error terms in Gaussian quadrature formulae with the Legendre weight function. Inspired by the work of H. Wang and L. Zhang [J. Sci. Comput., 75 (2018), pp. 457–477] and applying the results of S. Notaris [Math. Comp., 75 (2006), pp. 1217–1231], we determine an explicit formula for the kernel. This explicit expression is used for finding the points on ellipses where the maximum of the modulus of the kernel is attained. Effective error bounds for this quadrature formula for analytic integrands are derived.

Efficient time second-order SCQ formula combined with a mixed element method for a nonlinear time fractional wave model

&lt;abstract&gt;&lt;p&gt;In this article, a kind of nonlinear wave model with the Caputo fractional derivative is solved by an efficient algorithm, which is formulated by combining a time second-order shifted convolution quadrature (SCQ) formula in time and a mixed element method in space. The stability of numerical scheme is derived, and an optimal error result for unknown functions which include an original function and two auxiliary functions are proven. Further, the numerical tests are conducted to confirm the theoretical results.&lt;/p&gt;&lt;/abstract&gt;

On one method of constructing quadrature formulas for computing hypersingular integrals

On one method of constructing quadrature formulas for computing hypersingular integrals

The Ostrowski inequality for $ s $-convex functions in the third sense

&lt;abstract&gt;&lt;p&gt;In this paper, the Ostrowski inequality for $ s $-convex functions in the third sense is studied. By applying Hölder and power mean integral inequalities, the Ostrowski inequality is obtained for the functions, the absolute values of the powers of whose derivatives are $ s $-convex in the third sense. In addition, by means of these inequalities, an error estimate for a quadrature formula via Riemann sums and some relations involving means are given as applications.&lt;/p&gt;&lt;/abstract&gt;

Long time behavior of higher-order delay differential equation with vanishing proportional delay and its convergence analysis using spectral method

&lt;abstract&gt; &lt;p&gt;Delay differential equations (DDEs) are used to model some realistic systems as they provide some information about the past state of the systems in addition to the current state. These DDEs are used to analyze the long-time behavior of the system at both present and past state of such systems. Due to the oscillatory nature of DDEs their explicit solution is not possible and therefore one need to use some numerical approaches. In this article, we developed a higher-order numerical scheme for the approximate solution of higher-order functional differential equations of pantograph type with vanishing proportional delays. Some linear and functional transformations are used to change the given interval [0, T] into standard interval [-1, 1] in order to fully use the properties of orthogonal polynomials. It is assumed that the solution of the equation is smooth on the entire domain of interval of integration. The proposed scheme is employed to the equivalent integrated form of the given equation. A Legendre spectral collocation method relative to Gauss-Legendre quadrature formula is used to evaluate the integral term efficiently. A detail theoretical convergence analysis in L&lt;sub&gt;∞&lt;/sub&gt; norm is provided. Several numerical experiments were performed to confirm the theoretical results.&lt;/p&gt; &lt;/abstract&gt;

Метод полиномиальной сплайн-коллокации для решения слабо регулярных интегральных уравнений Вольтерра I рода

The polynomial spline collocation method is proposed for solution of Volterra integral equations of the first kind with special piecewise continuous kernels. The Gausstype quadrature formula is used to approximate integrals during the discretization of the proposed projection method. The estimate of accuracy of approximate solution is obtained. Stochastic arithmetics is also used based on the Controle et Estimation Stochastique des Arrondis de Calculs (CESTAC) method and the Control of Accuracy and Debugging for Numerical Applications (CADNA) library. Applying this approach it is possible to find optimal parameters of the projective method. The numerical examples are included to illustrate the efficiency of proposed novel collocation method.

Non-Newtonian blood flow model with the effect of different geometry of stenosis

The objective of this paper is to present a non-Newtonian blood flow model with the effect of different geometry of stenosis on various flow quantities. The Power-law model is considered to explore the non-Newtonian property of blood. Two-point Gauss quadrature formula is applied to obtain the numerical expressions of dimensionless flow resistance, skin-friction and flow rate. The variation of dimensionless flow resistance, skin-friction and flow rate with degree of stenosis, axial distance and power-law index is shown graphically. Moreover, the power-law index is adjusted to explore the non-Newtonian characteristics of blood. The importance of the present work has been carried out by comparing the results with other theories both numerically and graphically. It has been found that resistance to flow becomes maximum with total blockage of artery for different shape of stenosis.

Fitted operator method over gaussian quadrature formula for parabolic singularly perturbed convection-diffusion problem

Fitted operator method over gaussian quadrature formula for parabolic singularly perturbed convection-diffusion problem

"Optimal quadrature formulas for approximate solution of the rst kind singular integral equation with Cauchy kernel"

"In the present paper in $L_2^{(m)}(-1,1)$ space the optimal quadrature formulas with derivatives are constructed for approximate solution of a singular integral equation of the first kind with Cauchy kernel. Approximate solution of the singular integral equation is obtained applying the optimal quadrature formulas. Explicit forms of coefficients for the of optimal quadrature formulas are obtained. Some numerical results are presented."