Double Integral Research Articles

Post-quantum Ostrowski type integral inequalities for functions of two variables

<abstract><p>In this study, we give the notions about some new post-quantum partial derivatives and then use these derivatives to prove an integral equality via post-quantum double integrals. We establish some new post-quantum Ostrowski type inequalities for differentiable coordinated functions using the newly established equality. We also show that the results presented in this paper are the extensions of some existing results.</p></abstract>

Infinite triple series by integrals

A large class of infinite triple series are explicitly evaluated by computing definite double integrals. Several remarkable formulae are presented in terms of ? and ln 2.

Approximate solution of nonlinear fuzzy Fredholm integral equations using bivariate Bernstein polynomials with error estimation

<abstract><p>This paper is concerned with obtaining approximate solutions of fuzzy Fredholm integral equations using Picard iteration method and bivariate Bernstein polynomials. We first present the way to approximate the value of the multiple integral of any fuzzy-valued function based on the two dimensional Bernstein polynomials. Then, it is used to construct the numerical iterative method for finding the approximate solutions of two dimensional fuzzy integral equations. Also, the error analysis and numerical stability of the method are established for such fuzzy integral equations considered here in terms of supplementary Lipschitz condition. Finally, some numerical examples are considered to demonstrate the accuracy and the convergence of the method.</p></abstract>

A Numerical Method for Computing Double Integrals with Variable Upper Limits

We propose and justify a numerical method for computing the double integral with variable upper limits that leads to the variableness of the region of integration. Imposition of simple variables as functions for upper limits provides the form of triangles of integration region and variable in the external limit of integral leads to a continuous set of similar triangles. A variable grid is overlaid on the integration region. We consider three cases of changes of the grid for the division of the integration region into elementary volumes. The first is only the size of the imposed grid changes with the change of variable of the external upper limit. The second case is the number of division elements changes with the change of the external upper limit variable. In the third case, the grid size and the number of division elements change after fixing their multiplication. In these cases, the formulas for computing double integrals are obtained based on the application of cubatures in the internal region of integration and performing triangulation division along the variable boundary. The error of the method is determined by expanding the double integral into the Taylor series using Barrow’s theorem. Test of efficiency and reliability of the obtained formulas of the numerical method for three cases of ways of the division of integration region is carried out on examples of the double integration of sufficiently simple functions. Analysis of the obtained results shows that the smallest absolute and relative errors are obtained in the case of an increase of the number of division elements changes when the increase of variable of the external upper limit and the grid size is fixed.

Generation of waves due to bottom disturbances in a viscous fluid

The generation of two-dimensional surface waves due to various types of bottom disturbances such as underwater explosions, earthquakes, or volcanic eruptions is investigated here. Assuming linear theory the present problem is formulated as an initial value problem for the wave potential function ϕ and Stokes stream function ψ. Viscosity is considered. The physical model is illustrated by a sketch. Fourier and Laplace transform techniques are applied in the mathematical analysis to obtain the form of the free surface in terms of a multiple infinite integral. This integral is evaluated asymptotically by the method of steepest descent. The asymptotic form of the free surface is depicted graphically in some figures for different values of the viscosity and different types of ground disturbances. Appropriate conclusions are made.

A College Teacher’s Introduction to Jupyter Notebooks

This introduction to Jupyter Notebooks (JNBs) is an outcome of a year of learning, creating, and classroom testing JNBs in a variety of settings: a sabbatical with an urban violence reduction organization; an after school distance learning program for low to middle income high school students in Chicago; a general education computer-lab based math course; and finally two upper division courses for applied math majors at a liberal arts college. The examples include line graphs, histograms, and a pie chart of violent crime data created for a community organization, a high school/general education lab on computer music, a visualization of volumes computable by triple integrals, and creation of a .gif animation of a numerical solution to a 2-Dimensional PDE. In addition to executing and displaying output of computer code, JNBs can import data from files and external websites, interpret equations written in LaTeX, display PowerPoints in pdf format, and show videos posted on other websites. JNBs can also save output data, figures, and animations to files in a variety of formats. At a general education level, JNBs can enrich student learning experience by helping students overcome math anxiety through inclusion of the arts. At the undergraduate math major level, JNBs can create interesting visualizations/animations and support service learning projects. The appendix explains how to get started using JNBs. The learning curve for creating the JNBs described in this article is not that steep, and the low-hanging fruit from using JNBs for a wide-range of teaching contexts and for community service learning projects is plentiful.

Minimality of balls in the small volume regime for a general Gamow-type functional

Abstract We consider functionals given by the sum of the perimeter and the double integral of some kernel g : ℝ N × ℝ N → ℝ + {g:\mathbb{R}^{N}\times\mathbb{R}^{N}\to\mathbb{R}^{+}} , multiplied by a “mass parameter” ε. We show that, whenever g is admissible, radial and decreasing, the unique minimizer of this functional among sets of given volume is the ball as soon as ε ≪ 1 {\varepsilon\ll 1} .

Calculation of centroid of high dimensional fuzzy number and application

In this paper, the conception of centroid of $n$-dimensional fuzzy number is introduced viaregarding its membership function as the density function on its support set, and some properties of it are obtained. Compared with the mean of the multi dimensional fuzzy number, the centroid takes into account the overall relationship between the edge membership functions of the membership function of the multi dimensional fuzzy number. Therefore, it can approximate (characterize) the fuzzy number more objectively and reasonably than using the mean of the multi dimensional fuzzy number. The most important work of this paper is that for two special kinds of multi dimensional fuzzy numbers (fuzzy $n$-cell numbers and fuzzy $n$-ellipsoid numbers), we respectively give calculation formulas, which can be used conveniently in application since the formulas are based on a definite integral of the level set functions of the multi dimensional fuzzy number on the unit interval $[0,1]$, rather than the multiple integral of the membership function of the multi dimensional fuzzy number itself on its support set. Then, by using the calculation formulas, we obtain another special property of the centroid for fuzzy $n$-cell number and fuzzy $n$-ellipsoid number. Finally, as an example of application, by using the centroid of multi dimensional fuzzy number, we define a fuzzy order on $n$-dimensional fuzzy number space, which can be used to rank uncertain or imprecise multichannel digital information.

Quadruple Integral Involving the Logarithm and Product of Bessel Functions Expressed in Terms of the Lerch Function

In this paper, we have derived and evaluated a quadruple integral whose kernel involves the logarithm and product of Bessel functions of the first kind. A new quadruple integral representation of Catalan’s G and Apéry’s ζ(3) constants are produced. Some special cases of the result in terms of fundamental constants are evaluated. All the results in this work are new.

Double Integral Involving Logarithmic and Quotient Function with Powers Expressed in terms of the Lerch Function

In this work the authors use their contour integral method to derive the double integral given by $\int_{0}^{\infty}\int_{0}^{\infty}\frac{x^{m-1} y^{m+\frac{q}{2}-1} \log ^k(a x y)}{\left(x^q+1\right)^2 \left(y^q+1\right)^2}dxdy$ in terms of the Lerch function. This integral formula is then used to derive closed solutions in terms of fundamental constants and special functions. There are some useful results relating double integrals of certain kinds of functions to ordinary integrals for which we know no general reference. Thus a table of integral pairs is given for interested readers. All the results in this work are new.

On the rectangular mesh and the decomposition of a Green’s-function-based quadruple integral into elementary integrals

Computational electromagnetic problems require evaluating the electric and magnetic fields of the physical object under investigation, divided into elementary cells with a mesh. The partial element equivalent circuit (PEEC) method has recently received attention from academic and industry communities because it provides a circuit representation of the electromagnetic problem. The surface formulation, known as S-PEEC, requires computing quadruple integrals for each mesh patch. Several techniques have been developed to simplify the computational complexity of quadruple integrals but limited to triangular meshes as used in well-known methods such as the Method of Moments (MoM). However, in the S-PEEC method, the mesh can be rectangular and orthogonal, and new approaches must be investigated to simplify the quadruple integrals. This work proposes a numerical approach that treats the singularity and reduces the computational complexity of one of the two quadruple integrals used in the S-PEEC method. The accuracy and computational time are tested for representative parallel and orthogonal meshes.

Weighted Cebysev Type Inequalities for Double Integrals and Application

The purpose of this article is to generalize Cebysev type inequalities for double integrals involving a weight function.By using an integral transform that is a weighted Montgomery identity, we obtained a generalized form of weighted Cebysev type inequalities in $L_m,, mgeq 1$ norm of differentiable functions. Also, we give some applications of the probability density function.

Representation of Weierstrass integral via Poisson integrals

In our research, we have presented a second-order linear partial differential equation in polar coordinates. Considering this differential equation on the unit disk, we have obtained a one-dimensional heat equation. It is well-known that the heat equation can be solved taking into account the boundary condition for the general solution on the unit circle. In our paper, the boundary-value problem is solved using the well-known method called the separation of variables. As a result, the general solution to the boundary-value problem is presented in terms of the Fourier series. Then the expressions for the Fourier coefficients are used to transform the Fourier series expansion for the general solution to the boundary-value problem into the so-called Weierstrass integral, which is represented via the so-called Weierstrass kernel. A representation of the Weierstrass kernel via the infinite geometric series is derived by a way allowing a complicated function to be parameterized via a simplified function. The derivation of the corresponding parametrization is based on two well-known integrals. As a result, a complicated function of the natural argument is represented in the form of a double integral that contains a simplified function of the same natural argument. So, the double-integral representation of the Weierstrass kernel has been derived. To obtain this result, the integral representation of the so-called Dirac delta function is taken into account. The expression found for the Weierstrass kernel is substituted into the expression for the Weierstrass integral. As a result, it was found that the Weierstrass integral can be considered a double-integral that contains the Poisson and conjugate Poisson integrals.

The Hermite-Hadamard inequalities for $p$-convex functions

In this paper, the Hermite-Hadamard inequality for $p-$convex function is provided. Some integral inequalities for them are also presented. Also, based on the integral and double integral of $p-$convex sets, the new functions are defined and under certain conditions, $p-$convexity of these functions are shown. Some inequalities for these functions are expressed.

Multiple Series Representations of N-fold Mellin-Barnes Integrals.

Mellin-Barnes (MB) integrals are well-known objects appearing in many branches of mathematics and physics, ranging from hypergeometric functions theory to quantum field theory, solid-state physics, asymptotic theory, etc. Although MB integrals have been studied for more than one century, until now there has been no systematic computational technique of the multiple series representations of N-fold MB integrals for N>2. Relying on a simple geometrical analysis based on conic hulls, we show here a solution to this important problem. Our method can be applied to resonant (i.e., logarithmic) and nonresonant cases and, depending on the form of the MB integrand, it gives rise to convergent series representations or diverging asymptotic ones. When convergent series are obtained, the method also allows, in general, the determination of a single "master series" for each series representation, which considerably simplifies convergence studies and/or numerical checks. We provide, along with this Letter, a Mathematica implementation of our technique with examples of applications. Among them, we present the first evaluation of the hexagon and double box conformal Feynman integrals with unit propagator powers.

Nonvariational and singular double phase problems for the Baouendi-Grushin operator

In this paper we introduce a new double phase Baouendi-Grushin type operator with variable coefficients. We give basic properties of the corresponding functions space and prove a compactness result. In the second part, using topological argument, we prove the existence of weak solutions of some nonvariational problems in which this new operator is present. The present paper extends and complements some of our previous contributions related to double phase anisotropic variational integrals.

Integral equation fast solver with truncated and degenerated kernel for computing flaw signals in eddy current non-destructive testing

In this article, the kernel independent H-matrix method is applied as the mathematical framework. The dense matrix generated by the selected integral equation, without low frequency breakdown problem for solving 3D arbitrary shaped eddy current nondestructive evaluation (NDE) forward problems, is compressed by the truncated and degenerated (TD) kernel function method. It results in a significant reduction in computational burden with nearly linear complexity. The kernel functions (Green's function) are degenerated by Lagrange polynomials which leads to two advantages: firstly, the double integrals are separated in two single integrals, and secondly, they are represented in a factorized form with good accuracy control. Meanwhile, due to the nature of the kernel function with exponential decay in the lossy medium, it is truncated to improve the overall performance. The TD kernel function-based boundary element method (BEM) fits well for solving eddy current NDE problems over the adaptive cross approximation based BEM, especially for detections of flaws or slots in planar objects. Several numerical predictions calculated by the proposed method are compared with those achieved by others such as the experiment, analytical and semi-analytical methods from benchmarks. The robustness and efficiency are demonstrated with excellent agreements and reduced complexity.

Representation of Weierstrass integral via Poisson integrals

In our research, we have presented a second-order linear partial differential equation in polar coordinates. Considering this differential equation on the unit disk, we have obtained a one-dimensional heat equation. It is well-known that the heat equation can be solved taking into account the boundary condition for the general solution on the unit circle. In our paper, the boundary-value problem is solved using the well-known method called the separation of variables. As a result, the general solution to the boundary-value problem is presented in terms of the Fourier series. Then the expressions for the Fourier coefficients are used to transform the Fourier series expansion for the general solution to the boundary-value problem into the so-called Weierstrass integral, which is represented via the so-called Weierstrass kernel. A representation of the Weierstrass kernel via the infinite geometric series is derived by a way allowing a complicated function to be parameterized via a simplified function. The derivation of the corresponding parametrization is based on two well-known integrals. As a result, a complicated function of the natural argument is represented in the form of a double integral that contains a simplified function of the same natural argument. So, the double-integral representation of the Weierstrass kernel has been derived. To obtain this result, the integral representation of the so-called Dirac delta function is taken into account. The expression found for the Weierstrass kernel is substituted into the expression for the Weierstrass integral. As a result, it was found that the Weierstrass integral can be considered a double-integral that contains the Poisson and conjugate Poisson integrals.

Optimal integration of highly oscillating functions in general form

The development of information technology contributes to the improvement of mathematical models of phenomena and processes in many scientific areas of the technical direction. In particular, modern methods of digital signal and image processing use algorithms with new information operators. Cubature formulas are constructed for the approximate calculation of integrals of highly oscillating functions of many variables for various types of data. The paper deals with the estimation of the error in the numerical integration of highly oscillating functions of a general form on the class of differentiable functions of three variables in the case when information about the functions is given to their traces on the corresponding planes. The results obtained make it possible to research the quality of cubature formulas for the approximate calculation of triple integrals of highly oscillating functions of a general form.

Integration method of non-elementary exponential functions using iterated Fubinni integrals

The present work presents a new method of integration of non-elementary exponential functions where Fubinni's iterated integrals were used. In this research, some approximations were used in order to generalize the results obtained through mathematical series, in addition to integration methods and double integrals. In addition to the integration methods, the Taylor series was used, where the value found and compatible with the values ​​of the power series that are used to calculate the value of the exponential function demonstrated in the work was verified. In addition to the methods described, a comparison of the values ​​obtained by the series and the values ​​described in the method was improvised, where it was noticed that the higher the value of the variable, the closer the results show a stability for the variable greater than the value 4, described in table 01. The conclusions point to a great improvement, mainly for solving elliptic differential equations and statistical functions.