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This manuscript brings the derivation of influence functions for a three-dimensional full-space under bilinearly-distributed time-harmonic loads. The differential equations describing the medium are decomposed in terms of uncoupled vector fields. A double Fourier transform allows the system of equations to be solved algebraically in the transformed space, where the bilinear-loading boundary conditions are imposed. The resulting displacement and stress solutions are presented in terms of double improper integrals to be evaluated numerically. The manuscript brings selected results from the evaluation of these solutions. These influence functions can be used in boundary element models of elastodynamic problems to yield computationally-efficient solutions and improved representation of sharply-varying contact traction fields.

This study discusses the use of numerical integration in evaluating the improper integrals appearing as inverse integral transforms of non-analytic functions. These transforms appear while studying the response of various sources in an elastic medium through integral transform method. In these studies, the inverse Fourier transforms are solved numerically without bothering about the singularities and branch points in the corresponding integrands. References on numerical integration cited in relevant papers do not support such an evaluation but suggest contrary. Approximation of inverse Laplace transform integral into a series is used without following the essential restrictions and assumptions. Volume of the published papers using these dubious procedures has reached to an alarming level. The discussion presented aims to draw the attention of researchers as well as journals so as to stop this menace at the earliest possible.Keywords: Inverse Fourier transforms, inverse Laplace transforms, Romberg integration, improper integral, elastic waves

In this paper Green?s function for the semi-infinite strip (which is the two-dimensional Green?s function for a groove of infinite depth and length) is determined in the form of an improper integral, as opposed to the standard summation form. The integral itself, although rather complex, is found in a closed form. By using the derived Green?s function simple formulas are obtained for a single and two-wire line configurations inside the groove.

Calculation of hadronization, decay or scattering processes at non-zero temperatures and densities within the Nambu-Jona-Lasinio-like models requires some techniques for computation of Feynmann diagrams. Decomposition of Feynman diagrams at the one loop level leads to the appearance of elementary integrals with one, two, three, and four fermion lines. For example, evaluation of the $\pi\pi$ scattering amplitude requires calculating a box diagram with four fermion lines. In this work, the real and imaginary parts of the box integral at the one loop level are provided in the form suitable for numerical evaluation. The obtained expressions are applicable to any value of temperature, particle mass, and chemical potential. We pay special attention to the conditions for the existence of the appearing improper integrals and correct the results \cite{Klevansky} for the three fermion lines. As a result, we have obtained constraints on possible values of particle momenta. Among the expressions for the box integral, the general formulas for the integral with an arbitrary number of lines are derived for the case with zero or collinear fermion momenta., Comment: 22 pages, no figures. The second version of the paper is essentialy distinct from the first one. The puzzle with different values of the three line integral for different coordinate systems is solved

We study the eikonal approximation of the total cross section for the scattering of two unpolarized particles and obtain an approximate formula in the case where the eikonal function χ(b) is moderately small, |χ(b)| ≲ 0.1. We show that the total cross section is given by a series of improper integrals of the Born amplitude AB. The advantage of this representation compared with standard eikonal formulas is that these integrals contain no rapidly oscillating Bessel functions. We prove two theorems that allow relating the large-b asymptotic behavior of χ(b) to analytic properties of the Born amplitude and give several examples of applying these theorems. To check the effectiveness of the main formula, we use it to calculate the total cross section numerically for a selection of specific expressions for AB, choosing only Born amplitudes that result in moderately small eikonal functions and lead to the correct asymptotic behavior of χ(b). The numerical calculations show that if only the first three nonzero terms in it are taken into account, this formula approximates the total cross section with a relative error of O(10−5).

A series of operating (Laplace) non-standard images, the originals of which are absent in well-known reference books on operational calculus, are considered. By reducing one of the basic images to the Riemann-Mellin contour integral for the modified Bessel functions and analyzing the corresponding inversion formula using the approaches of the complex variable function theory, an analytical form of the original original is found, which is abrupt in nature with a break point. It is shown that analytical solutions of the corresponding mathematical models using the found originals have a wave character, which is expressed by the presence of the Heaviside step function in the solutions. The latter means that at any time there is a region of physical disturbance to the point of discontinuity and an unperturbed area after the point of discontinuity. The images studied are included in the operational solutions of mathematical models in many areas of applied mathematics. physics, thermomechanics, thermal physics, in particular in the theory of thermal shock of viscoelastic bodies, in the study of the thermal reaction of solids based on the classical Maxwell-Cattaneo-Lykov-Vernott phenomenology, taking into account the final rate of heat propagation. These models are needed to study the thermal reaction of relatively new consolidated structurally sensitive polymeric materials in structures exposed to high-intensity external influences. The analytical relations obtained for the originals and the original improper integrals resulting from them, containing combinations of Bessel functions, can be used in the general methodology of constructing and applying various mathematical models in a wide range of external influences on materials in many fields of science and technology.

PurposeA nonclassical method, usually called memory-free approach, has shown promising potential to release arithmetic complexity and meets high memory-storage requirements in solving fractional differential equations. Though many successful applications indicate the validity and effectiveness of memory-free methods, it has been much less understood in the rigorous theoretical basis. This study aims to focus on the theoretical basis of the memory-free Yuan–Agrawal (YA) method [Journal of Vibration and Acoustics 124 (2002), pp. 321-324].Design/methodology/approachMathematically, the YA method is based on the validity of two fundamental procedures. The first is to reverse the integration order of an improper quadrature deduced from the Caputo-type fractional derivative. And, the second concerns the passage to the limit under the integral sign of the improper quadrature.FindingsThough it suffices to verify the integration order reversibility, the uniform convergence of the improper integral is proved to be false. Alternatively, this paper proves that the integration order can still be reversed, as the target solution can be expanded as Taylor series on [0, ∞). Once the integration order is reversed, the paper presents a sufficient condition for the passage to the limit under the integral sign such that the target solution is continuous on [0, ∞). Both positive and counter examples are presented to illustrate and validate the theoretical analysis results.Originality/valueThis study presents some useful results for the real performance for the YA and some similar memory-free approaches. In addition, it opens a theoretical question on sufficient and necessary conditions, if any, for the validity of memory-free approaches.

An integral representation of solutions of the wave equation obtained earlier is studied. The integrand contains weighted localized solutions of the wave equation that depend on parameters, which are variables of integration. Dependent on parameters, a family of localized solutions is constructed from one solution by means of transformations of shift, scaling, and the Lorentz transform. Sufficient conditions are derived, which ensure the pointwise convergence of the obtained improper integral in the space of parameters. The convergence of this integral in ℒ2 norm is proved as well. Bibliography: 22 titles.

In a previous paper, the authors developed new rules for computing improper integrals which allow computer algebra systems (Cas) to deal with a wider range of improper integrals. The theory used in order to develop such rules where Laplace and Fourier transforms and the residue theorem. In this paper, we describe new rules for computing symbolic improper integrals using extensions of the residue theorem and analyze how some of the most important Cas could improve their improper integral computations using these rules. To achieve this goal, different tests are developed. The Cas considered have been evaluated using these tests. The obtained results show that all Cas involved, considering the new developed rules, could improve their capabilities for computing improper integrals. The results of the evaluations of the Cas are described providing a sorted list of the Cas depending on their scores.

Summary Calculation of electromagnetic fields in layered models of the medium leads to the need to calculate the direct and inverse Fourier-Bessel (Hankel) transformations. The calculation of such improper integrals is complicated by the oscillating nature of the integrand. As a rule, it is the product of a smooth function of bounded variation by a Bessel function of the first kind. This circumstance allows the first function to be approximated in such a way that the integral of its product by the Bessel function is calculated analytically. We have analyzed two calculation algorithms. One of them is based on the expansion in terms of eigenfunctions of the Hankel integral operator, the second is based on approximation by a linear combination of exponentials. Both approaches make it possible to present the results of calculating integrals in an analytical form that allows their further mathematical analysis.

The method of analytical preconditioning combines the discretization and the analytical regularization of a singular integral equation in a single step. In a recent paper by the author, such a method has been applied to a spectral domain integral equation formulation devised to analyze the propagation in polygonal cross-section microstrip lines, which are widely used as high-speed interconnects in monolithic microwave and millimeter waves integrated circuits. By choosing analytically Fourier transformable expansion functions reconstructing the behavior of the fields on the wedges, fast convergence is achieved, and the convolution integrals are expressed in closed form. However, the coefficient matrix elements are one-dimensional improper integrals of oscillating and, in the worst cases, slowly decaying functions. In this paper, a novel technique for the efficient evaluation of such kind of integrals is proposed. By means of a procedure based on Cauchy integral theorem, the general coefficient matrix element is written as a linear combination of fast converging integrals. As shown in the numerical results section, the proposed technique always outperforms the analytical asymptotic acceleration technique, especially when highly accurate solutions are required.

Two quadrature-based algorithms for computing the matrix fractional power $A^\alpha$ are presented in this paper. These algorithms are based on the double exponential (DE) formula, which is well-known for its effectiveness in computing improper integrals as well as in treating nearly arbitrary endpoint singularities. The DE formula transforms a given integral into another integral that is suited for the trapezoidal rule; in this process, the integral interval is transformed to the infinite interval. Therefore, it is necessary to truncate the infinite interval into an appropriate finite interval. In this paper, a truncation method, which is based on a truncation error analysis specialized to the computation of $A^\alpha$, is proposed. Then, two algorithms are presented -- one computes $A^\alpha$ with a fixed number of abscissas, and the other computes $A^\alpha$ adaptively. Subsequently, the convergence rate of the DE formula for Hermitian positive definite matrices is analyzed. The convergence rate analysis shows that the DE formula converges faster than the Gaussian quadrature when $A$ is ill-conditioned and $\alpha$ is a non-unit fraction. Numerical results show that our algorithms achieved the required accuracy and were faster than other algorithms in several situations., Comment: The title of the manuscript was changed. The former title is "Computing the matrix fractional power based on the double exponential formula"

Formulation of the classic Taylor series as an orthogonal concept based on identifying the expansions coefficients as differential transformation applied to a unique function; definition of operational orthogonality by analogy with the Hilbert space and identification of the nth derivative at a point based on the improper integral on the positive semiaxis ; deduction of inversion integrals for Laplace transforms for analytical functions

In a recent paper, I have shown that the electric field at the surface of a charged conducting sphere evaluates to half the field discontinuity across the surface of the sphere. This exact result has been found by solving an improper integral, which has been criticized by Assad in a very recent paper in this journal [6]. In this note, I present a simpler approach that yields a proper integral for that field. This integral also evaluates to exactly half the field discontinuity, contrarily to Assad's objections.

Accurate algorithms are proposed for approximation of integrals involving highly oscillatory Bessel function of the first kind over finite and infinite domains. Accordingly, Bessel oscillatory integrals having high oscillatory behavior are transformed into oscillatory integrals with Fourier kernel by using complex line integration technique. The transformed integrals contain an inner non-oscillatory improper integral and an outer highly oscillatory integral. A modified meshfree collocation method with Levin approach is considered to evaluate the transformed oscillatory type integrals numerically. The inner improper complex integrals are evaluated by either Gauss–Laguerre or multi-resolution quadrature. Inherited singularity of the meshfree collocation method at x = 0 is treated by a splitting technique. Error estimates of the proposed algorithms are derived theoretically in the inverse powers of ω and verified numerically.

Due to the merit of transforming fractional differential equations into ordinary differential equations, the Yuan and Agrawal method has gained a lot of research interests over the past decade. In this paper, this method is improved with major emphasis on enhancing the convergence rate. The key procedure is to transform fractional derivative into an improper integral, which is integrated by Gauss–Laguerre quadrature rule. However, the integration converges slowly due to the singularity and slow decay of the integrand. To solve these problems, we reproduce the integrand to circumvent the singularity and slow decay simultaneously. With the reproduced integrand, the convergence rate is estimated to be no slower than $$ \, O(n^{ - 2} ) $$ with $$ n $$ as the number of quadrature nodes. In addition, we utilize a generalized Gauss–Laguerre rule to further improve the accuracy. Numerical examples are presented to validate the rapid convergence rate of the improved method, without causing additional computational burden compared to the original approach.

In the paper we present closed and unified expressions for a sequence of improper integrals in terms of the beta function and the Wallis ratio. Hereafter, we derive integral representations for the Catalan numbers originating from combinatorics

[EN] This paper deals with the construction of mean square analytic-numerical solution of parabolic partial differential problems where both initial condition and coefficients are stochastic processes. By using a random Fourier transform, an infinite integral form of the solution stochastic process is firstly obtained. Afterwards, explicit expressions for the expectation and standard deviation of the solution are obtained. Since these expressions depend upon random improper integrals, which are not computable in an exact manner, random Gauss-Hermite quadrature formulae are introduced throughout an illustrative example., This work has been partially supported by the Spanish Ministerio de Economia y Competitividad grant MTM2013-41765-P.

An open type mixed quadrature rule is constructed blending the anti-Gauss 3-point rule with Steffensen's 4-point rule. The analytical convergence of the mixed rule is studied. An adaptive integration scheme is designed based on the mixed quadrature rule. A comparative study of the mixed quadrature rule and the Gauss‒Laguerre quadrature rule is given by evaluating several improper integrals of the form $\int\limits_{0}^{\infty}e^{-x}f(x)dx$. The advantage of implementing mixed quadrature rule in developing an efficient adaptive integration scheme is shown by evaluating some improper integrals.

The theory of integration to mathematical analysis is so important that many mathematicians continue to develop new theory to enlarge the class of integrable functions and simplify the Lebesgue theory integration. In this paper, by slight modifying the definition of the Henstock integral which was introduced by Jaroslav Kurzweil and Ralph Henstock, we present a new definition of integral on fractal sets. Furthermore, its integrability has been discussed, and the relationship between differentiation and integral is also established. As an example, the integral of Cantor function on Cantor set is calculated.

In this paper, we establish some new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two independent variables, and we present the applications to research the boundedness of solutions to retarded nonlinear Volterra-Fredholm type integral equations.

La modelación matemática es la actividad que consiste en representar, manipular y comunicar objetos del mundo real con contenidos matemáticos y que permitan la simulación de procesos complejos, generen hipótesis y sugieran experimentos o métodos de validación. La modelación matemática se presenta como estrategia didáctica que permite simular e interpretar diferentes problemas y situaciones de la vida real o académica, poniendo en evidencia diferentes condiciones de aplicación de los contenidos de los cursos de matemática universitaria. Específicamente en Cálculo II, la estrategia busca solventar la necesidad de mostrar y manipular aplicaciones de los contenidos de integrales impropias, polinomios de Taylor, coordenadas polares y secciones cónicas en problemas con ejemplos concretos en las áreas de ciencias biológicas e ingeniería. Dicha implementación respondió a un paradigma que busca la innovación y la mejora continua del proceso enseñanza y aprendizaje, lo cual es pertinente debido a la existencia de una plataforma de incorporación de las TIC (Tecnologías de la Información y la Comunicación) en el curso de Cálculo II y que favoreció el uso de software especializado para la creación y manipulación de los modelos matemáticos.

We suggest a simple method for reducing problems with an integral condition for evolution equations to a Volterra integral equation of the first kind. For Volterra equations of the convolution type, we indicate necessary and sufficient solvability conditions for the case in which the right-hand side lies in some classes of functions of finite smoothness. We use these conditions to construct examples of nonexistence of a local solution for the heat equation with an integral condition.

This technical note is concerned with the problem of stability analysis for time-delay systems. A new series of integral inequalities to bound a single integral term is presented by introducing some free matrices, which produces tighter bounds than some existing ones. Similarly, based on orthogonal polynomials defined in integral inner spaces, new series of multiple integral inequalities are presented as well, which include the existing double ones. To show the effectiveness of the proposed inequalities, their applications to stability analysis of systems with discrete and distributed delays are provided with numerical examples.

By some new analysis techniques, we generalize the results presented by Pachpatte in [Integral and Finite Difference Inequalities and Applications, volume 205, Elsevier, 2006] and and by S. D. Kendre in [Some nonlinear integral inequalities for Volterra–Fredholm integral equations, Adv. Inequal. App, 2014:Article 21, 2014.] to nonlinear retarded inequalities, and investigate also some new forms. Some applications are also presented in order to illustrate the usefulness of some of our results.

In the setting of a Dedekind complete Riesz space on which a conditional expectation is defined, we study the Ito integral. This integral obtains the integration of a stochastic process of a continuous parameter relative to a martingale. A considerable part of the paper is devoted to aspects of integration of a vector-valued function with respect to a vector measure. Here we use the Dobrakov integral that is, in our case, a generalization of the well known Bartle integral. We discuss natural and predictable processes to prove a general version of the Doob–Meyer decomposition of a submartingale. This decomposition provides an essential tool used in the definition of the stochastic integral. We derive the properties of the stochastic integral that are useful in developing the theory of stochastic processes in Riesz spaces.

In this paper we consider a master integral in four arbitrary parameters. The integrand involves the logarithmic function and the Gauss hypergeometric function, which in certain special cases the integral reduces to identities involving zeta functions. A relationship will also be created between the integral and Euler sums of arbitrary order and arbitrary argument. Many interesting new specific examples will be highlighted.

We consider linear functional equations of the third kind in L2 with arbitrary measurable coefficients and unbounded integral operators with kernels satisfying broad conditions. We propose methods for reducing these equations by linear continuous invertible transformations either to equivalent integral equations of the first kind with nuclear operators or to equivalent integral equations of the second kind with quasidegenerate Carleman kernels. To the integral equations obtained after the reduction, one can apply various exact and approximate methods of solution; in particular, the two approximate methods developed in this article.

In this paper, our main aim is to establish some new fractional integral inequalities involving Hadamard-type k-fractional integral operators recently given by Mubeen et al. Furthermore, the paper discusses some of their relevance with known results. Copyright © 2016 John Wiley & Sons, Ltd.

A parallelized numerical contour integral based method is proposed for counting interior eigenvalues in a given region on the complex plane. The proposed method is derived from descriptor system of power system linearized model and complex analysis theory, based on which the computation of eigenvalue number is converted to a set of matrix trace problems. The contour integral results are able to be utilized to detect missing target eigenvalues in partial eigenvalue methods and provide an approximate eigenvalue distribution along the integral curve. Efficient evaluation of integral function is implemented by exploiting the sparsity of descriptor systems. An adaptive integral point collocation strategy is proposed for numerically evaluating contour integral with moderate number of discretized points. As the computation of integral function is decoupled at each integral point, the proposed method features well parallel computing capability.

A new class of integral that corresponds to a nonadditive measure.The new integral is defined using an interaction operator.Several mathematical properties of this integral are demonstrated.A mathematical model using this integral is proposed and constructed.The mathematical model is applied to a real dataset. We propose an integral with respect to a nonadditive monotone measure. This integral is a generalization of the Lebesgue integral and also the Choquet integral. It has appropriate properties, and can be extensively applied to real data analysis.

The paper deals with a new object of study --- hyperbolic Hurwitz zeta function, which is given in the right \(\alpha\)-semiplane \( \alpha = \sigma + it \), \( \sigma> 1 \) by the equality $$ \zeta_H(\alpha; d, b) = \sum_{m \in \mathbb Z} \left(\, \overline{dm + b} \, \right)^{-\alpha}, $$ where \( d \neq0 \) and \( b \) --- any real number. Hyperbolic Hurwitz zeta function \( \zeta_H (\alpha; d, b) \), when \( \left\| \frac {b} {d} \right\|> 0 \) coincides with the hyperbolic zeta function of shifted one-dimensional lattice \( \zeta_H (\Lambda (d, b) | \alpha) \). The importance of this class of one-dimensional lattices is due to the fact that each Cartesian lattice is represented as a union of a finite number of Cartesian products of one-dimensional shifted lattices of the form \( \Lambda (d, b) = d \mathbb{Z} + b \). Cartesian products of one-dimensional shifted lattices are in substance shifted diagonal lattices, for which in this paper the simplest form of a functional equation for the hyperbolic zeta function of such lattices is given. The connection of the hyperbolic Hurwitz zeta function with the Hurwitz zeta function \( \zeta^* (\alpha; b)\) periodized by parameter \(b\) and with the ordinary Hurwitz zeta function \( \zeta (\alpha; b) \) is studied. New integral representations for these zeta functions and an analytic continuation to the left of the line \( \alpha = 1 + it \) are obtained. All considered hyperbolic zeta functions of lattices form an important class of Dirichlet series directly related to the development of the number-theoretical method in approximate analysis. For the study of such series the use of Abel's theorem is efficient, which gives an integral representation through improper integrals. Integration by parts of these improper integrals leads to improper integrals with Bernoulli polynomials, which are also studied in this paper.

In this paper, some new integral inequalities related to the bounded functions, involving hypergepmetric fractional integral operators, are established. Special cases of the main results are also pointed out.

Wave propagation and scattering occupy a large part of physical, mathematical and engineering sciences. The purpose of this chapter is to present the basic mathematical theory of certain aspects of wavefields, that is, waves and fields, as they occur under various physical situations. These are considered in both scalar or acoustical and vector or electromagnetic media, that is, in the context of Helmholtz’s and Maxwell’s equations. The major emphasis is on the mathematical aspects of Green’s functions, tensors and operators. In particular, the singularities involved are discussed at length. The basic mathematical concepts, tools and techniques, necessary for the presentation, are summarized in the beginning. It is shown that mathematical analyses reveal many subtleties hidden in the wavefields that would otherwise have gone unnoticed. Detailed derivations of the equations are provided whenever possible and necessary. Also, if there are alternative ways of solving a problem, these have been presented. Finally, copious remarks and notes are included for better explaining certain points.

The theory of integral calculations is employed in most fields of the natural sciences for computing the volumes of rotating bodies, areas with different shapes, distances between objects, and other applications. They can be used to explore energy signals to study earthquakes, the distribution of telecommunications signals, and other uses. In particular, many problems formulated as mathematical models use integral equations with variable boundaries. The most interesting applications use integrals where the boundaries are symmetric functions. In the present study, to solve the Volterra integral equations with variable boundaries, we propose multi-step methods such as hybrids, and we construct some simple algorithms for their application where the order of accuracy is p ⩽ 8 ( k = 1 ) .

A mixed multidimensional integral equation containing integral operators of various types is studied. The case in which the equation has one compact, self-adjoint, and strongly positive operator (with constant limits of integration) and two non-self-adjoint integral Volterra operators (with a variable upper limit of integration) is considered. To solve the equation, an effective projection method allowing one to obtain the result in a form with explicitly distinguished principal singularities is proposed.

In the paper, by the Cauchy integral formula in the theory of complex functions, the authors establish an integral representation for the reciprocal of the weighted geometric mean of two positive numbers. By the integral representation, the authors derive that the reciprocal of the weighted geometric mean of two positive numbers is a Stieltjes function and consequently a (logarithmically) completely monotonic function, present several integral formulas for some kinds of improper integrals, deduce several integral representations for some functions related to the weighted geometric mean, find an equivalent relation between integral representations for the weighted geometric mean and its reciprocal, connect the integral formulas with the central Delannoy numbers, and apply some integral representations and some integral formulas to compute some concrete improper integrals appeared in the theory of complex functions.

I propose that students pay more attention to the approximations that lead to integral formulas when they study applications of the definite integral. The goal is to focus students on the underlying concepts and the definition of the definite integral – and to steer them away from memorizing formulas. To address this goal, I have written some computer-based activities that guide students through the nuts and bolts of the approximations that eventually lead to integral formulas. I describe the activities and suggest a few changes as to how we might approach this material.

In this article, the definitions and basic properties of Riemann-Stieltjes integral are formalized in Mizar [1]. In the first section, we showed the preliminary definition. We proved also some properties of finite sequences of real numbers. In Sec. 2, we defined variation. Using the definition, we also defined bounded variation and total variation, and proved theorems about related properties. In Sec. 3, we defined Riemann-Stieltjes integral. Referring to the way of the article [7], we described the definitions. In the last section, we proved theorems about linearity of Riemann-Stieltjes integral. Because there are two types of linearity in Riemann-Stieltjes integral, we proved linearity in two ways. We showed the proof of theorems based on the description of the article [7]. These formalizations are based on [8], [5], [3], and [4].

Henstock–Kurzweil integral or gauge integral is the generalization of the Riemann integral. The functions which are not integrable because of singularity in the senses of Lebesgue or Riemann are gauge integrable. In this manuscript, we have generalized Fα-calculus using the gauge integral method for the integrating of the functions on fractal set subset of real-line where they have singularities. The suggested new method leads to the wider class of functions on the fractal subset of real-line that are *Fα-integrable. Using gauge function we define *Fα-derivative of functions their Fα-derivative is not exist. The reported results can be used for generalizing the fundamental theorem of Fα-calculus.

In this paper, we present some results on existence of solutions for a quadratic Volterra integral equation of fractional order in two independent variables. This equation is considered in the Banach space of real functions, defined, continuous and bounded on an unbounded interval. Moreover, we show that solutions of this integral equation are locally attractive. An example is provided to illustrate the theory.

The generalized Riemann integral of Pfeffer (1991) is defined on all bounded ${\rm BV$ subsets of $\mathbb R^n$, but it is additive only with respect to pairs of disjoint sets whose closures intersect in a set of $\sigma$-finite Hausdorff measure of codimension one. Imposing a stronger regularity condition on partitions of $\rm BV$ sets, we define a Riemann-type integral which satisfies the usual additivity condition and extends the integral of Pfeffer. The new integral is lipeomorphism-invariant and closed with respect to the formation of improper integrals. Its definition in $\mathbb R$ coincides with the Henstock-Kurzweil definition of the Denjoy-Perron integral.}

In this paper, we establish some Liouville type theorems for positive solutions of some integral equations and integral systems in $R^N$. The main technique we use is the method of moving planes in an integral form.

We present a method for the construction of integral identities that contain an undetermined function. Except for mild restrictions, this function can be chosen arbitrarily. Our method is illustrated by several examples leading to new integral identities.

The problem of finding mechanical and electrical fields in a homogeneous piezoelectric plate supported by a thin wedged-shaped cover plate is considered. Using the methods of the theory of analytic functions, the problem is reduced to a system of singular integro-differential equations. With the help of integral transformation, the exact solution of the posed by us problem is obtained. Keywords: Singular integral differential equation, Integral transformation, Piezoelectric medium, Riemann’s problem

Summary Integral inequalities have been widely used in stability analysis for systems with time-varying delay because they directly produce bounds for integral terms with respect to quadratic functions. This paper presents two general integral inequalities from which almost all of the existing integral inequalities can be obtained, such as Jensen inequality, the Wirtinger-based inequality, the Bessel–Legendre inequality, the Wirtinger-based double integral inequality, and the auxiliary function-based integral inequalities. Based on orthogonal polynomials defined in different inner spaces, various concrete single/multiple integral inequalities are obtained. They can produce more accurate bounds with more orthogonal polynomials considered. To show the effectiveness of the new inequalities, their applications to stability analysis for systems with time-varying delay are demonstrated with two numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.

Very often, in physical exercises, it is necessary to differentiate an integral with respect to a parameter. Generally we differentiate after the integral evaluation but, occasionally, it is desirable, from theoretical point of view, to interchange the order of differentiation and integration. In this letter, we examine in a simple way the derivative under the flux integral getting the result in a very simple manner that could be useful from didactic point of view.