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This study covers new relations among generalized Bernoulli, Euler and Apostol type numbers and polynomials, the Stirling, combinatorial numbers, and array polynomials. In order to give these relations, functional equations of generating functions can be used. [ABSTRACT FROM AUTHOR]

In this paper, we find the stability of the Cauchy additive functional equation, a-cubic and b-cubic functional equation and quartic functional equation in neutrosophic normed space via the direct method. [ABSTRACT FROM AUTHOR]

When optimizing the directivity and energy characteristics of the antennas with flat reflectors and the leaky-wave antennas, high-speed computational algorithms are required for a reliable analysis of the diffraction of electromagnetic waves on a conducting screen, accounting for the inhomogeneities with the dimensions commensurate with the wavelength. To calculate the diffraction field of an electromagnetic wave with an arbitrary-shaped front on a rectangular groove in the screen, a numerical model has been developed. This model is based on the approximation of the wave front by finite locally flat patches and the use of the authors' rigorous solution of the problem of locally flat wave diffraction on a conducting screen with a groove. The field above the screen is represented by the Fourier integral, while the field in the groove – by the sum of waveguide modes. The set of functional equations is obtained using the method of partial domains. Through a re-expansion of the modal functions of the field of partial domains in terms of the basis of the adjacent domain, the functional equations are reduced to a set of linear algebraic equations with respect to the complex amplitudes of the waveguide modes of the groove. The implementation of the main computational procedures is described. Based on the analysis of the particular results, the reliability of the developed model is proved. The influence of the quality of the wave front interpolation on the accuracy of the solution is studied. Recommendations on the frequency of a wave-front fragmentation are formulated. The proposed digital algorithm can be used to analyze and synthesize antennas with the flat perforated distributing and radiating systems. [ABSTRACT FROM AUTHOR]

Many problems in mechanics and mathematical physics are reduced to solving integral equations. Developing optimal approximate methods for solving such integral equations in functional spaces is one of the urgent problems of computational mathematics. In this paper, the solution of integral equations of fractional order, in particular, the analytical solution of the general Abel integral equation, is presented. It is devoted to constructing an optimal quadrature formula for approximating a general Abel integral equation in the Hilbert space real-valued functions. Here, we find the extremal function and calculate the square of the error functional norm for the quadrature formula. We also calculate the optimal coefficients for the quadrature formulas. The optimal quadrature formula is constructed by finding the optimal coefficients that give minimum to the norm of the error functional. [ABSTRACT FROM AUTHOR]

In the present article, the solution of the problem for choosing an optimal size range of linear pneumatic modules, widely used in the automation of discrete production processes, is considered. Choosing the optimal size range of translation modules is essential for good economic performance for both manufacturers and users of these devices. For solving the problem, the main parameters of the devices on which the optimization will be performed are selected – load capacity and stroke length, and their values are determined using series of preferred numbers. Module demand is forecast based on market research. A known optimality criterion is used, which represents the total costs of production and operation. On the basis of a known cost model and functional equations, an algorithm for solving the problem is developed, taking into account the presence of constraints on the values of the modules' parameters. The constraints on the values mean that the next element in the size range must have main parameters' values no less than the previous element. With the help of the developed tools, the optimal size ranges are determined for three different ways of defining the modules' demand: determined as a result of primary data collection from users, determined after linear regression of the principal components, and determined after recovery of the missing values. [ABSTRACT FROM AUTHOR]

Acoustical thermodynamic work is to estimate excess thermodynamic properties with coefficients of Redlich Kister polynomial equation in the studied functional liquid mixture combination (O-anisidine combined with Amyl acetate). Evaluated excess acoustical excess values such as Gibb's parameter (GE), total heat function or enthalpy (NE) & internal pressure (PE) are measured with experimental studied information. Further effects have additionally been beneficial to attain Redlich-Kister parameters from equation of polynomial and additionally to acquire their deviations and are analyzed to understand the molecular interactions among thing molecules at recognised temperatures variety from temperature 303.15K to 318.15K. [ABSTRACT FROM AUTHOR]

The main idea of this paper is to give some identities and applications of the two parametric kinds of Eulerian-type polynomials. By using generating functions and functional equations of these special polynomials, we derive some identities and relations associated with the two parametric kinds of Eulerian-type polynomials, the Hermite polynomials, and well-known combinatorial numbers and polynomials. Moreover, we give a remark on these special polynomials and functions. [ABSTRACT FROM AUTHOR]

The aim of this study is to define a higher-order extension of the numbers arising from the finite sums containing higher powers of binomial coefficients. By using functional equations of the generating functions for these numbers, we derive some identities and relations between these numbers and λ-array polynomials. Finally, we give some remarks and observations on relations among the aforementioned numbers, λ-array polynomials and Eulerian type splines with their applications. [ABSTRACT FROM AUTHOR]

In this paper, we study Hyers-Ulam Rassias stability of additive and quintic mixed type of functional equation Gg(x, y)=g(3x+y)-5g(2x+y)+g(2x-y)+10g(x+y)-5g(x-y)=10g(y)+4g(2x)-8g(x) in the non-Archimedean normed space. [ABSTRACT FROM AUTHOR]

This paper deals with the study of the generalized Hyers-Ulam-Rassias stability of the following additive-quadratic functional equation f(2x+y)+f(2x-y)=f(x+y)+f(x-y)+2f(2x)-2f(x)in non-archimedean quasi-Banach space by using Hyers direct method. [ABSTRACT FROM AUTHOR]

The article is devoted to the lateral-torsional buckling of elastic beams loaded with a potential moment. Such problems were previously investigated in a linearized formulation. The prebuckling deformations of the beam is not considered in this setting. We use the nonlinear geometrically exact beam theory and an exact non-linear solution in our article. The exact spatial stability functional and stability equations are obtained by the variation calculus. The exact value of a critical moment for the cantilever beam was obtained. [ABSTRACT FROM AUTHOR]

The development of a unified approach to solving the problems of the dynamics and interaction of multiply connected structurally-inhomogeneous shell structures, which are an arbitrary composition of multilayer shells of revolution and rings, the creation and implementation of an appropriate software package with a high level of automation of all stages of calculations oriented to computer use significantly increase the design efficiency and are a major scientific problem of great economic importance. This paper is dedicated to solving this problem. The study and prediction of the deformation properties of the materials studied in the work are possible based on mathematical modeling of deformation and relaxation processes. In this article, we give an algorithm for solving a nonlinear functional equation with complex variables resulting from mathematical modeling of problems concerning the properties of a deformable solid. [ABSTRACT FROM AUTHOR]

we study the existence of piecewise (PC)-mild solutions for non-local fractional impulsive functional integrodifferential equations with finite delay. That is, we study the existence of mild solutions for non-local impulsive fractional semi-linear integrodifferential equations. [ABSTRACT FROM AUTHOR]

We establish the existence and controllability results for impulsive neutral mixed-type functional integrodifferential systems with infinite delay. The results are obtained using the theory of fractional powers of operators and Sadovski's fixed point theorem. That is the principle motivation behind this chapter is to research the existing results of the following impulsive neutral mixed functional integrodifferential equations with infinite delay. [ABSTRACT FROM AUTHOR]

We are concerned with the existence of mild solutions for the impulsive neutral mixed functional integrodifferential equations with infinite delay. The results are obtained using the theory of fractional powers of operators and Sadovskii's fixed point theorem. we establish existence results of the following mixed functional integrodifferential equations with infinite delay [ABSTRACT FROM AUTHOR]

We establish the controllability for a class of abstract nonlocal impulsive neutral mixed-type functional integrodifferential equations with finite delay in a Banach space. Some sufficient conditions for controllability are obtained by using the Monch fixed point theorem via measures of non-compactness and semigroup theory. Particularly, we do not assume the compactness of the evolution system. That is, we are going to establish at less some sufficient conditions for the controllability of an impulsive neutral mixed-type functional integrodifferential system with finite delay and non-local conditions. [ABSTRACT FROM AUTHOR]

In this work, the authors solve some stability problems of quartic functional equation in ℒ – fuzzy normed space(ℒ – FNS) and non-Archimedean ℒ – fuzzy normed space(ℒ – FNS). Also, we generalize the stability results of quadratic functional equation in fuzzy Banach space(FBS) within a any t′ – norm. [ABSTRACT FROM AUTHOR]

In this work, we investigate two-dimensional nonlinear Hammerstein-Fredholm fuzzy functional integral equation (2D-NHFFFIE. We propose an efficient iterative method of successive approximations by using fuzzy variant of the cubature rule of the rectangle for classes of fuzzy number-valued functions of Lipschitz type for two-dimensional case to approximate the solution. We prove the convergence and numerical stability of the presented method with respect to the choice of the first iteration. Finally, some numerical experiment confirm the theoretical results and illustrate the accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

A fractional diffusion equation with the presence of a drift and a delay of a general form is considered. For this problem, a family of grid schemes with weights is constructed based on the L1-method for approximating the fractional derivative and applying the piecewise constant interpolation of discrete history. The algorithm is reduced to solving linear systems with a tridiagonal matrix. The order of the residual of the method without interpolation and the order of the residual of the method with interpolation are investigated. The stability conditions of the algorithm are obtained. A theorem is proved that under stability conditions the method has the first order with respect to the time-step and the second order with respect to the space-step. [ABSTRACT FROM AUTHOR]

A space-fractional two-dimensional diffusion equation with a delay of a general form is considered. For this problem, the ADI method is constructed based on the shifted Grunwald-Letnikov formula for approximating the fractional derivatives and applying the piecewise constant interpolation of discrete history. The method convergence order is obtained. The results of numerical experiments are presented. [ABSTRACT FROM AUTHOR]

We consider the initial-boundary value problem for nonlinear time fractional partial differential equations. The source of the problem are population models. Both space differential operator and inhomogeneity function contain nonlinearity. We have constructed a nonlinear implicit difference scheme based on the L1 method for approximating the fractional derivative and the idea of interpolation and extrapolation of the discrete history of the model. The conditions under which the method converges are studied. To solve a nonlinear system at each step, the Newton method is used, with which the method is reduced to solving a sequence of linear systems of a tridiagonal structure. The results of numerical experiments are presented. [ABSTRACT FROM AUTHOR]

The article deals with a deterministic variant of the dynamic task of assigning labor resources. There are labor resources, wishing to find a job, and many enterprises offering vacancies. Appointment of workers can be done in an optimal way: for a given initial state of the process, maximize the total income of the enterprises. The problem can be solved using the dynamic programming technique - Bellman functional equations. A numerical example is given. [ABSTRACT FROM AUTHOR]

In this paper, we solve the following quartic functional equation f ( x + y 2 − z) + f ( y + z 2 − x) + f ( z + x 2 − y) = 9 16 (f (x − y) + f (y − z) + f (z − x)) originating from the sum of the medians of a triangle, and prove the Ulam-Hyers stability of the quartic functional equation in fuzzy normed space by the direct and fixed point methods. An application of this functional equation is also provided. [ABSTRACT FROM AUTHOR]

The purpose of this current work, we introduced the finite dimensional quadratic functional equation of the form φ ( ∑ i = 1 m s i) + ∑ 1 ≤ i < j ≤ m φ (− s i − s j + ∑ l = 1 i ≠ j ≠ 1 m s 1) = ( − m 2 + 9 m − 18 2) ∑ 1 ≤ i < j ≤ m φ (s i − s j) + ( m 3 − 9 m 2 + 26 m − 16 2) ∑ i = 1 m φ (s i) when m is a positive integer with n is greater than three. And also we investigated its Hyers-Ulam stabilities in Banach spaces through two different techniques. [ABSTRACT FROM AUTHOR]

In this paper, we introduced m-dimensional quadratic functional equation of the form ∑ 1 ≤ i < j < k < l ≤ m φ (s i + s j + s k + s l) = ( m 2 − 5 m + 6 2) ∑ i = 1 ; i ≠ j m φ (s i + s j) − ( m 3 − 6 m 2 + 11 m − 6 3) ∑ i = 1 m ( φ (s i) + φ (− s i) 2) here m is a non-negative integer with N — {0,1,2.3}. Moreover, we derive its general solution and established Hyers-Ulam stability results in fuzzy normed space through direct and fixed point methods. [ABSTRACT FROM AUTHOR]

In this paper, the existence, uniqueness and controllability for impulsive fuzzy neutral functional integrodifferential equations are studied using Banach fixed point theorem. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to give identities and relations including the Frobenius-Euler polynomials, the Fubini type numbers and polynomials, the central factorial numbers, the λ-array polynomials and the other well-known combinatorial numbers. By using generating functions and their functional equations, we also give some formulas and relations related to these numbers and polynomials. [ABSTRACT FROM AUTHOR]

In “Construction of some new families of Apostol-type numbers and polynomials via Dirichlet character and p-adic q-integrals Turk J Math (2018) 42: 557-577”, we defined some new families of numbers and polynomials related to the Dirichlet character of a finite abelian group. In this paper, we investigate some properties and relations for these numbers and polynomials with their series representations and generating functions. By using functional equations of generating functions with their differential equations and series representations of these numbers and polynomials, we derive some formulas including recurrence relations, combinatorial numbers, combinatorial sums and special numbers and polynomials. By using p-adic integral method, we also derive p-adic series representations of these series representations of these numbers and polynomials. [ABSTRACT FROM AUTHOR]

In this article, we propose a new mixed type functional equation derived from additive and quartic mappings of the form aq(2u + v) + aq(2u − v) + aq(u + 2v) + aq(u − 2v) = 8[aq(u + v) + aq(u − v)] + aq(2u) − 5aq(u) + 7aq(−u) + 2aq(2v) − 5aq(−v) − 9aq(v) and validate its classical stabilities in Banach space. [ABSTRACT FROM AUTHOR]

We will investigate the superstability of a generalized trigonometric functional equation from the Pexideried functional equation: f (x + y) − f (x − y) = λ · g(x)h(y), which is a trigonometric functional equation mixed by the sine and cosine function. [ABSTRACT FROM AUTHOR]

In the present communication, the exact solutions of the (2+1)-dimensional Boiti-Leon-Pempinelli System is obtained by using the first integral method. Some of the solutions are recovered as reported earlier in the literature by the exp-function method and some new solutions are also furnished. [ABSTRACT FROM AUTHOR]

Many types of fuzzy numbers were studied in the literature and applied in many mathematical fields. Three of these types, the triangular, the trapezoidal and the hexagonal fuzzy number are used widely, especially in mathematical programming. A piecewise linear fuzzy number (PLFN), which generalizes these three types, is introduced in this paper. Based on the definition of the ranking function with total integral value, a ranking function for this piecewise linear fuzzy number is proposed. Arithmetic operations with their properties are introduced on the trapezoidal fuzzy number as a special case, then a generalization of these operations on the PLFN is given. It is also shown that the proposed operations on the PLFN are accepted generalizations of the conventional arithmetic operations on real numbers. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give interpolation function for the families of numbers which are associated with not only the Apostol-Bernoulli numbers, but also the Apostol-Euler numbers and the Apostol-Genocchi numbers. We investigate some properties of these functions and these numbers. Moreover, we give some formulas including these functions and these numbers. Finally, we also give some observations related to well-known families of zeta-type functions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give some combinatorial sums, identities and relations related to a family of combinatorial numbers and the Bernstein type basis functions. With the help of generating functions for the combinatorial numbers from this aforementioned family, we give some functional equations. By using these functional equations, we obtain several relationships including these combinatorial numbers and the Bernstein type basis functions. Moreover, we provide not only some identities, but also applications including combinatorial sums, integrals, the combinatorial numbers, the extended Bersntein basis functions, and the beta function. [ABSTRACT FROM AUTHOR]

In this paper, we determine new type (p, q)- Bernstein polynomials which different from the (p, q)-Bernstein polynomials of Mursaleen et.al.([11]). After that, by the similar motivation from ([16]), we obtain some new functional equations and properties with related to these new polynomials arising from (p, q)- calculus. [ABSTRACT FROM AUTHOR]

The Szegö kernel plays an important role in conformal mapping and satisfies a boundary Kerzman-Stein integral equation. The series representation for the Szegö kernel for an annulus region is well known. In this paper, the Adomian decomposition method is applied to solve the boundary Kerzman-Stein integral equation. The method provides another series form for the Szegö kernel for the annulus region. The two series are shown to be equivalent. Some numerical examples are also presented to compare the speed of convergence of the two series. [ABSTRACT FROM AUTHOR]

In this paper, we construct a non-Volterra Gaussian Quadratic Stochastic Operators defined on the set of real numbers R and show non-ergodicity of this transformation. [ABSTRACT FROM AUTHOR]

In this paper, the reliability of the two methods from the family of fourth-order implicit finite difference schemes, namely the fourth-order Crank-Nicolson approximation scheme and the fourth-order standard implicit approximation scheme in solving the diffusion equation has been studied. The corresponding system of linear approximation equations for both fourth-order schemes were derived and then solved iteratively. In addition, for the purpose of assessing the efficiency of both schemes, comparison was made by considering three parameters, which are the iteration numbers, the computation time and the maximum error. Based on the results of the numerical experiments, it shows that the performance of the fourth-order Crank-Nicolson scheme is more efficient compared to the fourth-order standard implicit scheme in solving one-dimensional (1D) diffusion equation. [ABSTRACT FROM AUTHOR]

In this study, the Two-point Explicit Group Successive Over Relaxation (2-EGSOR) iterative method is used to solve a system of linear equations generated from the discretization of system of second kind Fredholm integral equations. The discretization process is done by using quadrature scheme. The formulation and implementation of proposed iterative methods are also presented. Then numerical experiments are carried out onto three test problems to verify the effectiveness of the Gauss-Seidel (GS), Successive Over Relaxation (SOR) and 2- EGSOR methods. The numerical results shown that the 2-EGSOR iterative method with quadrature scheme is more efficient compared to GS and SOR methods. [ABSTRACT FROM AUTHOR]

A new quadratic-reciprocal functional equation f ( ( k + 1 ) x + k y ) + f ( ( k + 1 ) x − k y ) = 2 f ( x ) f ( y ) [ ( k + 1 ) 2 f ( y ) + k 2 f ( x ) ] [ ( k + 1 ) 2 f ( y ) − k 2 f ( x ) ] 2 is introduced. The Hyers-Ulam stability for the quadratic-reciprocal functional equations is proved in Banach spaces using the direct method and the fixed point method, respectively. [ABSTRACT FROM AUTHOR]

We consider various problems for functions f such that ... where p,q are real numbers and f(p) is the p th derivative of f. The connection with functional differential equation is studied. Some of the results may be useful in teaching algebra and analysis. [ABSTRACT FROM AUTHOR]

In this paper, Painlev'e equation I problem was solved approximately using Daftardar-Gejji and Jafari Method (DJM) with initial conditions. The results obtained through this iterative method, is compared with that obtained by other methods such as Adomian Decomposition Method (ADM), Homotopy Pertubation Method (HPM) and Variational Iteration Method (VIM). The numerical results show that there is no significant difference. It has been found that the results obtained are in fully agreement. [ABSTRACT FROM AUTHOR]

In the paper, we propose a correct and efficient semi-analytical approach to solve initial value problem for systems of functional differential equations with delay. The idea is to combine the method of steps and differential transformation method (DTM). In the latter, formulas for proportional arguments and nonlinear terms are used. An example of using this technique for a system with constant and proportional delays is presented. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give a new modification of the generating functions for the generalized second kind Apostol Euler type polynomials and numbers. We give a recusance relations of these numbers and polynomials. By applying differential operator to these gnarling functions, we derive some identities and relations for these numbers and polynomials. [ABSTRACT FROM AUTHOR]

In this article, by using generating functions method we derive some new identities and relations related to the Fibonacci numbers, Lucas numbers, Bernoulli numbers, Euler numbers. We also give a relation between Lucas numbers of order k and the numbers y3(n, k; λ; a, b) which were defined by the second author [7]. [ABSTRACT FROM AUTHOR]

In this paper, we give some identities and relations using some families of the special numbers with their generating functions. We investigate some properties of these numbers. Finally, we state combinatorial interpretations for the combinatorial type numbers. We also express remarks and comments on our new numbers and their identities. [ABSTRACT FROM AUTHOR]

The numerical solution of a class of boundary value problems for nonlinear fractional differential equations is discussed. Using a suitable integral equation reformulation of the boundary value problem and spline collocation techniques on graded grids, high order methods for solving fractional boundary value problems are constructed. [ABSTRACT FROM AUTHOR]

In this paper a functional equation for the fractional derivative of the Riemann ζ function is presented. The fractional derivative of ζ is computed by a generalization of the Grünwald-Letnikov fractional operator, which satisfies the generalized Leibniz rule. It is applied to the asymmetric functional equation of ζ in order to obtain the result sought. Moreover, further properties of this fractional derivative are proposed and discussed. [ABSTRACT FROM AUTHOR]

The relation between the Ahlfors map and Szegö kernel S (z, a) is classical. The Szegö kernel is a solution of a Fredholm integral equation of the second kind with the Kerzman-Stein kernel. The exact zeros of the Ahlfors map are known for a particular family of doubly connected regions and a particular triply connected region. This paper presents a numerical method for computing the zeros of the Ahlfors map of any bounded multiply connected regions with smooth boundaries. The method depends on the values of S (z(t), a), S'(z(t), a) and θ'(t), where θ(t) is the boundary correspondence function of Ahlfors map. A formula is derived for computing S'(z(t), a). An integral equation for θ'(t) is used for finding the zeros of Ahlfors map. The numerical examples presented here demonstrate the method. [ABSTRACT FROM AUTHOR]

In the current paper we propose a new formulation of a local boundary value problem for a one-dimensional wave equation in a rectangular domain in which boundary conditions are given on the whole boundary. We prove the well-posedness of the boundary value problem in the classical and generalized senses. The proof of the well-posedness of the formulated problem is reduced to question of the existence and uniqueness of solutions of the corresponding functional equations. [ABSTRACT FROM AUTHOR]