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We study variational approximations of a dual pair of mathematical programming problems in terms of epi/hypo-convergence and inside epi/hypo-convergence of approximating Lagrange functions of the pair. First, the Painlevé -Kuratowski convergence of approximate saddle points of approximating Lagrange functions is established under the inside epi/hypo-convergence of these approximating Lagrange functions. From this, we obtain a couple of solutions of the pair of problems and a strong duality. Under a stronger variational convergence called ancillary tight epi/hypo-convergence, we obtain the Painle vé-Kuratowski convergence of approximate minsup-points and approximate maxinf-points of approximating Lagrange functions (when approximate saddle points are not necessary to exist).

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The important issue of the aggregation preference is how to determine the weights associated with different ranking places and DEA models play an important role in this subject. DEA models use assignments of the same aggregate value (equal to unity) to evaluate multiple alternatives as efficient. Furthermore, overly diverse weights can appear, thus, the efficiency of different alternatives obtained by different sets of weights may be unable to be compared and ranked on the same basis. In order to solve two above problems, and rank all the alternatives on the same scale, in this paper, we propose a multiple objective programming (MOP) approach for generating a common set of weights in the DEA framework. Also, we develop a novel model to make a maximum discriminating among candidates’ rankings. Additionally, we present two scenarios to provide suitable strategies for solving the proposed MOP model.

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In this paper, we develop a numerical method to solve a famous free boundary PDE called the one dimensional Stefan problem.
First, we rewrite the PDE as a variational inequality problem (VIP). Using the linear finite element method, we discretize the variational inequality and achieve a linear complementarity problem (LCP). We present some existence and uniqueness theorems for solutions of the variational inequalities and free boundary problems. Finally we solve the LCP numerically by applying a modification of the active set strategy.

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In this article, the properties of single-term Walsh series are presented and utilized for solving the nonlinear Volterra-Hammerstein integral equations of second kind. The interval [0;1) is divided tomequal subintervals,mis a positive integer number. The midpoint of each subinterval is chosen as a suitable collocation point. By the method the computations of integral equations reduce into some nonlinear algebraic equations. The method is computationally attractive, and gives a continuous approximate solution. An analysis for the convergence of method is presented.The efficiency and accuracy of the method are demonstrated through illustrative examples. Some comparisons aremade with the existing results.

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The main goal of this paper is to propose interval network data envelopment analysis (INDEA) model for performance evaluation of network decision making units (DMUs) with two-stage network structure under data uncertainty. It should be explained that for dealing with uncertainty of data, an interval programming method as a popular uncertainty programming approach is applied. Also, to show the applicability of proposed model, INDEA approach is implemented for performance measurement and ranking of 10 insurance companies from Iranian insurance industry. Note that insurance companies are undoubtedly one of the most important pillars of the financial markets, whose great performance will drive the economy of the country. The empirical results indicate that the proposed INDEA is capable to be utilized to assess the performance of two-stage DMUs in the presence of interval data.

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Our aim in this paper, is applying Adams-Moulton algorithm to find the geodesics as the answers of the classical system of ordinary differential equations on a 2-dimensional surface for which a Riemannian metric is defined.

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The objective of this work is to investigate the infl uence of boundary conditions at depth soil on the development of methods to determine the soil′s apparent thermal diffusivity based on solution of inverse problems of a heat-transfer equation. Experimental investigations were carried out to establish the influence of boundary conditions at depth in soil on the solution of inverse problems of modeling of heat transfer in soils. For this purpose, 1 soil profile in the land at different depths (x=0, 5, 10, 15,  20, 40, 60 cm) thermal sensors (Temperature recorder Elitech RC-4) have been installed to measure soil temperatures depending on time and depths. Based on these data, the apparent  thermal diffusivity in soils was calculated using the classical (layered) and proposed (point) methods developed for the case with one and two harmonics, and they were compared and the calculated characteristics were compared with the experimental results. It was found that the proposed point methods best reflect the movement of heat in the soil profile. 

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‎A numerical approach based on Bernstein polynomials is presented to unravel optimal control of nonlinear systems. The operational matrices of differentiation, integration and product are introduced. Then, these matrices are implemented to decrease the solution of nonlinear optimal control problem to the solution of the quadratic programming problem which can be solved with many algorithms and softwares. This method is easy to implement it with an accurate solution. Some examples are included to demonstrate the validity and applicability of the technique.

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In this paper we prove the Riesz-Thorian interpolation theo-rem for weighted Orlicz and weighted Morrey Spaces.

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In this paper‎, ‎the polynomial differential quadrature method (PDQM) is implemented to find the numerical solution of the generalized Black-Scholes partial differential equation‎. ‎The PDQM reduces the problem into a system of first order non-linear differential equations and then‎, ‎the obtained system is solved by optimal four-stage‎, ‎order three strong stability-preserving time-stepping Runge-Kutta (SSP-RK43) scheme‎. ‎Numerical examples are given to illustrate the efficiency of the proposed method‎.

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In economics, a production function relates the outputs of a production process to the inputs of the production. Generally, the production function is not available due to the complexity of the production process, the changes in production technology. Therefore, we have to consider an approximation of the production function. Data Envelopment Analysis (DEA) is a non-parametric methodology for obtaining an approximation of the production function and assessing the relative efficiency of economic units. Sensitivity analysis and sustainability evaluation of Decision Making Units (DMUs) are as the most important concerns of Decision Makers (DM). This study considers the sustainability radius of economic performance of DMUs and then proposes some approaches combined with sensitivity analysis for determining the sustainability radius of cost efficiency, revenue efficiency and profit efficiency of units. The proposed approaches eliminate the unit under evaluation from the observed data and disturb the data of it, based on the sensitivity analysis, to determine the sustainability radius of cost efficiency, revenue efficiency and profit efficiency of decision making units. Potential application of our proposed methods is illustrated with a dataset consisting of 21 medical centers in Taiwan.

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‎The existence and number of limit cycles is an important problem ‎in the study of ordinary differential equations and dynamical‎

‎systems‎. ‎In this work we consider $2$-dimensional predator-prey‎ ‎system and‎, ‎using Poincarchr('39'){e}-Bendixson theorem and LaSallechr('39')s‎ ‎invariance principle‎, ‎present some new necessary and some new‎ ‎sufficient conditions for the existence and nonexistence of limit‎

‎cycles of the system‎. ‎These results extend and improve the‎ ‎previous results in this subject‎. ‎Local or global stability of the‎

‎rest points of a system is also an important issue in the study of‎ ‎the equations and systems‎. ‎In this paper a sufficient condition‎

‎about global stability of a critical point of the system will also‎ ‎be presented‎. ‎Our results are sharp and are applicable for‎

‎predator-prey systems with functional response which is function‎ ‎of prey and predator‎. ‎At the end of the manuscript‎, ‎some examples‎

‎of well-known predator-prey systems are provided to illustrate our‎ ‎results‎.

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In this work, we will first propose an optimal three-step without-memory
method for solving nonlinear equations. Then, by introducing the self-accelerating
parameters, the with-memory-methods have been built. They have a fifty-nine
percentage improvement in the convergence order. The proposed methods have
not the problems of calculating the function derivative. We use these Steffensen-
type methods to solve nonlinear equations with simple zeroes with the appropri-
ate initial approximation of the root. we have solved a few nonlinear problems
to justify the theoretical study. Finally, are described the dynamics of the with-
memory method for complex polynomials of degree two.

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The construction of a nonnegative matrix for a given set of eigenvalues is one of the objectives of this paper. The generalization of the cases discussed in the previous papers as well as finding a recursive solution for the Suleimanova spectrum are other points that are studied in this paper.

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‎Nonlinear problems in partial differential equations are open problems in many field of mathematics and engineering‎. ‎So associated with the structure of the problems‎, ‎many analytical and numerical methods are obtained‎. ‎We show that the differential transformation method is an appropriate method for the Dullin-Gottwald-Holm equation‎ (DGH), ‎which is a nonlinear partial differential equation arise in many physical phenomenon‎. ‎Hence in this paper‎, ‎the differential transform method (DTM) is applied to the Dullin-Gottwald-Holm equation‎. ‎We obtain the exact solutions of Dullin-Gottwald-Holm equation by using the DTM‎. ‎In addition‎, ‎we give some examples to illustrate the sufficiency of the method for solving such nonlinear partial differential equations‎. ‎These results show that this technique is easy to apply and provide a suitable method for solving differential equations‎. ‎To our best knowledge‎, ‎the theorem presented in Section 2 has been not introduced previously‎. ‎We presented and proved this new theorem which can be very effective for formulating the nonlinear forms of partial differential equations‎.

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In any Bayesian inference problem, the posterior distribution is a product of the
likelihood and the prior: thus, it is a ected by both in cases where one possesses little or no
information about the target parameters in advance. In the case of an objective Bayesian
analysis, the resulting posterior should be expected to be universally agreed upon by ev-
eryone, whereas . subjective Bayesianism would argue that probability corresponds to the
degree of personal belief. In this paper, we consider Bayesian estimation of two-parameter
exponential distribution using the Bayes approach needs a prior distribution for parame-
ters. However, it is dicult to use the joint prior distributions. Sometimes, by using linear
transformation of reliability function of two-parameter exponential distribution in order to
get simple linear regression model to estimation of parameters. Here, we propose to make
Bayesian inferences for the parameters using non-informative priors, namely the (depen-
dent and independent) Je reys' prior and the reference prior. The Bayesian estimation was
assessed using the Monte Carlo method. The criteria mean square error was determined
evaluate the possible impact of prior speci cation on estimation. Finally, an application on
a real dataset illustrated the developed procedures.

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The value of an auxiliary parameter incorporated into the well-known variational iteration method (VIM) to obtain solutions of wave equations in unbounded domains is discussed in this article. The suggested method, namely the optimal variational iteration method, is investigated for convergence. Furthermore, the proposed method is tested on one-dimensional and two-dimensional wave equations in unbounded domains in order to better understand the solution mechanism and choose the best auxiliary parameter.Comparisons with results from the standard variational iteration procedure demonstrate that the auxiliary parameter is very useful in tracking the convergence field of the approximate solution.

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The aim of this paper is to investigate the existence and uniqueness of solution for a class of nonlinear integro-differential equations known as Hammerstein type. We study fractional equations in the Banach space whose derivative is of the Caputo type. The existence of solution is studied by using the Schauder's fixed point theorem, and the uniqueness is established via a generalization of the Banach fixed point theorem. Finally, an example is given to illustrate the analytical findings.

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In the current paper, we study an inverse problem of identifying a time-dependent forcing term in the one-dimensional wave equation. We have the information of the wave displacement at two different instants of time and two sensor locations of space along with a dynamic type boundary condition. We prove the unique solvibility of the problem under some regularity and consistency conditions. Then, an approximate solution of the given inverse problem based upon deploying the Ritz technique along with the the collocation method is presented which converts the problem to a linear system of algebraic equations. The method takes advantage of the Tikhonov regularization technique to solve the linear system of equations that is not well-conditioned in order to achieve stable solutions. Numerical findings are also included to support the claim that the presented method is reliable in finding accurate and stable solutions.
 

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‎In this paper‎, ‎a new modified line search Armijo is used in the diagonal discrete gradient bundle method to solve large-scale non-smooth optimization problems‎. ‎The new principle causes the step in each iteration to be longer‎, ‎which reduces the number of iterations‎, ‎evaluations‎, ‎and the computational time‎. ‎In other words‎, ‎the efficiency and performance of the method are improved‎. ‎We prove that the diagonal discrete gradient bundle method converges with the proposed monotone line search principle for semi-smooth functions‎, ‎which are not necessarily differentiable or convex‎. ‎In addition‎, ‎the numerical results confirm the efficiency of the proposed correction‎.