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In the study of some real cases, it is possible to encounter well-defined geometric conditions, of an industrial or design type—for example, the case of a specific volume within each of several holes. In most of these cases, it is recommended to fulfil a function defined in a domain in which information is missing in one or more sub-domains (holes) of the global set, where the function data are not known. The problem of filling holes or completing a surface in three dimensions appears in many fields of computing, such as computer-aided geometric design (CAGD). A method to solve the shape-preserving variational spline approximation problem for hole filling in generalized offset surfaces is presented. The existence and uniqueness of the solution of the studied method are established, as well as the computation, and certain convergence results are analyzed. A graphic and numerical example complete this study to demonstrate the effectiveness of the presented method. This manuscript presents the resolution of a complicated problem due to the study of some criteria that can be traduced via an approximation problem related to generalized offset surfaces with holes and also the preservation of the shape of such surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we propose a variational method in order to solve Bratu's problem for two dimensions in an adequate space of biquadratic spline functions. The solution is obtained by resolving a sequence of boundary value problems. We study some characterizations of the functions of such sequence and we express them as some linear combination of biquadratic spline bases functions. We finish by showing some numerical and graphical examples in order to prove the validity and the effectiveness of our method. [ABSTRACT FROM AUTHOR]

We present an approximation problem of surfaces of a generalized offset surface with offset variable distances and directions. Such approximating surface fits some given data points and minimizes a Sobolev's semi-norm of order 3. The study of the new results, from a mathematical point of view, carefully establishing the proof of the convergence between the generalized offset surface and its approximating spline in an adequate parametric bicubic spline space. Moreover, the approximating spline function is computed and an estimation of the relative error is introduced. Finally, some numerical and graphic examples are shown in order to prove the useful and the effectiveness of our method. [ABSTRACT FROM AUTHOR]

In this work we present a new method to solve the statistical difficult when economic series are spliced. Therefore, we discuss the scope of some splicing tools in the literature, namely the splicing by variation and linear interpolation methods. These methods carry on some problems of non-linearity. In this way, certain internal inconsistency or structural incongruity can appear when the economic series are spliced. This is a serious problem for the System of National Accounts. We introduce an approximation method for statistical splicing of economic series by smoothing quadratic splines. The proposed technique is linear and thus structurally congruent. Finally, we show the effectiveness of our method by the results of the splicing of the GDP (Gross Domestic Product) of Venezuela between 1950 and 2012, and the corresponding economics activities. [ABSTRACT FROM AUTHOR]

Abstract We present a smoothing method of a set of points to be approximated from a given boundary value problem for the modified Helmholtz equation. A solution of such problem will be called MH-spline. Two convergence results are carefully established. Some numerical and graphic examples are shown in order to test the validity of the method. [ABSTRACT FROM AUTHOR]

This article deals with a numerical approximation method using an evolutionary partial differential equation (PDE) by discrete variational splines in a finite element space. To formulate the problem, we need an evolutionary PDE equation with respect to the time and the position, certain boundary conditions and a set of approximating points. We show the existence and uniqueness of the solution and we study a computational method to compute such a solution. Moreover, we established a convergence result with respect to the time and the position. We provided several numerical and graphic examples of approximation in order to show the validity and effectiveness of the presented method.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 5-18, 2018 [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the construction and characterization of discrete variational PDE splines. To formulate the problem, we need an evolutionary PDE equation, certain boundary conditions and a set of points to approximate. We show the existence and the uniqueness of the solution of such a problem and we establish a convergence result of a discrete variational evolutionary PDE spline to a given function. We present several numerical and graphical examples of construction and approximation of surfaces in order to prove the validity of the presented method. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a new approach for solving second kind Fredholm integral equation system. The method is based on the minimization of a suitable functional in a certain Sobolev space. We study the convergence of the method and we present some numerical examples in order to show the validity of the proposed method. [Copyright &y& Elsevier]

Abstract: In this paper, we present an interpolation method for curves from a data set by means of the optimization of the parameters of a quadratic functional in a space of parametric cubic spline functions. The existence and the uniqueness of this problem are shown. Moreover, a convergence result of the method is established in order to justify the method presented. The aforementioned functional involves some real non-negative parameters; the optimal parametric curve is obtained by the suitable optimization of these parameters. Finally, we analyze some numerical and graphical examples in order to show the efficiency of our method. [Copyright &y& Elsevier]

Abstract: This paper addresses the problem of constructing some free-form curves and surfaces from given to different types of data: exact and noisy data. We extend the theory of -splines over a bounded domain for noisy data to the smoothing variational vector splines. Both results of convergence for respectively the exact and noisy data are established, as soon as some estimations of errors are given. [Copyright &y& Elsevier]

This paper deals with a shape preserving method of interpolating positive data at points of the plane in R2. We formulate a problem in order to define a positive interpolation variational spline in the Sobolev space Hm(Ω), where Ω is a non-empty bounded set of R2, by minimizing the semi-norm of order m, and we discrete such problem in a finite element space. An algorithm allows us to compute the resulting function. Some convergence theorems are established. The error is of the order O(1/N)m when N tends to +∞, being N the number of the interpolating points. Some numerical and graphical examples are given in order to test the validity of this method. [Copyright &y& Elsevier]