Search

In this paper, we investigate sparse representations of approximation to identity via time–space fractional heat equations: \[ \left\{ \begin{aligned} & \partial^{\beta}_{t}u(t,x)=-\nu(-\Delta)^{\alpha/2}u(t,x),\quad (t,x)\in\mathbb R^{1+n}_{+};\cr & u(0,x)=f(x),\ x\in\mathbb R^{n}. \end{aligned}\right. \] { ∂ t β u (t , x) = − ν (− Δ) α / 2 u (t , x) , (t , x) ∈ R + 1 + n ; u (0 , x) = f (x) , x ∈ R n. Due to the time-fractional derivative, the semigroup property is invalid for the solutions $ u(\cdot,\cdot) $ u (⋅ , ⋅) to the above problem. This deficiency makes it difficult to verify the boundary vanishing condition of $ u(\cdot,\cdot) $ u (⋅ , ⋅) , which is essential for getting the sparse representations. We develop a new method to avoid using the semigroup property. The analogous results are obtained for stochastic time–space fractional heat equations. As an application, we apply the adaptive Fourier decomposition to establish sparse representations of the solutions to the concerned equations. [ABSTRACT FROM AUTHOR]

Cauchy–Pompeiu and Bochner–Martinelli–Koppelman integral representation formulas in Clifford and complex analysis are related to Euclidean Dirac operator D , operator ∂ ¯ , and two associated kernels. The article analyzes couples (D , Φ) , where D is a homogeneous first-order differential operator on R n , n ≥ 2 , with constant coefficients in a Banach algebra A , and Φ : R n \ { 0 } → A is a smooth kernel homogeneous of degree 1 - n . Operator-kernel couples have integral representation formulas, which in addition to expected components include remainders. The two main results identify operator-kernel couples that yield Cauchy–Pompeiu and Bochner–Martinelli–Koppelman formulas. The formulas are used to derive quantitative Hartogs–Rosenthal theorems for such couples. The concepts of Dirac and semi-Dirac pairs of differential operators point out the role played by Laplace operators as part of the general approach. [ABSTRACT FROM AUTHOR]

The theory of fractal surfaces, in its basic setting, asserts the existence of a bivariate continuous function defined on a triangular domain. The extant literature on the construction of fractal surfaces over triangular domains use certain assumptions for the construction and deal primarily with the interpolation aspects. Working in the framework of fractal surfaces over triangular domains, this note has a two-fold target. Firstly, to revamp the existing constructions of fractal surfaces over triangular domains, and secondly to connect the idea of fractal surfaces further with the theory of approximation. To this end, in the same spirit of the so-called α -fractal functions on intervals and hyperrectangles, we study a salient subclass of fractal surfaces, which provides a parameterized family of bivariate fractal functions corresponding to a fixed continuous function defined on a triangular domain. Some elementary properties of the single-valued (linear and nonlinear) and multi-valued fractal operators associated with the α -fractal function formalism of the bivariate fractal functions on a triangular domain are recorded. A fractal approximation process, that is a sequence of single-valued fractal operators converging strongly to the identity operator on C (Δ , R) , the space of all real-valued continuous functions defined on a triangular domain Δ , is obtained. An approximation class of fractal functions, referred to as the fractal polynomials, is hinted at. The notion of α -fractal function and associated single-valued fractal operator in conjunction with appropriate stability results for Schauder bases provide Schauder bases consisting of self-referential functions for C (Δ , R) . [ABSTRACT FROM AUTHOR]

The paper proposes a general quasi-interpolation scheme for high-dimensional function approximation. To facilitate error analysis, we view our quasi-interpolation as a two-step procedure. In the first step, we approximate a target function by a purpose-built convolution operator (with an error term referred to as convolution error). In the second step, we discretize the underlying convolution operator using certain quadrature rule at the given sampling data sites (with an error term called discretization error). The final approximation error is obtained as an optimally balanced sum of these two errors, which in turn views our quasi-interpolation as a regularization technique that balances convolution error and discretization error. As a concrete example, we construct a sparse grid quasi-interpolation scheme for high-dimensional function approximation. Both theoretical analysis and numerical implementations provide evidence that our quasi-interpolation scheme is robust and is capable of mitigating the curse of dimensionality for approximating high-dimensional functions. [ABSTRACT FROM AUTHOR]

The primary objective of this research article is to introduce and study an approximation operator involving the Tricomi function by using Korovkin’s theorem and a conventional method based on the modulus of continuity. In Lipschitz-type spaces, we demonstrate the rate of convergence, and we are also able to determine the convergence properties of our operators. In addition, we illustrate the convergence of our proposed operators using various graphs and error-estimating tables for numerical instances. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with Triebel–Lizorkin–Morrey spaces E u , p , q s (Ω) of positive smoothness s defined on (special or bounded) Lipschitz domains Ω ⊂ R d as well as on R d . For those spaces we prove new equivalent characterizations in terms of local oscillations which hold as long as some standard conditions on the parameters are fulfilled. As a byproduct, we also obtain novel characterizations of E u , p , q s (Ω) using differences of higher order. Special cases include standard Triebel–Lizorkin spaces F p , q s (Ω) and hence classical L p -Sobolev spaces H p s (Ω) . [ABSTRACT FROM AUTHOR]

In this paper, we introduce the nonlinear Meyer-König and Zeller operators based on q-integers. Firstly, we describe the q-Meyer-König and Zeller operators of max-product type. Then, we give an error estimation for the q-Meyer-König and Zeller operators of max-product kind by using a modulus of continuity. [ABSTRACT FROM AUTHOR]

We introduce the novel concept of integral Read–Bajraktarević (iRB) operator and discuss some of its properties. We show that this iRB operator generalizes the known Read–Bajraktarević (RB) operator and we derive conditions for the fixed point of the iRB operator to belong to certain function spaces. [ABSTRACT FROM AUTHOR]

Interpolation-based methods are well-established and effective approaches for the efficient generation of accurate reduced-order surrogate models. Common challenges for such methods are the automatic selection of good or even optimal interpolation points and the appropriate size of the reduced-order model. An approach that addresses the first problem for linear, unstructured systems is the iterative rational Krylov algorithm (IRKA), which computes optimal interpolation points through iterative updates by solving linear eigenvalue problems. However, in the case of preserving internal system structures, optimal interpolation points are unknown, and heuristics based on nonlinear eigenvalue problems result in numbers of potential interpolation points that typically exceed the reasonable size of reduced-order systems. In our work, we propose a projection-based iterative interpolation method inspired by IRKA for generally structured systems to adaptively compute near-optimal interpolation points as well as an appropriate size for the reduced-order system. Additionally, the iterative updates of the interpolation points can be chosen such that the reduced-order model provides an accurate approximation in specified frequency ranges of interest. For such applications, our new approach outperforms the established methods in terms of accuracy and computational effort. We show this in numerical examples with different structures. [ABSTRACT FROM AUTHOR]

In this paper we consider a new class of RBF (Radial Basis Function) neural networks, in which smoothing factors are replaced with shifts. We prove under certain conditions on the activation function that these networks are capable of approximating any continuous multivariate function on any compact subset of the d-dimensional Euclidean space. For RBF networks with finitely many fixed centroids we describe conditions guaranteeing approximation with arbitrary precision. [ABSTRACT FROM AUTHOR]

The degree of approximation (in Hölder metric) of a periodic function belonging to a specific Hölder class by its Abel–Poisson means of Fourier series and of the conjugate function by the conjugate Abel–Poisson means of the conjugate series of the Fourier series is obtained. [ABSTRACT FROM AUTHOR]

In this article, we construct a generalized Kantorovich operator based on Hermite polynomials of three variable to approximate integrable functions. For the operator, we derive the moment generating function (m.g.f.) and a few direct findings associated with m.g.f. We continue by examining the quantitative asymptotic formula and convergence in relation to the modulus of continuity and pace at which our operator converges has shown graphically. Then we capture new discrete type operator based on composition of operators and studied its local convergence and a Voronovskaja theorem in weighted space and some direct results. [ABSTRACT FROM AUTHOR]

Parameter space reduction has been proved to be a crucial tool to speed-up the execution of many numerical tasks such as optimization, inverse problems, sensitivity analysis, and surrogate models’ design, especially when in presence of high-dimensional parametrized systems. In this work we propose a new method called local active subspaces (LAS), which explores the synergies of active subspaces with supervised clustering techniques in order to carry out a more efficient dimension reduction in the parameter space. The clustering is performed without losing the input–output relations by introducing a distance metric induced by the global active subspace. We present two possible clustering algorithms: K-medoids and a hierarchical top–down approach, which is able to impose a variety of subdivision criteria specifically tailored for parameter space reduction tasks. This method is particularly useful for the community working on surrogate modelling. Frequently, the parameter space presents subdomains where the objective function of interest varies less on average along different directions. So, it could be approximated more accurately if restricted to those subdomains and studied separately. We tested the new method over several numerical experiments of increasing complexity, we show how to deal with vectorial outputs, and how to classify the different regions with respect to the LAS dimension. Employing this classification technique as a preprocessing step in the parameter space, or output space in case of vectorial outputs, brings remarkable results for the purpose of surrogate modelling. [ABSTRACT FROM AUTHOR]

In this paper, we study convergence of a family of Neural Network operators of the type Kantorovich with sigmoidal activation functions. Such operators are multivariate. We study the problem of convergence in the setting of Lebesgue space with variable exponent L p (·) (R) with 1 ≤ p - ≤ p (x) ≤ p + < + ∞ , ∀ x ∈ R . Also, the pointwise and uniform convergence for functions belonging to suitable spaces are proved. In particular, our results apply to L p -spaces. Multivariate Neural Network approximation finds applications, typically in neurocomputing processes. [ABSTRACT FROM AUTHOR]

The set of smooth functions is not dense in Morrey spaces. To address the density issue in Morrey spaces, Zorko spaces are defined by utilizing the difference of a function of first order. In this paper, we propose a subspace of Morrey spaces which is defined using the difference of a function of second order. Approximation properties in the new subspace are investigated and the relation with Zorko spaces is studied via properties of smoothness spaces. [ABSTRACT FROM AUTHOR]

The aim of present article is to introduce the Szász-integral type sequences of operators in terms of Hermite polynomials and gamma function. Further, we calculate some estimates at test functions and central moments. Moreover, we discuss uniform convergence theorem and order of approximation via Korovkin theorem and first order modulus of smoothness respectively. Next, we study pointwise approximation results in view of Peetre's K-functional, second order modulus of smoothness and Lipschitz type space. Lastly, bivariate version of these sequences of operators are introduced. Moreover, their rate of convergence and order of approximation are investigated. [ABSTRACT FROM AUTHOR]

In this study, we take into account the of modified Szász–Mirakyan–Kantorovich operators to obtain their rate of convergence using the modulus of continuity and for the functions in Lipschitz space. Then, we obtain the statistical convergence of this form. In addition, we determine the weighted statistical convergence and compare it with the statistical one for the same operator. Medical applications and traditional mathematics; one way to get a close approximation of the Riemann integrable functions is through the use of the Kantorovich modification of positive linear operators. The use of Kantorovich operators is tremendously helpful from a medical point of view. Their application is shown as an approximation of the rate of convergence in respect of modulus of continuity. [ABSTRACT FROM AUTHOR]

We provide a novel approach to approximate bounded Lipschitz domains via a sequence of smooth, bounded domains. The flexibility of our method allows either inner or outer approximations of Lipschitz domains which also possess weakly defined curvatures, namely, domains whose boundary can be locally described as the graph of a function belonging to the Sobolev space W 2 , q for some q ≥ 1 . The sequences of approximating sets is also characterized by uniform isocapacitary estimates with respect to the initial domain Ω . [ABSTRACT FROM AUTHOR]

We calculate the rate of convergence of the double rational Fourier series for regular, bounded, measurable, and two-variable functions. The rectangular oscillation of the two-variable function is used to quantify this rate. Additionally, we give an approximation of convergence rate of the double rational Fourier series for continuous functions with generalized bounded variation. [ABSTRACT FROM AUTHOR]

This study simulates the coronavirus infection (Covid-19) which is given as a mathematical model based on a set of ordinary differential equations. We provide this numerical treatment and simulation by using the finite element method (FEM). The proposed model is reduced to a set of algebraic equations using the FEM. The goal of this research is to stop and slow the spread of a sickness that is ravaging the globe. A susceptible person may also transfer immediately to the quarantined class after the exposed person has been quarantined or moved to one of the contaminated classes. This model was employed by the researchers to account for both asymptomatic and symptomatic infected people. The obtained findings are compared to those results by using the Runge–Kutta method of the fourth order (RK4). Finally, we calculate the residual error function of the approximation to validate the FEM. [ABSTRACT FROM AUTHOR]

In this paper, we propose an approximation technique to compute multiple integrals of a non-negative real continuous function over a hyper-rectangle Ω of ℝn. The main idea is to use a reducing transformation procedure obtained by using α-dense curves. First, the region Ωf whose measure represents the value of the integral, is densified by a specific curve ℓα(t) of finite length. Therefore, the multiple integral can be approached by a simple integral corresponding to ℓα (t). Some numerical examples are given.

This article explores a Chlodowsky-type extension of Szász operators using the general-Appell polynomials. The convergence properties of these operators are established by employing the universal Korovkin-type property, and the order of approximation is determined using the classical modulus of continuity. Additionally, the weighted B -statistical convergence and statistically weighted B -summability properties of the operators are derived. Theoretical results are supported by numerical and graphical examples. [ABSTRACT FROM AUTHOR]

The present paper deals with the study of the approximation properties of the well-known sampling Kantorovich (SK) operators in "Sobolev-like settings". More precisely, a convergence theorem in case of functions belonging to the usual Sobolev spaces for the SK operators has been established. In order to get such a result, suitable Strang-Fix type conditions have been required on the kernel functions defining the above sampling type series. As a consequence, certain open problems related to the convergence in variation for the SK operators have been solved. Then, we considered the above operators in a fractional-type setting. It is well-known that, in the literature, several notions of fractional Sobolev spaces are available, such as, the Gagliardo Sobolev spaces (GSs) defined by means of the Gagliardo semi-norm, or the weak Riemann-Liouville Sobolev spaces (wRLSs) defined by the weak (left and right) Riemann-Liouville fractional derivatives and so on. Here, in order to face the above convergence problem, we introduced a new definition of fractional Sobolev spaces, that we called the tight fractional Sobolev spaces (tfSs) and generated as the intersection of the GSs and the symmetric Sobolev spaces (i.e., that given by the intersection of the left and the right wRLSs). In the latter setting, we obtain one of the main results of the paper, that is a convergence theorem for the SK operators with respect to a suitable norm on tfSs. [ABSTRACT FROM AUTHOR]

Sleeve functions are generalizations of the well-established ridge functions that play a major role in the theory of partial differential equation, medical imaging, statistics, and neural networks. Where ridge functions are non-linear, univariate functions of the distance to hyperplanes, sleeve functions are based on the squared distance to lower-dimensional manifolds. The present work is a first step to study general sleeve functions by starting with sleeve functions based on finite-length curves. To capture these curve-based sleeve functions, we propose and study a two-step method, where first the outer univariate function—the profile—is recovered, and second, the underlying curve is represented by a polygonal chain. Introducing a concept of well-separation, we ensure that the proposed method always terminates and approximates the true sleeve function with a certain quality. Investigating the local geometry, we study an inexact version of our method and show its success under certain conditions. [ABSTRACT FROM AUTHOR]

In recent years, the modified Extended Direct Algebraic Method (mEDAM) has demonstrated to be an effective method for finding novel soliton solutions to nonlinear Fractional Partial Differential Equations that appear in the fields of science and engineering. In this study, mEDAM is used to explore special soliton solutions for the (3+1)-Fractional Modified Zakharov–Kuznetsov Equation (FMZKE) arising in electrical engineering which is first mathematically modeled through the implementation of Kirchhoff's Law to the nonlinear electrical transmission line circuit. The wave behaviours of various soliton solutions are graphically represented using three-dimensional (3D) graphs which provide a clear and thorough explanation of the usefulness and high performance of the suggested method. The acquired results offer helpful insights to the behavior and dynamics of the FMZKE, leading to a more deeply comprehending of the model and its applications in various fields. [ABSTRACT FROM AUTHOR]

In the present paper we study the perturbed sampling Kantorovich operators in the general context of the modular spaces. After proving a convergence result for continuous functions with compact support, by using both a modular inequality and a density approach, we establish the main result of modular convergence for these operators. Further, we show several instances of modular spaces in which these results can be applied. In particular, we show some applications in Musielak–Orlicz spaces and in Orlicz spaces and we also consider the case of a modular functional that does not have an integral representation generating a space, which can not be reduced to previous mentioned ones. [ABSTRACT FROM AUTHOR]

A novel approach to zipper fractal interpolation theory for functions of several variables is presented. Multivariate zipper fractal functions are constructed and then perturbed through free choices of base functions, scaling functions, and a binary matrix called signature to obtain their zipper α-fractal versions. In particular, we propose a multivariate Bernstein zipper fractal function and study its coordinate-wise monotonicity which depends on the values of signature. We derive bounds for the graph of a multivariate zipper fractal function by imposing conditions on the scaling factors and the Hölder exponent of the associated germ function and base function. The box dimension result for multivariate Bernstein zipper fractal function is derived. Finally, we study some constrained approximation properties for multivariate zipper Bernstein fractal functions. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a vector-valued fractal interpolation function in a more general setting by using the Rakotch fixed point theory and the iterated function system. We also show the existence of the Borel probability measure, widely known as a fractal measure, supported on the graph of this vector-valued fractal interpolation function. We obtain the Hausdorff dimension and the box-counting dimension of the graph of this more general fractal function using some function spaces. We obtain bounds for the fractal dimension of the graph of the Riemann-Liouville fractional integral of this fractal function. Using our techniques, we also calculate the fractal dimensions of the graphs of some fractal interpolation functions. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of the fractal interpolation function corresponding to a general data set using the Rakotch contraction theory and iterated function system. We also prove the existence of the fractal measure supported on the graph of the fractal interpolation function. We emphasize the fact that our theory covers the fractal interpolation theory for finite cases, countably infinite cases, and many more. We establish dimensional results for the graph of the fractal interpolation function for the general data sets. [ABSTRACT FROM AUTHOR]

We study approximation properties of multi-resolution analyses in the context of matrix-valued function spaces. Here, we generalize the notions of approximation order and density order given by the reference [de Boor C, DeVore RA, Ron A. Approximation from shift-invariant subspaces of L 2 (R d). Trans Am Math Soc. 1994;341(2):787–806]. Indeed, we prove a characterization of the closed subspaces generated by the shifts of a single matrix-valued function that provide approximation order and/or density order α ≥ 0. To give our conditions, we need the classical notion of approximate continuity. As a consequence, we prove necessary and sufficient conditions on a matrix-valued function to be a scaling function in a multi-resolution analysis. [ABSTRACT FROM AUTHOR]

Here we examine the multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or ℝN , N ∈ ℕ, by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last four types. These approximations are derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Fréchet derivatives. Our multivariate operators are defined by using a multidimensional density function induced by a hyperbolic tangent like sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.

We show that the Peetre K -functional between the space L p with 0 < p < 1 and the corresponding smooth function space W p ψ generated by the Weyl-type differential operator ψ (D) , where ψ is a homogeneous function of any positive order, is identically zero. The proof of the main results is based on the properties of the de la Vallée Poussin kernels and the quadrature formulas for trigonometric polynomials and entire functions of exponential type. [ABSTRACT FROM AUTHOR]

A polar coordinate transformation is considered, which transforms the complex geometries into a unit disc. Some basic properties of the polar coordinate transformation are given. As applications, we consider the elliptic equation in two-dimensional complex geometries. The existence and uniqueness of the weak solution are proved, the Fourier–Legendre spectral-Galerkin scheme is constructed and the optimal convergence of numerical solutions under H 1 -norm is analyzed. The proposed method is very effective and easy to implement for problems in 2D complex geometries. Numerical results are presented to demonstrate the high accuracy of our spectral-Galerkin method. [ABSTRACT FROM AUTHOR]

Abstract Here, we present multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or $${\mathbb {R}}^{N},$$ R N , $$N\in {\mathbb {N}},$$ N ∈ N , by the multivariate normalized, quasi-interpolation, Kantorovich-type and quadrature-type neural network operators. We examine also the case of approximation by iterated operators of the last four types. These approximations are achieved by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high-order Fréchet derivatives. Our multivariate operators are defined using a multidimensional density function induced by the Richards’s curve, which is a generalized logistic function. The approximations are pointwise, uniform and $$L_{p}.$$ L p . The related feed-forward neural network is with one hidden layer.

A multi-domain Galerkin method is presented for the Burgers equation, whose solutions behave differently at positive infinity and negative infinity. The original problem is transformed into a problem with general variable coefficients and homogeneous boundary conditions at infinities by using the auxiliary function, which is capable of being reformulated as a suitable variational formulation that is the most convenient for analysing numerical error. As an application, the composite spectral scheme with numerical integration is provided for the underlying problem on the whole line, which is easy to deal with problems concerning general variable coefficients. Some composite generalized Laguerre–Legendre interpolations on the whole line are established. The convergence and stability of the proposed algorithm are proved. Numerical results show the efficiency of this approach and agree well with the theoretical analysis. [ABSTRACT FROM AUTHOR]