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In this paper, using operational rules, we derive some explicit expansion formulas associated with Brenke type polynomials. In particular, the corresponding connection and linearization coefficients are expressed either in terms of double sums or integral representations or sometimes in closed hypergeometric terms. The derived results are applied to Generalized Gould–Hopper and q-Laguerre polynomials. Some well-known duplication and convolution formulas are deduced as particular cases. [ABSTRACT FROM AUTHOR]

The accurate calculation of reaction rates from experimental data is crucial for understanding and characterizing chemical processes. However, the presence of noise in experimental data can introduce errors in rate calculations. In this study, we introduced a novel approach that utilizes the Legendre series expansion method to directly derive smooth reaction rates from noisy experimental data, eliminating the need for numerical differentiation methods. This approach proves to be highly effective in handling noisy thermogravimetric analysis (TGA) data obtained from the thermal decomposition of specific polymers. We demonstrated the robustness and reliability of this method and provided Gnu Octave codes as a free alternative to MATLAB, making the implementation more accessible. Furthermore, the smooth reaction rates obtained were used to evaluate the activation energy using the Friedman isoconversional method. The results showed excellent agreement with those obtained using the Vyazovkin integral method. Additionally, the proposed method can be applied to obtain smooth derivative thermogravimetric (DTG) curves using noisy TGA data set. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of the α -fractal function and fractal approximation for a set-valued continuous map defined on a closed and bounded interval of real numbers. Also, we study some properties of such fractal functions. Further, we estimate the perturbation error between the given continuous function and its α -fractal function. Additionally, we define a new graph of a set-valued function different from the standard graph introduced in the literature and establish some bounds on the fractal dimension of the newly defined graph of some special classes of set-valued functions. Also, we explain the need to define this new graph with examples. In the sequel, we prove that this new graph of an α -fractal function is an attractor of an iterated function system. [ABSTRACT FROM AUTHOR]

In this paper, optimal estimates on the decaying rates of Jacobi expansion coefficients are obtained by fractional calculus for functions with algebraic and logarithmic singularities. This is inspired by the fact that integer-order derivatives fail to deal with singularity of fractional-type, while fractional calculus can. To this end, we first introduce new fractional Sobolev spaces defined as the range of the L p -space under the Riemann-Liouville fractional integral. The connection between these new spaces and classical fractional-order Sobolev spaces is then elucidated. Under this framework, the optimal decaying rate of Jacobi expansion coefficients is obtained, based on which the projection errors under different norms are given. This work is expected to introduce fractional calculus into traditional fields in approximation theory and to explore the possibility in solving classical problems by this ‘new’ tool. [ABSTRACT FROM AUTHOR]

This study is devoted to the numerical investigation of linear and nonlinear hyperbolic telegraph equation. We have proposed a wavelet collocation method based on Legendre polynomials for approximating the solution. Both the spatial and temporal variables, along with their derivatives, are approximated using the Legendre wavelet and its integration. The present approach is simple, consistent and straightforward. To assure the theoretical consistency of the method, an estimate for the upper bound of the error norm is provided. We have proved an exponential order of convergence which is better than the methods available in the literature. Some numerical experiments are carried out to justify the theoretical results and the outcomes confirm the computational efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this manuscript, a new family of (λ , μ )-Bernstein–Kantorovich operators are introduced. A Kovovkin type approximation theorem is obtained, the rate of convergence and A-statistical convergence are investigated by using the modulus of smoothness, Steklov mean and Korovkin-type statistical approximation theorem, graphical representations and numerical examples are also presented to compare the newly defined ones with other forms. Finally, an asymptotic estimate on the rate of convergence for some absolutely continuous functions are derived by using the Bojainc–Cheng decomposition method and some analysis techniques. [ABSTRACT FROM AUTHOR]

Variable-order time fractional Volterra–Fredholm integral partial differential equations with weakly singular kernels are taken into account as results of modeling diverse physical phenomena. In order to determine the approximate solution of this category of functional equation, a matrix collocation method by utilizing the hybrid Bernstein polynomials and block-pulse functions is proffered so that their two-dimensional operational matrices of the integration are derived to avoid the integration operation. Therefore, the speed of computations will increase. Convergence, error estimation, existence and uniqueness of the solution are inspected. The approach discussed is executed on several experimental examples and the results illustrate the effectiveness, applicability and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

Composition detection and semantic line detection are important research topics in computer vision and play an important auxiliary role in the analysis of image esthetics. However, at present, few researchers have considered the internal relationship between these two related tasks for comprehensive research. In order to solve this problem, we propose a synchronous detection network of composition class and semantic lines based on cross-attention, which can realize the mutual supervision and guidance between composition class detection and semantic line detection, to improve the accuracy of each other’s detection. First, the pre-trained composition detection model and the pre-trained semantic line detection model as two teacher models to provide data labels of composition and semantic line information for the student model. Then, we train a student model with the help of the teacher model. The student model adopts the multi-task learning architecture by combining soft and hard parameter sharing, as we propose. At the same time, we develop a cross-attention module to ensure that both tasks get the help and supervision they need from each other. Experimental results show that our method can draw semantic lines while detecting composition classes, which increases the interpretability of composition class detection. Our composition detection accuracy reaches 92.57%, and for benchmark semantic lines, the accuracy of our AUC_A metric can reach 92.00%. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the analysis and detailed study of a procedure to generate both the weighted arithmetic and harmonic means of n positive real numbers. Together with this interpretation, we prove some relevant properties that will allow us to define numerical approximation methods in several dimensions adapted to discontinuities. [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 establish two direct estimates by K-functionals of the rate of approximation by the Kantorovich operators in variable exponent Lebesgue spaces. They extend known results in the non-variable exponent Lebesgue spaces. The approach applied heavily relies on the boundedness of the Hardy–Littlewood maximal operator. [ABSTRACT FROM AUTHOR]

This paper demonstrates the optimality of an interpolation set employed in derivative-free trust-region methods. This set is optimal in the sense that it minimizes the constant of well-poisedness in a ball centred at the starting point. It is chosen as the default initial interpolation set by many derivative-free trust-region methods based on underdetermined quadratic interpolation, including NEWUOA, BOBYQA, LINCOA, and COBYQA. Our analysis provides a theoretical justification for this choice. [ABSTRACT FROM AUTHOR]

This paper presents a numerical method based on an operational matrix of Legendre polynomials for resolving the class of time fractional diffusion (TFD) equations. The operational matrix of fractional order derivatives of the Legendre polynomials is derived as a product of matrices. The collocation method together with the operational matrix of Legendre polynomials are employed to transform the TFD equations into a set of algebraic equations. The perturbation method is applied to show the stability of the discussed method. The accuracy of the suggested method is validated using numerical experiments. The solution obtained by this method is in excellent agreement with the exact solution for the integer order of derivatives and is more precise than the solution obtained by the existing method in which Bernstein polynomials are taken as the basis polynomials. [ABSTRACT FROM AUTHOR]

To solve fractional partial differential equations (FPDEs) under various physical conditions, this study developed a novel method known as the Hermite wavelet method employing the functional integration matrix. The method that has been suggested is based on the Hermite wavelet collocation process. To determine the solution of the fractional differential equations, the Caputo fractional derivative operator of order α ∈ (0 , 1 ] is used. With the use of appropriate grid points, this method converts FPDEs into a system of nonlinear algebraic equations. We achieve a solution by solving these nonlinear algebraic equations by the Newton–Raphson method. Tables and graphs show that the suggested method produces superior results. We provide various illustrative examples to establish the effectiveness of the suggested concept, and the outcomes support the applicability of the suggested strategy. Obtained results are numerically expressed in terms of absolute errors. Finally, convergence analyses are discussed as some theorem with proof. [ABSTRACT FROM AUTHOR]

This study is devoted to designing two hybrid computational algorithms to find approximate solutions for a class of singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters. In our approaches, the time discretization is first performed by the well-known Rothe method and Taylor series procedures, which reduce the underlying model problem into a sequence of boundary value problems (BVPs). Hence, a matrix collocation technique based on novel shifted Delannoy functions (SDFs) is employed to solve each BVP at each time step. We show that our proposed hybrid approximate techniques are uniformly convergent in order O(Δτs+M−12){\mathcal{O}}\left(\Delta {\tau }^{s}+{M}^{-\tfrac{1}{2}}) for s=1,2s=1,2, where Δτ\Delta \tau is the time step and MM is the number of SDFs used in the approximation. Numerical simulations are performed to clarify the good alignment between numerical and theoretical findings. The computational results are more accurate as compared with those of existing numerical values in the literature.

In this study, we explore the effectiveness of elliptic partial differential equations (PDEs) in two and three dimensional space based on Helmholtz equation for simulations of acoustic sound in a very complex environment of propagation. For this purpose, we use an advance and robust numerical technique by utilizing the properties of shifted Chebyshev spectral collocation method. This technique is an extension of the traditional Chebyshev polynomials, incorporating a shift in their argument to enhance flexibility across a wider domain, while retaining an extraordinary numerical characteristic such as orthogonality and spectral convergence making them exceptionally effective in finding the approximate solutions. The exponential order of convergence of the proposed approach is shown both through theoretical and numerical approaches. We provide a number of numerical experiments to verify the theoretical results. The spectral convergence has been substantially enhanced by these numerical examples. The exponential order is further validated by numerical error behaviour in both [Formula: see text] and [Formula: see text] norms.

The main object of this paper is to construct a new class of modified (p , q) -Gamma-type operators. For this new class of operators, in section one, the general moments are find; in section two, the Korovkin-type theorem and some direct results are proved by considering the modulus of continuity and modulus of smoothness and their behavior in Lipschitz-type spaces. In section three, some results in the weighted spaces are given, and in the end, some shape-preserving properties are proven. [ABSTRACT FROM AUTHOR]

Over the several decades, approximating functions in infinite dimensions from samples has gained increasing attention in computational science and engineering, especially in computational uncertainty quantification. This is primarily due to the relevance of functions that are solutions to parametric differential equations in various fields, e.g. chemistry, economics, engineering, and physics. While acquiring accurate and reliable approximations of such functions is inherently difficult, current benchmark methods exploit the fact that such functions often belong to certain classes of holomorphic functions to get algebraic convergence rates in infinite dimensions with respect to the number of (potentially adaptive) samples m. Our work focuses on providing theoretical approximation guarantees for the class of so-called (b , ε) -holomorphic functions, demonstrating that these algebraic rates are the best possible for Banach-valued functions in infinite dimensions. We establish lower bounds using a reduction to a discrete problem in combination with the theory of m-widths, Gelfand widths and Kolmogorov widths. We study two cases, known and unknown anisotropy, in which the relative importance of the variables is known and unknown, respectively. A key conclusion of our paper is that in the latter setting, approximation from finite samples is impossible without some inherent ordering of the variables, even if the samples are chosen adaptively. Finally, in both cases, we demonstrate near-optimal, non-adaptive (random) sampling and recovery strategies which achieve close to same rates as the lower bounds. [ABSTRACT FROM AUTHOR]

Linear approximation methods suffer from Gibbs oscillations when approximating functions with jumps. Essentially non oscillatory subcell-resolution (ENO-SR) is a local technique avoiding oscillations and with a full order of accuracy, but a loss of regularity of the approximant appears. The goal of this paper is to introduce a new approach having both properties of full accuracy and regularity. In order to obtain it, we propose a three-stage algorithm: first, the data is smoothed by subtracting an appropriate non-smooth data sequence; then a chosen high order linear approximation operator is applied to the smoothed data and finally, an approximation with the proper jump or corner (jump in the first order derivative) discontinuity structure is reinstated by correcting the smooth approximation with the non-smooth element used in the first stage. This new procedure can be applied as subdivision scheme to design curves and surfaces both in point-value and in cell-average contexts. Using the proposed algorithm, we are able to construct approximations with high precision, with high piecewise regularity, and without smearing nor oscillations in the presence of discontinuities. These are desired properties in real applications as computer aided design or car design, among others. [ABSTRACT FROM AUTHOR]

In this note, we study some approximation properties on a class of special Lototsky–Bernstein bases. We focus on approximation of | x | on [ − 1 , 1 ] by an approximation process generated from fixed points on Lototsky–Bernstein bases. Our first result shows that the approximation procedure p n (x) to | x | preserves good shapes on [ − 1 , 1 ] . Moreover, some convergence results and inequalities are derived. Our second main result states that the rate convergence of the approximation is O (n − 2) . [ABSTRACT FROM AUTHOR]

We present an effective practical approach for solving multi-order nonlinear fractional differential equations. Our method uses integrated Bernoulli polynomials and comes with a comprehensive convergence analysis. The integrated Bernoulli polynomials are combined with the collocation and simple iteration methods to approximate the solutions. We have provided several numerical examples to demonstrate the effectiveness, strength, and flexibility of our method. The results obtained from implementing the method have been compared with exact solutions and results obtained from other methods mentioned in the articles. [ABSTRACT FROM AUTHOR]

Let u be a moment functional associated with the Hermite, Laguerre, or Jacobi classical orthogonal polynomials. We study approximation by polynomials in H r (u) , the Sobolev space consisting of functions whose derivatives of consecutive orders up to r belong to the L 2 space associated with u . This requires the simultaneous approximation of a function f and its consecutive derivatives up to order N ⩽ r . We explicitly construct orthogonal polynomials that achieve such simultaneous approximation and provide error estimates in terms of E n (f (r)) , the error of best approximation of f (r) in L 2 (u) . [ABSTRACT FROM AUTHOR]

We study the solution set to multivariate Chebyshev approximation problem, focussing on the ill-posed case when the uniqueness of solutions can not be established via strict polynomial separation. We obtain an upper bound on the dimension of the solution set and show that nonuniqueness is generic for ill-posed problems on discrete domains. Moreover, given a prescribed set of points of minimal and maximal deviation we construct a function for which the dimension of the set of best approximating polynomials is maximal for any choice of domain. We also present several examples that illustrate the aforementioned phenomena, demonstrate practical application of our results and propose a number of open questions. [ABSTRACT FROM AUTHOR]

The present paper aims to propose an approximation method of Caputo fractional operator using discretization based on quadrature theory to minimize the error function for an Artificial Neural Network (ANN) with higher convergence rate. In the proposed work authors have verified the suitability of multilayer feed forward ANN architecture to get an estimated solution of fractional order differential equations. The back propagation algorithm and an unsupervised learning, was imposed in order to minimize the error function including the optimizing the network parameters such as weights of synapses and the biases. The algorithm uses truncated power series to replace the unknown function in the fractional differential equations. The novelty of the present work is an approximation of a fractional operator using discretization based on quadrature theory to construct an error function to provide an appropriate estimated solution of the nonlinear fractional differential equations. Four illustrative examples with different orders of nonlinear fractional differential equations are unraveled to approve the validity of the model along with demonstration of effectiveness and fast convergence of the proposed method.

A new numerical approximation method for a class of Gaussian random fields on compact connected oriented Riemannian manifolds is introduced. This class of random fields is characterized by the Laplace–Beltrami operator on the manifold. A Galerkin approximation is combined with a polynomial approximation using Chebyshev series. This so-called Galerkin–Chebyshev approximation scheme yields efficient and generic sampling algorithms for Gaussian random fields on manifolds. Strong and weak orders of convergence for the Galerkin approximation and strong convergence orders for the Galerkin–Chebyshev approximation are shown and confirmed through numerical experiments. [ABSTRACT FROM AUTHOR]

This paper generates a novel approach called the clique polynomial method (CPM) using the clique polynomials raised in graph theory and used for solving the fractional order PDE. The fractional derivative is defined in terms of the Caputo fractional sense and the fractional partial differential equations (FPDE) are converted into nonlinear algebraic equations and collocated with suitable grid points in the current approach. The convergence analysis for the proposed scheme is constructed and the technique proved to be uniformly convegant. We applied the method for solving four problems to justify the proposed technique. Tables and graphs reveal that this new approach yield better results. Some theorems are discussed with proof. [ABSTRACT FROM AUTHOR]

The best polynomial approximation and the Chebyshev approximation are both important in numerical analysis. In tradition, the best approximation is regarded as better than the Chebyshev approximation, because it is usually considered in the uniform norm. However, it is not always superior to the latter, as noticed by Trefethen (Math. Today 47:184–188, 2011) for the function f (x) = | x - 0.25 | . Recently, Wang (arxiv, 2106.03456, 2021) observed a similar phenomenon for a function with an algebraic singularity and proved it in theory. In this paper, we find that for functions with logarithmic endpoint singularities, the pointwise errors of the Chebyshev approximation are smaller than those of the best approximations of the same degree, except only at the narrow boundary layer. The pointwise error for the Chebyshev series truncated at the degree n is O (n - κ) ( κ = min { 2 γ + 1 , 2 δ + 1 } ), but increases as O (n - (κ - 1)) at the endpoint singularities. Theorems are given to explain this effect. [ABSTRACT FROM AUTHOR]

In the given note, we present the Durrmeyer modification of Lupaş type operators having Pochhammer k - symbol. We study convergence, asymptotic result and error of approximation in terms of first order modulus of continuity for these operators. Also, we estimate the approximation behavior of these operators in two variables in terms of complete and partial modulus of continuity. At last, we unveil some numerical examples with graphs to verify our results. [ABSTRACT FROM AUTHOR]