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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.

We consider the numerical evaluation of a class of double integrals with respect to a pair of self-similar measures over a self-similar fractal set (the attractor of an iterated function system), with a weakly singular integrand of logarithmic or algebraic type. In a recent paper (Gibbs et al. Numer. Algorithms 92 , 2071–2124 2023), it was shown that when the fractal set is “disjoint” in a certain sense (an example being the Cantor set), the self-similarity of the measures, combined with the homogeneity properties of the integrand, can be exploited to express the singular integral exactly in terms of regular integrals, which can be readily approximated numerically. In this paper, we present a methodology for extending these results to cases where the fractal is non-disjoint but non-overlapping (in the sense that the open set condition holds). Our approach applies to many well-known examples including the Sierpinski triangle, the Vicsek fractal, the Sierpinski carpet, and the Koch snowflake.

Based on Chebyshev spectral collocation and numerical techniques for handling highly oscillatory integrals, we propose a numerical method for a class of highly oscillatory Volterra integral equations frequently encountered in engineering applications. Specifically, we interpolate the unknown function at Chebyshev points, and substitute these points into the integral equation, resulting in a system of linear equations. The highly oscillatory integrals are treated using either the numerical steepest descent method or the Filon-Clenshaw-Curtis method. Additionally, we derive an error estimation formula for this method using error analysis techniques and validate the convergence and effectiveness of the proposed approach through numerical examples.

We present a method for computing nearly singular integrals that occur when single or double layer surface integrals, for harmonic potentials or Stokes flow, are evaluated at points nearby. Such values could be needed in solving an integral equation when one surface is close to another or to obtain values at grid points. We replace the singular kernel with a regularized version having a length parameter δ in order to control discretization error. Analysis near the singularity leads to an expression for the error due to regularization which has terms with unknown coefficients multiplying known quantities. By computing the integral with three choices of δ , we can solve for an extrapolated value that has regularization error reduced to O ( δ 5 ) , uniformly for target points on or near the surface. In examples with δ / h constant and moderate resolution, we observe total error about O ( h 5 ) close to the surface. For convergence as h → 0 , we can choose δ proportional to h q with q < 1 to ensure the discretization error is dominated by the regularization error. With q = 4 / 5 , we find errors about O ( h 4 ) . For harmonic potentials, we extend the approach to a version with O ( δ 7 ) regularization; it typically has smaller errors, but the order of accuracy is less predictable.

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.

We establish weighted Hermite-Hadamard-type inequalities for some classes of differentiable functions without assuming any symmetry property on the weight function. Next, we apply our obtained results to the approximation of some classes of weighted integrals.

This paper deals with the computation of the Hankel transform by means of the sinc rule applied after a special exponential transformation. An error analysis, particularly suitable for meromorphic functions, together with the parameter selection strategy, is considered. A prototype algorithm for automatic integration is also presented.

Simplices are the fundamental domain when integrating over convex polytopes. The aim of this work is to establish a novel framework of Monte Carlo integration over simplices, throughout from sampling to variance reduction. Namely, we develop a uniform sampling method on the standard simplex consisting of two independent procedures and construct theories on change of measure on each of the two independent elements in the developed sampling technique with a view towards variance reduction by importance sampling. We provide illustrative figures and numerical results to support our theoretical findings and demonstrate the strong potential of the developed framework for effective implementation and acceleration of Monte Carlo integration over simplices.

Objectives In this paper, a numerical scheme is designed for solving singularly perturbed Fredholm integro-differential equation. The scheme is constructed via the exact (non-standard) finite difference method to approximate the differential part and the composite Simpson’s 1/3 rule for the integral part of the equation. Result The stability and uniform convergence analysis are demonstrated using solution bound and the truncation error bound. For three model examples, the maximum absolute error and the rate of convergence for different values of the perturbation parameter and mesh size are tabulated. The computational result shows, the proposed method is second-order uniformly convergent which is in a right agreement with the theoretical result.