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Relevant bibliographies by topics / Weakly singular kernel

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Journal articles

Academic literature on the topic 'Weakly singular kernel'

Author: Grafiati

Published: 4 June 2021

Last updated: 14 June 2025

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Journal articles on the topic "Weakly singular kernel"

1

Usmonov, Botir. "A Numerical Solution of Hereditary Equations with a Weakly Singular Kernel for Vibration Analysis of Viscoelastic Systems / Vienâdojumu Ar Vâjo Singulâro Kodolu Skaitliskais Risinâjums Iedzimto Viskoelastîgo Sistçmu Vibrâciju Analîzei." Proceedings of the Latvian Academy of Sciences. Section B. Natural, Exact, and Applied Sciences. 69, no. 6 (2015): 326–30. http://dx.doi.org/10.1515/prolas-2015-0048.

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Abstract Viscoelastic, or composite materials that are hereditary deformable, have been characterised by exponential and weakly singular kernels in a hereditary equation. An exponential kernel is easy to be numerically implemented, but does not well describe complex vibratory behaviour of a hereditary deformable system. On the other hand, a weakly singular kernel is known to describe the complex vibratory behaviour, but is nontrivial to be numerically implemented. This study presents a numerical formulation for solving a hereditary equation with a weakly singular kernel. Recursive algebraic equations, which are numerically solvable, are formulated by using the Galerkin method enhanced by a numerical integration and elimination of weak singularity. Numerical experiments showed that the present approach with a weakly singular kernel is well fitted into a realistic vibratory behaviour of a hereditary deformable system under dynamic loads, as compared to the same approach with an exponential kernel.

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2

Bijura, Angelina. "Singularly perturbed Volterra integral equations with weakly singular kernels." International Journal of Mathematics and Mathematical Sciences 30, no. 3 (2002): 129–43. http://dx.doi.org/10.1155/s016117120201325x.

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We consider finding asymptotic solutions of the singularly perturbed linear Volterra integral equations with weakly singular kernels. An interesting aspect of these problems is that the discontinuity of the kernel causes layer solutions to decay algebraically rather than exponentially within the initial (boundary) layer. To analyse this phenomenon, the paper demonstrates the similarity that these solutions have to a special function called the Mittag-Leffler function.

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3

Szufla, Stanisław. "On the Volterra integral equation with weakly singular kernel." Mathematica Bohemica 131, no. 3 (2006): 225–31. http://dx.doi.org/10.21136/mb.2006.134139.

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4

Zheng, Kelong, Wenqiang Feng, and Chunxiang Guo. "Some New Nonlinear Weakly Singular Inequalities and Applications to Volterra-Type Difference Equation." Abstract and Applied Analysis 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/912874.

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Some new nonlinear weakly singular difference inequalities are discussed, which generalize some known weakly singular inequalities and can be used in the analysis of nonlinear Volterra-type difference equations with weakly singular kernel. An application to the upper bound of solutions of a nonlinear difference equation is also presented.

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5

Gao, Jing, and Yao-Lin Jiang. "A periodic wavelet method for the second kind of the logarithmic integral equation." Bulletin of the Australian Mathematical Society 76, no. 3 (2007): 321–36. http://dx.doi.org/10.1017/s0004972700039721.

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A periodic wavelet Galerkin method is presented in this paper to solve a weakly singular integral equations with emphasis on the second kind of Fredholm integral equations. The kernel function, which includes of a smooth part and a log weakly singular part, is discretised by the periodic Daubechies wavelets. The wavelet compression strategy and the hyperbolic cross approximation technique are used to approximate the weakly singular and smooth kernel functions. Meanwhile, the sparse matrix of systems can be correspondingly obtained. The bi-conjugate gradient iterative method is used to solve the resulting algebraic equation systems. Especially, the analytical computational formulae are presented for the log weakly singular kernel. The computational error for the representative matrix is also evaluated. The convergence rate of this algorithm is O (N-p log(N)), where p is the vanishing moment of the periodic Daubechies wavelets. Numerical experiments are provided to illustrate the correctness of the theory presented here.

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6

Aliev, N., and S. Mohammad Hosseini. "A Regularization of Fredholm type singular integral equations." International Journal of Mathematics and Mathematical Sciences 26, no. 2 (2001): 123–28. http://dx.doi.org/10.1155/s0161171201010286.

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We present a method to regularize first and second kind integral equations of Fredholm type with singular kernel. By appropriate application of the Poincaré-Bertrand formula we change such integral equations into a second kind Fredholm's integral equation with at most weakly singular kernel.

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7

Biazar, Jafar. "RBFs for Integral Equations with a Weakly Singular Kernel." American Journal of Applied Mathematics 3, no. 6 (2015): 250. http://dx.doi.org/10.11648/j.ajam.20150306.12.

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8

Avdonin, Sergei, and Luciano Pandolfi. "A linear algorithm for the identification of a weakly singular relaxation kernel using two boundary measurements." Journal of Inverse and Ill-posed Problems 26, no. 2 (2018): 299–310. http://dx.doi.org/10.1515/jiip-2016-0064.

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AbstractWe consider a distributed system of a type which is encountered in the study of diffusion processes with memory and in viscoelasticity. The key feature of such a system is the persistence in the future of the past actions due the memory described via a certain relaxation kernel; see below. The parameters of the kernel have to be inferred from experimental measurements. Our main result in this paper is that by using two boundary measurements, the identification of a relaxation kernel which is a linear combination of Abel kernels (as often assumed in applications) can be reduced to the solution of a (linear) deconvolution problem.

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9

Ghoochani-Shirvan, Rezvan, Jafar Saberi-Nadjafi, and Morteza Gachpazan. "An Analytical and Approximate Solution for Nonlinear Volterra Partial Integro-Differential Equations with a Weakly Singular Kernel Using the Fractional Differential Transform Method." International Journal of Differential Equations 2018 (2018): 1–10. http://dx.doi.org/10.1155/2018/7237680.

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An analytical-approximate method is proposed for a type of nonlinear Volterra partial integro-differential equations with a weakly singular kernel. This method is based on the fractional differential transform method (FDTM). The approximate solutions of these equations are calculated in the form of a finite series with easily computable terms. The analytic solution is represented by an infinite series. We state and prove a theorem regarding an integral equation with a weak kernel by using the fractional differential transform method. The result of the theorem will be used to solve a weakly singular Volterra integral equation later on.

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10

Rehman, Sumaira, Arvet Pedas, and Gennadi Vainikko. "FAST SOLVERS OF WEAKLY SINGULAR INTEGRAL EQUATIONS OF THE SECOND KIND." Mathematical Modelling and Analysis 23, no. 4 (2018): 639–64. http://dx.doi.org/10.3846/mma.2018.039.

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We discuss the bounds of fast solving weakly singular Fredholm integral equations of the second kind with a possible diagonal singularity of the kernel and certain boundary singularities of the derivatives of the free term when the information about the smooth coefficient functions in the kernel and about the free term is restricted to a given number of sample values. In this situation, a fast/quasifast solver is constructed. Thus the complexity of weakly singular integral equations occurs to be close to that of equations with smooth data without singularities. Our construction of fast/quasifast solvers is based on the periodization of the problem.

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