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Journal articles on the topic 'Weakly singular kernel'

Author: Grafiati

Published: 4 June 2021

Last updated: 14 June 2025

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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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11

Brewer, Dennis W., and Robert K. Powers. "Parameter identification in a Volterra equation with weakly singular kernel." Journal of Integral Equations and Applications 2, no. 3 (1990): 353–73. http://dx.doi.org/10.1216/jiea/1181075568.

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12

Usmonov, Botir, and Quvvatali Rakhimov. "Vibration analysis of airfoil on hereditary deformable suspensions." E3S Web of Conferences 97 (2019): 06006. http://dx.doi.org/10.1051/e3sconf/20199706006.

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This paper describes the analyses of the nonlinear vibrations and dynamic stability of an airfoil on hereditary-deformable suspensions. The model is based on two-degree-of-freedom structure in geometrically nonlinear statements. It provides justification for the choice of the weakly singular Abelian type kernel, with rheological parameters. To solve problems of viscoelastic system with weakly singular kernels of relaxation, a numerical method has been used, based on quadrature formulae. With a combination of the Galerkin and the presented method, problems of nonlinear vibrations and dynamic stability in viscoelastic two-degree-of-freedom structure have been solved. A comparison of the results obtained via this method is also presented. In all problems, the convergence of the Galerkin method has been investigated. The implications of material viscoelasticity on vibration and dynamic stability are presented graphically.

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13

Al-Humedi, Hameeda Oda. "The Reproducing Kernel Hilbert Space Method for Solving System of Linear Weakly Singular Volterra Integral Equations." JOURNAL OF ADVANCES IN MATHEMATICS 15 (November 14, 2018): 8070–80. http://dx.doi.org/10.24297/jam.v15i0.7869.

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The exact solutions of a system of linear weakly singular Volterra integral equations (VIE) have been a difficult to find. The aim of this paper is to apply reproducing kernel Hilbert space (RKHS) method to find the approximate solutions to this type of systems. At first, we used Taylor's expansion to omit the singularity. From an expansion the given system of linear weakly singular VIE is transform into a system of linear ordinary differential equations (LODEs). The approximate solutions are represent in the form of series in the reproducing kernel space . By comparing with the exact solutions of two examples, we saw that RKHS is a powerful, easy to apply and full efficiency in scientific applications to build a solution without linearization and turbulence or discretization.

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14

Guebbai, Hamza, and Laurence Grammont. "A new degenerate kernel method for a weakly singular integral equation." Applied Mathematics and Computation 230 (March 2014): 414–27. http://dx.doi.org/10.1016/j.amc.2013.12.102.

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15

Kaneko, Hideaki, Richard Noren, and Yuesheng Xu. "Regularity of the solution of Hammerstein equations with weakly singular kernel." Integral Equations and Operator Theory 13, no. 5 (1990): 660–70. http://dx.doi.org/10.1007/bf01732317.

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16

Banerjea, Sudeshna, Barnali Dutta, and A. Chakrabarti. "Solution of Singular Integral Equations Involving Logarithmically Singular Kernel with an Application in a Water Wave Problem." ISRN Applied Mathematics 2011 (May 12, 2011): 1–16. http://dx.doi.org/10.5402/2011/341564.

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A direct function theoretic method is employed to solve certain weakly singular integral equations arising in the study of scattering of surface water waves by vertical barriers with gaps. Such integral equations possess logarithmically singular kernel, and a direct function theoretic method is shown to produce their solutions involving singular integrals of similar types instead of the stronger Cauchy-type singular integrals used by previous workers. Two specific ranges of integration are examined in detail, which involve the following: Case(i) two disjoint finite intervals (0,a)∪(b,c) and (a,b,c being finite ) and Case(ii) a finite union of n disjoint intervals. The connection of such integral equations for Case(i), with a particular water wave scattering problem, is explained clearly, and the important quantities of practical interest (the reflection and transmission coefficients) are determined numerically by using the solution of the associated weakly singular integral equation.

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17

Alvandi, Azizallah, and Mahmoud Paripour. "The Combined Reproducing Kernel Method and Taylor Series for Solving Weakly Singular Fredholm Integral Equations." International Journal of Advances in Applied Sciences 5, no. 3 (2016): 109. http://dx.doi.org/10.11591/ijaas.v5.i3.pp109-117.

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<p>In this paper, a numerical method is proposed for solving weakly singular Fredholm integral equations in Hilbert reproducing kernel space (RKHS). The Taylor series is used to remove singularity and reproducing kernel function are used as a basis. The effectiveness and stability of the numerical scheme is illustrated through two numerical examples.</p>

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18

Diaz, Katharine Perkins. "The Szego Kernel as a Singular Integral Kernel on a Family of Weakly Pseudoconvex Domains." Transactions of the American Mathematical Society 304, no. 1 (1987): 141. http://dx.doi.org/10.2307/2000708.

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19

Diaz, Katharine Perkins. "The Szegő kernel as a singular integral kernel on a family of weakly pseudoconvex domains." Transactions of the American Mathematical Society 304, no. 1 (1987): 141. http://dx.doi.org/10.1090/s0002-9947-1987-0906810-4.

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20

Wei, Yunxia, and Yanping Chen. "Convergence Analysis of the Spectral Methods for Weakly Singular Volterra Integro-Differential Equations with Smooth Solutions." Advances in Applied Mathematics and Mechanics 4, no. 1 (2012): 1–20. http://dx.doi.org/10.4208/aamm.10-m1055.

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AbstractThe theory of a class of spectral methods is extended to Volterra integro-differential equations which contain a weakly singular kernel (t - s)->* with 0< μ <1. In this work, we consider the case when the underlying solutions of weakly singular Volterra integro-differential equations are sufficiently smooth. We provide a rigorous error analysis for the spectral methods, which shows that both the errors of approximate solutions and the errors of approximate derivatives of the solutions decay exponentially inL°°-norm and weightedL2-norm. The numerical examples are given to illustrate the theoretical results.

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21

Shao, Jing. "New Integral Inequalities with Weakly Singular Kernel for Discontinuous Functions and Their Applications to Impulsive Fractional Differential Systems." Journal of Applied Mathematics 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/252946.

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Some new integral inequalities with weakly singular kernel for discontinuous functions are established using the method of successive iteration and properties of Mittag-Leffler function, which can be used in the qualitative analysis of the solutions to certain impulsive fractional differential systems.

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22

Shoukralla, E. S., and M. A. Markos. "The economized monic Chebyshev polynomials for solving weakly singular Fredholm integral equations of the first kind." Asian-European Journal of Mathematics 13, no. 01 (2018): 2050030. http://dx.doi.org/10.1142/s1793557120500308.

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This paper presents a numerical method for solving a certain class of Fredholm integral equations of the first kind, whose unknown function is singular at the end-points of the integration domain, and has a weakly singular logarithmic kernel with analytical treatments of the singularity. To achieve this goal, the kernel is parametrized, and the unknown function is assumed to be in the form of a product of two functions; the first is a badly-behaved known function, while the other is a regular unknown function. These two functions are approximated by using the economized monic Chebyshev polynomials of the same degree, while the given potential function is approximated by monic Chebyshev polynomials of the same degree. Further, the two parametric functions associated to the parametrized kernel are expanded into Taylor polynomials of the first degree about the singular parameter, and an asymptotic expression is created, so that the obtained improper integrals of the integral operator become convergent integrals. Thus, and after using a set of collocation points, the required numerical solution is found to be equivalent to the solution of a linear system of algebraic equations. From the illustrated example, it turns out that the proposed method minimizes the computational time and gives a high order accuracy.

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23

Wu, Qinghua. "The Approximate Solution of Fredholm Integral Equations with Oscillatory Trigonometric Kernels." Journal of Applied Mathematics 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/172327.

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A method for approximating the solution of weakly singular Fredholm integral equation of the second kind with highly oscillatory trigonometric kernel is presented. The unknown function is approximated by expansion of Chebychev polynomial and the coefficients are determinated by classical collocation method. Due to the highly oscillatory kernels of integral equation, the discretised collocation equation will give rise to the computation of oscillatory integrals. These integrals are calculated by using recursion formula derived from the fundamental recurrence relation of Chebyshev polynomial. The effectiveness and accuracy of the proposed method are tested by numerical examples.

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24

Yang, Yin. "JACOBI SPECTRAL GALERKIN METHODS FOR VOLTERRA INTEGRAL EQUATIONS WITH WEAKLY SINGULAR KERNEL." Bulletin of the Korean Mathematical Society 53, no. 1 (2016): 247–62. http://dx.doi.org/10.4134/bkms.2016.53.1.247.

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25

Dutkiewicz, Aldona. "On the functional-integral equation of Volterra type with weakly singular kernel." Publications de l'Institut Math?matique (Belgrade) 83, no. 97 (2008): 57–63. http://dx.doi.org/10.2298/pim0897057d.

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We give sufficient conditions for the existence of Lp-solution of a Volterra functional-integral equation in a Banach space. Our assumptions and proofs are expressed in terms of measures of noncompactness.

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26

Mastroianni, Giuseppe, and Siegfried Prössdorf. "A Quadrature Method for Cauchy Integral Equations with Weakly Singular Perturbation Kernel." Journal of Integral Equations and Applications 4, no. 2 (1992): 205–28. http://dx.doi.org/10.1216/jiea/1181075682.

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27

Lima, Pedro, and Teresa Diogo. "An extrapolation method for a Volterra integral equation with weakly singular kernel." Applied Numerical Mathematics 24, no. 2-3 (1997): 131–48. http://dx.doi.org/10.1016/s0168-9274(97)00016-0.

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28

Biazar, Jafar. "FD-RBF for Partial Integro-Differential Equations with a Weakly Singular Kernel." Applied and Computational Mathematics 4, no. 6 (2015): 445. http://dx.doi.org/10.11648/j.acm.20150406.17.

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29

Jumarhon, B., and M. Pidcock. "On a Nonlinear Volterra Integro-differential Equation with a Weakly Singular Kernel." ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 76, no. 6 (1996): 357–60. http://dx.doi.org/10.1002/zamm.19960760611.

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30

Bahşi, M. Mustafa, Mehmet Çevik, and Mehmet Sezer. "Jacobi polynomial solutions of Volterra integro-differential equations with weakly singular kernel." New Trends in Mathematical Science 3, no. 6 (2018): 24–38. http://dx.doi.org/10.20852/ntmsci.2018.291.

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31

Alvandi, Azizallah, and Mahmoud Paripour. "Reproducing kernel method for a class of weakly singular Fredholm integral equations." Journal of Taibah University for Science 12, no. 4 (2018): 409–14. http://dx.doi.org/10.1080/16583655.2018.1474841.

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32

Chen, Zhong, and YingZhen Lin. "The exact solution of a linear integral equation with weakly singular kernel." Journal of Mathematical Analysis and Applications 344, no. 2 (2008): 726–34. http://dx.doi.org/10.1016/j.jmaa.2008.03.023.

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33

Shi, Guodong, Yanlei Gong, and Mingxu Yi. "Alternative Legendre Polynomials Method for Nonlinear Fractional Integro-Differential Equations with Weakly Singular Kernel." Journal of Mathematics 2021 (July 9, 2021): 1–13. http://dx.doi.org/10.1155/2021/9968237.

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In this paper, we present a numerical scheme for finding numerical solution of a class of weakly singular nonlinear fractional integro-differential equations. This method exploits the alternative Legendre polynomials. An operational matrix, based on the alternative Legendre polynomials, is derived to be approximated the singular kernels of this class of the equations. The operational matrices of integration and product together with the derived operational matrix are utilized to transform nonlinear fractional integro-differential equations to the nonlinear system of algebraic equations. Furthermore, the proposed method has also been analyzed for convergence, particularly in context of error analysis. Moreover, results of essential numerical applications have also been documented in a graphical as well as tabular form to elaborate the effectiveness and accuracy of the proposed method.

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34

Dutkiewicz, Aldona, and Stanislaw Szufla. "Kneser’s theorem for weak solutions of an integral equation with weakly singular kernel." Publications de l'Institut Math?matique (Belgrade) 77, no. 91 (2005): 87–92. http://dx.doi.org/10.2298/pim0591087d.

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35

Xu, Hongbin Chen and Da. "A Compact Difference Scheme for an Evolution Equation with a Weakly Singular Kernel." Numerical Mathematics: Theory, Methods and Applications 5, no. 4 (2012): 559–72. http://dx.doi.org/10.4208/nmtma.2012.m11032.

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36

Chen, Yunxia Wei and Yanping. "A Spectral Method for Neutral Volterra Integro-Differential Equation with Weakly Singular Kernel." Numerical Mathematics: Theory, Methods and Applications 6, no. 2 (2013): 424–46. http://dx.doi.org/10.4208/nmtma.2013.1125nm.

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37

Huang, Li, Yu Lin Zhao, and Liang Tang. "Solving the Volterra Integral Equations with Weakly Singular Kernel by Taylor Expansion Methods." Applied Mechanics and Materials 220-223 (November 2012): 2129–32. http://dx.doi.org/10.4028/www.scientific.net/amm.220-223.2129.

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In this paper, we propose a Taylor expansion method for solving (approximately) linear Volterra integral equations with weakly singular kernel. By means of the nth-order Taylor expansion of the unknown function at an arbitrary point, the Volterra integral equation can be converted approximately to a system of equations for the unknown function itself and its n derivatives. This method gives a simple and closed form solution for the integral equation. In addition, some illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method.

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38

Zhu, Li, and Yanxin Wang. "Numerical solutions of Volterra integral equation with weakly singular kernel using SCW method." Applied Mathematics and Computation 260 (June 2015): 63–70. http://dx.doi.org/10.1016/j.amc.2015.03.065.

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39

Benrabia, Noureddine, and Hamza Guebbai. "On the regularization method for Fredholm integral equations with odd weakly singular kernel." Computational and Applied Mathematics 37, no. 4 (2018): 5162–74. http://dx.doi.org/10.1007/s40314-018-0625-3.

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40

Adolfsson, Klas, Mikael Enelund, and Stig Larsson. "Adaptive discretization of an integro-differential equation with a weakly singular convolution kernel." Computer Methods in Applied Mechanics and Engineering 192, no. 51-52 (2003): 5285–304. http://dx.doi.org/10.1016/j.cma.2003.09.001.

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41

Zaeri, Sedigheh, Habibollah Saeedi, and Mohammad Izadi. "Fractional integration operator for numerical solution of the integro-partial time fractional diffusion heat equation with weakly singular kernel." Asian-European Journal of Mathematics 10, no. 04 (2017): 1750071. http://dx.doi.org/10.1142/s1793557117500711.

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In this paper, an approximate solution for solving weakly singular kernel partial integro-differential equations with time fractional order is proposed. The method is based on using a second-order time difference approximation followed by applying the fractional integral operator and piecewise linear interpolation to compute the singularity of the kernel that appear in the discretization process. The stability of the method is also considered in the sense of von Neumann analysis. Numerical examples are solved to demonstrate the validity and applicability of the presented technique.

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42

Alvandi, Azizallah, Mahmoud Paripour, and Lutz Angermann. "The combined reproducing kernel method and Taylor series to solve nonlinear Abel’s integral equations with weakly singular kernel." Cogent Mathematics 3, no. 1 (2016): 1250705. http://dx.doi.org/10.1080/23311835.2016.1250705.

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43

McLean, W., and V. Thomée. "Numerical solution of an evolution equation with a positive-type memory term." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 35, no. 1 (1993): 23–70. http://dx.doi.org/10.1017/s0334270000007268.

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AbstractWe study the numerical solution of an initial-boundary value problem for a Volterra type integro-differential equation, in which the integral operator is a convolution product of a positive-definite kernel and an elliptic partial-differential operator. The equation is discretised in space by the Galerkin finite-element method and in time by finite differences in combination with various quadrature rules which preserve the positive character of the memory term. Special attention is paid to the case of a weakly singular kernel. Error estimates are derived and numerical experiments reported.

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44

Kim, Chang Ho, and U. Jin Choi. "Spectral collocation methods for a partial integro-differential equation with a weakly singular kernel." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 39, no. 3 (1998): 408–30. http://dx.doi.org/10.1017/s0334270000009474.

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AbstractWe propose and analyze the spectral collocation approximation for the partial integro-differential equations with a weakly singular kernel. The space discretization is based on the pseudo-spectral method, which is a collocation method at the Gauss-Lobatto quadrature points. We prove unconditional stability and obtain the optimal error bounds which depend on the time step, the degree of polynomial and the Sobolev regularity of the solution.

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45

Zhang, Yiao Yong, and Hua Feng Wu. "Convergence Analysis of the Spectral Method for Second-Kind Volterra Integral Equations with a Weakly Singular Kernel." Applied Mechanics and Materials 263-266 (December 2012): 3313–16. http://dx.doi.org/10.4028/www.scientific.net/amm.263-266.3313.

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The Legendre spectral Galerkin method for Volterra integral equations of the second kind with a weakly singular kernel is proposed in this paper. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors (in the L2 -norm and the L∞ -norm ) will decay exponentially provided that the source function is sufficiently smooth. Numerical examples are given to illustrate the theoretical results.

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46

Tang, Tao. "A finite difference scheme for partial integro-differential equations with a weakly singular kernel." Applied Numerical Mathematics 11, no. 4 (1993): 309–19. http://dx.doi.org/10.1016/0168-9274(93)90012-g.

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47

Chen, Zhong, and Wei Jiang. "The exact solution of a class of Volterra integral equation with weakly singular kernel." Applied Mathematics and Computation 217, no. 18 (2011): 7515–19. http://dx.doi.org/10.1016/j.amc.2011.02.059.

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48

Kulkarni, Rekha P., and Akshay S. Rane. "Asymptotic expansions for approximate eigenvalues of integral operators with a weakly singular periodic kernel." International Journal of Convergence Computing 2, no. 1 (2016): 79. http://dx.doi.org/10.1504/ijconvc.2016.080402.

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49

Chen, C., V. Thom{ée, and L. B. Wahlbin. "Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel." Mathematics of Computation 58, no. 198 (1992): 587. http://dx.doi.org/10.1090/s0025-5718-1992-1122059-2.

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50

Wang, Yulan, Temuer Chaolu, and Zhong Chen. "Using reproducing kernel for solving a class of singular weakly nonlinear boundary value problems." International Journal of Computer Mathematics 87, no. 2 (2010): 367–80. http://dx.doi.org/10.1080/00207160802047640.

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