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Journal articles on the topic 'Weakly singular kernel'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 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 eq

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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

Darania, Parviz, Bahaa Hussain Alrikabi, and Saeed Pishbin. "Development of the Nystr\"{o}m Method for Weakly Singular Functional Integral Equations." European Journal of Pure and Applied Mathematics 18, no. 2 (2025): 5704. https://doi.org/10.29020/nybg.ejpam.v18i2.5704.

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In this research, we apply the standard product integration method (Nystr\"{o}m method) for solving the delay nonlinear weakly singular Volterra integral equations. Typically, in weakly singular integral equations, the singularity of the kernel leads to the derivatives of the solution becoming singular at the boundary of the domain. The Chelyshkov polynomials serving as orthogonal polynomials, find application in numerical integration. Here, we use roots of these polynomials to make Lagrange interpolating polynomial for approximating the kernel functions in weakly singular functional integral

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5

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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6

Dhunde, Ranjit R. "Double Laplace Transform Method for Solving Fractional Fourth-Order Partial Integro-Differential Equations with Weakly Singular Kernel." Indian Journal Of Science And Technology 17, no. 36 (2024): 3712–18. http://dx.doi.org/10.17485/ijst/v17i36.2005.

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Objectives: To investigates the solutions of fourth order partial integro-differential equations with high-order non-integer derivatives and weakly singular kernels. Methods: Weakly singular kernels present challenges in both analytical and numerical treatments due to their intricate behaviour near singular points. In this article, we introduce a novel approach utilizing the double Laplace transform method to effectively address these challenges. Findings: By solving a series of precise and understandable examples, the double Laplace transform clearly transforms the fractional partial integro-

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7

Bazine, Imane, and Samir Lemita. "Fredholm integro-differential equation with weak singularities: a combined analytical and numerical study." STUDIES IN ENGINEERING AND EXACT SCIENCES 5, no. 2 (2024): e9901. http://dx.doi.org/10.54021/seesv5n2-413.

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This study is dedicated to the analytical and numerical investigation of a nonlinear Fredholm integro-differential equation, specifically one that is characterized by weakly singular kernels. The primary challenge addressed is the weak singularity in the kernel, which complicates the solution process. To tackle this, we employ a combination of two techniques: the Nyström method and the product integration method. The Nyström method is used for approximating the solution to the Fredholm integro-differential equation, while the product integration technique is applied to handle the singular beha

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8

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 th

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9

Ranjit, R. Dhunde. "Double Laplace Transform Method for Solving Fractional Fourth-Order Partial Integro-Differential Equations with Weakly Singular Kernel." Indian Journal of Science and Technology 17, no. 36 (2024): 3712–18. https://doi.org/10.17485/IJST/v17i36.2005.

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Abstract <strong>Objectives:</strong> To investigates the solutions of fourth order partial integro-differential equations with high-order non-integer derivatives and weakly singular kernels. <strong>Methods:</strong> Weakly singular kernels present challenges in both analytical and numerical treatments due to their intricate behaviour near singular points. In this article, we introduce a novel approach utilizing the double Laplace transform method to effectively address these challenges. <strong>Findings:</strong> By solving a series of precise and understandable exam

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10

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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11

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 sin

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12

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 s

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13

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/quasifa

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14

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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15

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 st

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16

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 solution

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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

Al-Khaled, Kamel, Isam Al-Darabsah, Amer Darweesh, and Amro Alshare. "Analytical and Numerical Treatment of Evolutionary Time-Fractional Partial Integro-Differential Equations with Singular Memory Kernels." Fractal and Fractional 9, no. 6 (2025): 392. https://doi.org/10.3390/fractalfract9060392.

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Evolution equations with fractional-time derivatives and singular memory kernels are used for modeling phenomena exhibiting hereditary properties, as they effectively incorporate memory effects into their formulation. Time-fractional partial integro-differential equations (FPIDEs) represent a significant class of such evolution equations and are widely used in diverse scientific and engineering fields. In this study, we use the sinc-collocation and iterative Laplace transform methods to solve a specific FPIDE with a weakly singular kernel. Specifically, the sinc-collocation method is applied t

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19

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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20

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

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21

Sedka, Ilyes, Ammar Khellaf, Samir Lemita, and Mahammed Zine Aissaoui. "New algorithm on linearization-discretization solving systems of nonlinear integro-differential Fredholm equations." Boletim da Sociedade Paranaense de Matemática 42 (April 19, 2024): 1–17. http://dx.doi.org/10.5269/bspm.63480.

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This article deals with a new strategy for solving a certain type of nonlinear integro-differential Fredholm equations with a weakly singular kernel. We build our new algorithm starting with the linearization phase using Newton's iterative process, then with the discretization phase we apply the Kantorovich's projection method. The discretized linear scheme will be approximated by the product integration method in the weak singular terms, and the other regular integrals will be approximated by the Nyström method. The process of convergence of our new algorithm is carried out under certain pred

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22

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 illus

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23

Zhou, Haopan, Jun Zhou, and Hongbin Chen. "The Optimal L2-Norm Error Estimate of a Weak Galerkin Finite Element Method for a Multi-Dimensional Evolution Equation with a Weakly Singular Kernel." Fractal and Fractional 9, no. 6 (2025): 368. https://doi.org/10.3390/fractalfract9060368.

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This paper proposes a weak Galerkin (WG) finite element method for solving a multi-dimensional evolution equation with a weakly singular kernel. The temporal discretization employs the backward Euler scheme, while the fractional integral term is approximated via a piecewise constant function method. A fully discrete scheme is constructed by integrating the WG finite element approach for spatial discretization. L2-norm stability and convergence analysis of the fully discrete scheme are rigorously established. Numerical experiments are conducted to validate the theoretical findings and demonstra

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24

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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25

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. Th

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26

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 polynom

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27

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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28

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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29

Jia, Jinhong, Zhiwei Yang, and Hong Wang. "Analysis of a nonlocal diffusion model with a weakly singular kernel." Applied Mathematics Letters 154 (August 2024): 109103. http://dx.doi.org/10.1016/j.aml.2024.109103.

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30

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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31

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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32

Shoukralla, Emil Sobhi, Alfaisal Abdelhamid Hasan, Sara Abdelhamid, and Ahmed Yehia Sayed. "Approximate Solutions to Weakly Singular Fredholm Integral Equations of the Second Kind using Shifted Legendre Polynomials of the First Kind." Journal of Advanced Research in Applied Sciences and Engineering Technology 50, no. 2 (2024): 76–89. http://dx.doi.org/10.37934/araset.50.2.7689.

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In this research, we presented a simple approach to approximate the second type of linear weakly singular and non-singular Fredholm integral. Shifted Legendre polynomials of the first kind in matrix-vector forms were used to construct the approach. The singularity of the kernel was removed analytically. Theorems regarding the convergence of the estimations of the error norm and the mean were proved. The numerical examples demonstrated the method's uniqueness and precision.

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33

Touati, Sami, Mohamed-Zine Aissaoui, Samir Lemita, and Hamza Guebbai. "Investigation approach for a nonlinear singular Fredholm integro-differential equation." Boletim da Sociedade Paranaense de Matemática 40 (January 30, 2022): 1–11. http://dx.doi.org/10.5269/bspm.46898.

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In this paper, we examine the existence and uniqueness of the solution of nonlinear integro-differential Fredholm equation with a weakly singular kernel. Then, we develop an iterative scheme to approach this solution using the product integration method. Finally, we conclude with a numerical tests to show the effectiveness of the proposed method.

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34

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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35

BEN DAHMANE, Raouia, Khanssa BEN DAHMANE, and Hanane KABOUL. "Extending The Product Integration Method." All Sciences Abstracts 1, no. 4 (2023): 2. http://dx.doi.org/10.59287/as-abstracts.1223.

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We examine a weakly singular kernel Fredholm integral equation of the second kind in1([ ] R) and provide adequate conditions for the solution’s existence and uniqueness. Furthermore, weexpand the product integration technique initially proposed in 1([ ] R) and adapt it to 1([ ][ ] R). To demonstrate its effectiveness, we present numerical evidence along with an application inAstrophysics.

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36

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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37

Zhang, Chao, Zhipeng Liu, Sheng Chen, and DongYa Tao. "New spectral element method for Volterra integral equations with weakly singular kernel." Journal of Computational and Applied Mathematics 404 (April 2022): 113902. http://dx.doi.org/10.1016/j.cam.2021.113902.

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38

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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39

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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40

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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41

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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42

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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43

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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44

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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45

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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46

Ameen, Ismail Gad, Dumitru Baleanu, and Hussien Shafei Hussien. "Efficient method for solving nonlinear weakly singular kernel fractional integro-differential equations." AIMS Mathematics 9, no. 6 (2024): 15819–36. http://dx.doi.org/10.3934/math.2024764.

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<abstract><p>This paper introduced an efficient method to obtain the solution of linear and nonlinear weakly singular kernel fractional integro-differential equations (WSKFIDEs). It used Riemann-Liouville fractional integration (R-LFI) to remove singularities and approximated the regularized problem with a combined approach using the generalized fractional step-Mittag-Leffler function (GFSMLF) and operational integral fractional Mittag matrix (OIFMM) method. The resulting algebraic equations were turned into an optimization problem. We also proved the method's accuracy in approxima

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47

Emil Sobhi Shoukralla, Nermin Abdelsatar Saber, and Ahmed Yehia Sayed. "The Numerical Solutions of weakly singular Fredholm integral equations of the Second kind Using Chebyshev Polynomials of the Second Kind." Journal of Advanced Research in Applied Sciences and Engineering Technology 44, no. 1 (2024): 22–30. http://dx.doi.org/10.37934/araset.44.1.2230.

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In this study, the second kind Chebyshev Polynomials were utilized to acquire interpolated solutions for the second kind Fredholm integral equations with weakly singular kernel. To accomplish this, the data, unknown, and kernel functions were converted into matrix form, and consequently we completely isolated the singularity of the kernel. The primary benefit of this method is the ability to change the form of integral equation to an equivalent algebraic system, which is easier to solve. The effectiveness of our technique was evaluated by applying it to three illustrated examples, and it was o

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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 report

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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. Furthe

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Ibrahim, Zahraa A., and Nabaa N. Hasan. "Approximation Solution of Fuzzy Singular Volterra Integral Equation by Non-Polynomial Spline." Ibn AL-Haitham Journal For Pure and Applied Sciences 36, no. 1 (2023): 407–14. http://dx.doi.org/10.30526/36.1.2860.

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A non-polynomial spline (NPS) is an approximation method that relies on the triangular and polynomial parts, so the method has infinite derivatives of the triangular part of the NPS to compensate for the loss of smoothness inherited by the polynomial. In this paper, we propose polynomial-free linear and quadratic spline types to solve fuzzy Volterra integral equations (FVIE) of the 2nd kind with the weakly singular kernel (FVIEWSK) and Abel's type kernel. The linear type algorithm gives four parameters to form a linear spline. In comparison, the quadratic type algorithm gives five parameters t

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