Reproducing kernel method for a class of weakly singular Fredholm integral equations

Alvandi, Azizallah; Paripour, Mahmoud

doi:10.1080/16583655.2018.1474841

Cited by 9 publications

(7 citation statements)

References 16 publications

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“…The accuracy of the RKHSM for the problem is controllable. Many scientific properties of the RKHSM can be seen in [30][31][32][33][34][35][36][37][38][39][40].…”

Section: Numerical Experimentsmentioning

confidence: 99%

Reproducing kernel Hilbert space method for the solutions of generalized Kuramoto–Sivashinsky equation

Akgül

Bonyah

2019

Journal of Taibah University for Science

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Reproducing kernel Hilbert space method is given for the solution of generalized Kuramoto-Sivashinsky equation. Reproducing kernel functions are obtained to get the solution of the generalized Kuramoto-Sivashinsky equation. Two examples have been introduced to prove the accuracy of the method. The obtained results show that the reproducing kernel Hilbert space method gives approximate analytical solutions which are very close to the exact solution of the generalized Kuramoto-Sivashinsky equation, which demonstrates the power of the proposed technique. We prove the efficiency of the reproducing kernel Hilbert space method in this paper.

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“…The accuracy of the RKHSM for the problem is controllable. Many scientific properties of the RKHSM can be seen in [30][31][32][33][34][35][36][37][38][39][40].…”

Section: Numerical Experimentsmentioning

confidence: 99%

Reproducing kernel Hilbert space method for the solutions of generalized Kuramoto–Sivashinsky equation

Akgül

Bonyah

2019

Journal of Taibah University for Science

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“…The mentioned scheme above is an efficient method of solving nonlinear equations [31][32][33]. However, in implementing this algorithm on a computer, { ( )} ∞ =1 is not quite orthogonal, due to rounding errors.…”

Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

Application of Reproducing Kernel Hilbert Space Method for Solving a Class of Nonlinear Integral Equations

Javan

Abbasbandy

Araghi

2017

Mathematical Problems in Engineering

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A new approach based on the Reproducing Kernel Hilbert Space Method is proposed to approximate the solution of the second-kind nonlinear integral equations. In this case, the Gram-Schmidt process is substituted by another process so that a satisfactory result is obtained. In this method, the solution is expressed in the form of a series. Furthermore, the convergence of the proposed technique is proved. In order to illustrate the effectiveness and efficiency of the method, four sample integral equations arising in electromagnetics are solved via the given algorithm.

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“…In this study, we focus on the numerical solution of the second kind of Fredholm equations with another type of kernel's singularity, which needs another technique different from those for the singular logarithmic kernels of Fredholm equations of the first kind. Many methods for solving weakly singular Fredholm integral equations of the second kind have been published [14][15][16][17][18][19][20][21][22][23][24][25]. For example, Yin et al [14], used the Jacobi-Gauss quadrature formula to approximate the integral operator in the numerical implementation of the spectral collocation method and established the spectral Chebyshev collocation method for solving Fredholm integral equations of the second kind with the weakly singular kernel.…”

Section: Introductionmentioning

confidence: 99%

Computational method for solving weakly singular Fredholm integral equations of the second kind using an advanced barycentric Lagrange interpolation formula

Shoukralla

Saber

Sayed

2021

Adv. Model. and Simul. in Eng. Sci.

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In this study, we applied an advanced barycentric Lagrange interpolation formula to find the interpolate solutions of weakly singular Fredholm integral equations of the second kind. The kernel is interpolated twice concerning both variables and then is transformed into the product of five matrices; two of them are monomial basis matrices. To isolate the singularity of the kernel, we developed two techniques based on a good choice of different two sets of nodes to be distributed over the integration domain. Each set is specific to one of the kernel arguments so that the kernel values never become zero or imaginary. The significant advantage of thetwo presented techniques is the ability to gain access to an algebraic linear system equivalent to the interpolant solution without applying the collocation method. Moreover, the convergence in the mean of the interpolant solution and the maximum error norm estimation are studied. The interpolate solutions of the illustrated four examples are found strongly converging uniformly to the exact solutions.

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Reproducing kernel method for a class of weakly singular Fredholm integral equations

Cited by 9 publications

References 16 publications

Reproducing kernel Hilbert space method for the solutions of generalized Kuramoto–Sivashinsky equation

Reproducing kernel Hilbert space method for the solutions of generalized Kuramoto–Sivashinsky equation

Application of Reproducing Kernel Hilbert Space Method for Solving a Class of Nonlinear Integral Equations

Computational method for solving weakly singular Fredholm integral equations of the second kind using an advanced barycentric Lagrange interpolation formula

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