A projection method for solving Cauchy singular integro-differential equations

Mennouni, Abdelaziz

doi:10.1016/j.aml.2011.11.012

Cited by 14 publications

(11 citation statements)

References 3 publications

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“…The main idea of [10] is to propose a collocation method for solving singular integro-differential equations with logarithmic kernel using airfoil polynomials. The goal of [9] is to numerically solve the Cauchy integro-differential equations using the projection method based on the Legendre polynomials. In this paper, we introduce three methods to solve two classes of integral equations of the second kind.…”

Section: Introductionmentioning

confidence: 99%

Three methods to solve two classes of integral equations of the second kind

2022

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Three methods to solve two classes of integral equations of the second kind are introduced in this paper. Firstly, two Kantorovich methods are proposed and examined to numerically solving an integral equation appearing from mathematical modeling in biology. We use a sequence of orthogonal nite rank projections. The rst method is based on general grid projections. The second one is established by using the shifted Legendre polynomials. We establish a new convergence analysis and we prove the associated theorems. Secondly, a new Nystrom method is introduced for solving Fredholm integral equation of the second kind.

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Section: Introductionmentioning

confidence: 99%

Three methods to solve two classes of integral equations of the second kind

2022

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“…Therefore, an approximation and numerical processing is necessary. Most of the equations found, used and studied are the equations where the derivative of the unknown function is outside the integral sign, and scientists have built several methods to estimate the solution like the Homotopy perturbation method [5], Fractional order operational matrix methods the Adomain decomposition method [6] and another method using Legendre scaling functions [7,8], Embedded pseudo-Runge-Kutta methods [9], Volterra-Runge-Kutta method [10], the iterative variational method [11] and the projection methods [12].…”

Section: Introductionmentioning

confidence: 99%

Solving linear Fredholm integro-differential equation by Nyström method

Tair¹,

Guebbai²,

Segni³

et al. 2021

J. Appl. Math. Comput. Mech.

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The study of the solution's existence and uniqueness for the linear integrodifferential Fredholm equation and the application of the Nyström method to approximate the solution is what we will present in this paper. We use the Neumann theorem to construct a sufficient condition that ensures the solution's existence and uniqueness of our problem in the Banach space C 1 [a, b]. We have applied the Nyström method based on the trapezoidal rule to avoid adding other conditions in order to the approximation method's convergence. The Nyström method discretizes the integro-differential equation into solving a linear system. Only with the existence and uniqueness condition, we show the solution's existence and uniqueness of the linear system and the convergence of the numerical solution to the exact solution in infinite norm sense. We present two theorems to give a good estimate of the error. Also, to show the efficiency and accuracy of the Nyström method, some numerical examples will be provided at the end of this work.

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“…Integral equation perform role effectively in many fields of science and engineering. Recently, there are a lot of orthonormal basis function that have been used to find an approximate solution, mention Fourier functions [2], Legendre polynomials [21] and wavelets [10,12,13,16,17,19,20,26]. Although, the wavelet bases are one of the most interesting basis, especially for large scale problems, in which the kernel can be constituted as sparse matrix.…”

Section: Introductionmentioning

confidence: 99%

A new class of fredholm integral equations of the second kind with non symmetric kernel: solving by wavelets method

Mennouni¹,

Ramdani²,

Zennir

2021

bspm

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In this paper, we present an ecient modication of the wavelets method to solve a new class of Fredholm integral equations of the second kind with non symmetric kernel. This -analytical method based on orthonormal wavelet basis, as a consequence three systems are obtained, a Toeplitz system and two systems with condition number close to 1. Since the preconditioned conjugate gradient normal equation residual (CGNR) and preconditioned conjugate gradient normal equation error (CGNE) methods are applicable, we can solve the systems in O(2n log(n)) operations, by using the fast wavelet transform and the fast Fourier transform.

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A projection method for solving Cauchy singular integro-differential equations

Cited by 14 publications

References 3 publications

Three methods to solve two classes of integral equations of the second kind

Three methods to solve two classes of integral equations of the second kind

Solving linear Fredholm integro-differential equation by Nyström method

A new class of fredholm integral equations of the second kind with non symmetric kernel: solving by wavelets method

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