Solving Cauchy integral equations of the first kind by the Adomian decomposition method

Bougoffa, Lazhar; Mennouni, Abdelaziz; Rach, Randolph

doi:10.1016/j.amc.2012.10.046

Cited by 15 publications

(9 citation statements)

References 21 publications

(9 reference statements)

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“…These problems appeared in the theory of airfoils in fluid mechanics [1,3,11,20]. The programs have been provided by Mathematica 8.…”

Section: Numerical Illustrationsmentioning

confidence: 99%

“…Also, Wazwaz in [21] and Delves in [2] used this method for solving Fredholm and Volterra integral equations of the first kind. In [1], Bougoffa et al applied the regularization method with parameter to solve a given Cauchy integral equations of the first kind.…”

Section: Introductionmentioning

confidence: 99%

Fibonacci-regularization method for solving Cauchy integral equations of the first kind

Araghi

Noeiaghdam

2017

Ain Shams Engineering Journal

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In this paper, a novel scheme is proposed to solve the first kind Cauchy integral equation over a finite interval. For this purpose, the regularization method is considered. Then, the collocation method with Fibonacci base function is applied to solve the obtained second kind singular integral equation. Also, the error estimate of the proposed scheme is discussed. Finally, some sample Cauchy integral equations stem from the theory of airfoils in fluid mechanics are presented and solved to illustrate the importance and applicability of the given algorithm. The tables in the examples show the efficiency of the method. Ó 2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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“…The Adomian decomposition method (ADM) is one of iterative and applicable methods for solving various problems such as the Klein-Gordon equation [29], Triki-Biswas equation [30], the problem of boundary layer convective heat transfer [31], integral equations (IEs) of the first and second kinds with hypersingular kernels [32,33], the Volterra integral form of the Lane-Emden equations with initial values and boundary conditions [34], Cauchy IEs of the first kind [35], linear and nonlinear IEs [36] and partial differential equations [37].…”

Section: Introductionmentioning

confidence: 99%

The Numerical Validation of the Adomian Decomposition Method for Solving Volterra Integral Equation with Discontinuous Kernels Using the CESTAC Method

Noeiaghdam¹,

Sidorov²,

Wazwaz³

et al. 2021

Mathematics

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The aim of this paper is to present a new method and the tool to validate the numerical results of the Volterra integral equation with discontinuous kernels in linear and non-linear forms obtained from the Adomian decomposition method. Because of disadvantages of the traditional absolute error to show the accuracy of the mathematical methods which is based on the floating point arithmetic, we apply the stochastic arithmetic and new condition to study the efficiency of the method which is based on two successive approximations. Thus the CESTAC method (Controle et Estimation Stochastique des Arrondis de Calculs) and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library are employed. Finding the optimal iteration of the method, optimal approximation and the optimal error are some of advantages of the stochastic arithmetic, the CESTAC method and the CADNA library in comparison with the floating point arithmetic and usual packages. The theorems are proved to show the convergence analysis of the Adomian decomposition method for solving the mentioned problem. Also, the main theorem of the CESTAC method is presented which shows the equality between the number of common significant digits between exact and approximate solutions and two successive approximations.This makes in possible to apply the new termination criterion instead of absolute error. Several examples in both linear and nonlinear cases are solved and the numerical results for the stochastic arithmetic and the floating-point arithmetic are compared to demonstrate the accuracy of the novel method.

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“…Indeed, Abel's integral equation is one of the most famous equations that frequently appear in many engineering problems and physical properties such as heat conduction, semiconductors, chemical reactions, and metallurgy (see, e.g., [6,7]). Besides, over the past few years, many numerical methods for solving Abel's integral equation have been developed, such as collocation methods [8], product integration methods [9,10], fractional multistep methods [11][12][13], methods based on wavelets [14][15][16], backward Euler methods [9], Adomian decomposition method [17], and Tau approximation method [18].…”

Section: Introductionmentioning

confidence: 99%

Solution of Singular Integral Equations via Riemann–Liouville Fractional Integrals

Alqudah

Mohammed

Abdeljawad

2020

Mathematical Problems in Engineering

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In this attempt, we introduce a new technique to solve main generalized Abel’s integral equations and generalized weakly singular Volterra integral equations analytically. This technique is based on the Adomian decomposition method, Laplace transform method, and Ψ-Riemann–Liouville fractional integrals. Finally, some examples are proposed and they illustrate the rapidness of our new technical method.

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“…Mennouni and Guedjiba (2011) have also used the successive approximation method to solve the airfoil equation. The Adomain decomposition method was also used after regularization to solve airfoil equation by Bougoffa et al (2013). Setia (2014) has proposed a Bernstein polynomial-based method to solve the classical airfoil equation.…”

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confidence: 99%

Approximate solution of first kind singular integral equation with generalized kernel using Legendre multiwavelets

2019

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In this study, numerical solution of a class of Cauchy type singular integral equations of the first kind with generalized kernels (CSIEFKGK) is obtained using Legendre multiwavelets (LMW). Instead of using complex function theory, the appropriate weight function of the solution is obtained by applying LMW-based numerical scheme. We evaluate the matrix representation of generalized kernel with and without the weight factor using recurrence relations and some elementary methods. Using the multiscale representation of the integral operator, the equation is converted into a system of linear equations. The numerical solution of CSIEFKGK is obtained by solving this systems. Finally, a number of examples including applications of crack problems are considered to illustrate the efficiency of the method developed here. Keywords Cauchy type singular integral equation of the first kind • Generalized kernel • Legendre multiwavelets • Wavelet Galerkin method • Multiscale approximation Mathematics Subject Classification 65R20 • 65D15 • 65T60 • 45E05 • 42C40 1 Introduction Singular integral equation (SIE) arises in various areas of mathematical physics and engineering. The Cauchy type singular integral equation of the first kind with the generalized kernel (CSIEFKGK) arises from the formulation of a large class of mixed BVPs in mathe-Communicated by Hui Liang.

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Solving Cauchy integral equations of the first kind by the Adomian decomposition method

Cited by 15 publications

References 21 publications

Fibonacci-regularization method for solving Cauchy integral equations of the first kind

The Numerical Validation of the Adomian Decomposition Method for Solving Volterra Integral Equation with Discontinuous Kernels Using the CESTAC Method

Solution of Singular Integral Equations via Riemann–Liouville Fractional Integrals

Approximate solution of first kind singular integral equation with generalized kernel using Legendre multiwavelets

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