Generalized Jacobi method for linear bounded operators system
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Generalized Jacobi method for linear bounded operators system Samir Lemita1 · Hamza Guebbai1 · M. Z. Aissaoui1
Received: 5 June 2017 / Revised: 14 December 2017 / Accepted: 16 December 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
Abstract In this paper, we construct a generalization of the Jacobi iterative method adapted to a system of linear bounded operators. This new method is used with product integration method to solve a linear Fredholm integral equation of the second kind with a weakly singular kernel. The convergence analysis is proved. The numerical tests developed show its effectiveness. Keywords Fredholm equation of the second kind · Weakly singular kernel · Jacobi method · Bounded operators matrix · Product integration method Mathematics Subject Classification 45B05 · 65F10 · 65J10
1 Introduction Numerical approximation of linear functional problems as differential or integral equations conventionally begins with discretization methods, that leads to linear systems, where the size of the matrices obtained depends on the order of convergence. Consequently, to get a small error of the approximate solutions of differential or integral equations, we solve a huge system. Most of the time, this latter system cannot be solved directly , but there
Communicated by Michele Benzi.
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Hamza Guebbai [email protected] Samir Lemita [email protected] M. Z. Aissaoui [email protected]
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Laboratoire de Mathématiques Appliquées et de Modélisation, Faculté de Mathématiques et de l’Informatique et des Sciences de la Matiére, Université 8 Mai 1945 Guelma, B.P. 401, Guelma 24000, Algeria
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are many iterative methods that should be used as the Jacobi or Gauss–Seidel iterative scheme to approach their solutions. However, to confirm the convergence of the latter iterative schemes, the corresponding matrix should verify the row strictly diagonally dominant property (Saad 2003). Many authors (for example Salkuyeh 2007; Dafchahi 2008; Vatti and Eneyew 2011; Vatti and Gonfa 2011; Laskar and Behera 2014) developed other methods called generalized Jacobi (Gauss–Seidel), refinement of Jacobi (Gauss–Seidel), and refinement of generalized Jacobi (Gauss–Seidel), respectively. The results show that methods are more efficient than conventional Jacobi (Gauss–Seidel) method. Also, other techniques have been developed by Li and Sun (2000), Zhang et al. (2005), Zou and Jiang (2011). Finally, we point out that all the previous methods were adapted to linear algebraic systems. In this paper, we construct a generalization of the Jacobi iterative method adapted to a system of linear bounded operators as the following form: ⎧ λu 1 = T11 u 1 + T12 u 2 + · · · + T1N u N + f 1 , ⎪ ⎪ ⎪ λu = T u + T u + · · · + T u + f , ⎨ 2 21 1 22 2 2N N 2 () .. .. ⎪ ⎪ . . ⎪ ⎩ λu N = TN 1 u 1 + TN 2 u 2 + · · · + TN N u N + f N , where Ti j 1≤i, j≤N is a family of bounded operators. To show the effectiveness of this new version of generalization Jacobi method, we apply this method to solve o
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