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Solving linear Fredholm integro-differential equation by Nyström method

Tair Boutheina, Guebbai Hamza, Segni Sami, Ghiat Mourad

Journal of Applied Mathematics and Computational Mechanics

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2021

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Vol. 20, nr 3

53--64

EN

The study of the solution’s existence and uniqueness for the linear integro-differential 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 C1 [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.

2

A new optimization method based on Perry's idea through the use of the matrix power

Hafaidia Imane, Benrabia Noureddine, Ghiat Mourad, Guebbai Hamza

Journal of Applied Mathematics and Computational Mechanics

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2021

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Vol. 20, nr 4

29--41

EN

The purpose of this paper is to present a new conjugate gradient method for solving unconstrained nonlinear optimization problems, based on Perry’s idea. An accelerated adaptive algorithm is proposed, where our search direction satisfies the sufficient descent condition. The global convergence is analyzed using the spectral analysis. The numerical results are described for a set of standard test problems, and it is shown that the performance of the proposed method is better than that of the CG-DESCENT, the mBFGS and the SPDOC.

3

Solving a nonlinear Volterra-Fredholm integro-differential equation with weakly singular kernels

Touati Sami, Lemita Samir, Ghiat Mourad, Aissaoui Mohamed-Zine

Fasciculi Mathematici

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2019

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nr 62

155--168

EN

The aim of the current work is to investigate the numerical study of an integro-differential nonlinear Volterra-Fredholm equation with a weakly singular kernels. Our approximation technique is based on the product integration method in conjunction with an iterative scheme. The existence and uniqueness of the solution have been proved. We conclude the paper with numerical examples to illustrate the effectiveness of our method.