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Papers by rebiha zeghdane
International journal of computing science and mathematics, 2024
Expert Systems with Applications, Feb 29, 2024
There is a growing need of stochastic integral equations (SIEs) to investigate the behavior of co... more There is a growing need of stochastic integral equations (SIEs) to investigate the behavior of complex dynamical systems. Since real-world phenomena frequently dependent on noise sources, modeling them naturally necessitates the use of SIEs. As most SIEs cannot be solved explicitly, thus the behaviors of the studied systems are investigated using approximate solutions of their SIEs. Despite the fact that this problem has been soundly investigated and numerous methods have been presented, the practice demonstrated that obtaining satisfied approximations is not always guaranteed, necessitating the development of new effective techniques. This paper gives a new technique for solving nonlinear Itô-Volterra SIEs by reducing them to linear or nonlinear algebraic systems via the power of a combination of generalized Lagrange functions and Jacobi-Gauss collocation points. The accuracy and reliability of the new technique are evaluated and compared with the existing techniques. Moreover, sufficient conditions to make the estimate error tends to zero are given. The new technique shows surprisingly high efficiency over the existing techniques in terms of computational efficiency and approximation capability. The accuracy of the solution based on the new technique is much higher than that via the existing techniques. The required time of the new technique is much less than that of the existing techniques, where, in some circumstances, the existing techniques take more than 20 times as long as the new technique.
Numerical Algorithms, Dec 28, 2023
The paper introduces an acurrate numerical appraoch based on orthonormal Bernoulli polynomials fo... more The paper introduces an acurrate numerical appraoch based on orthonormal Bernoulli polynomials for solving parabolic partial integro- differential equations (PIDEs). This type of equations arises in physics and engineering. Some operational matrix are given for these polynomials and are also used to obtain the numerical solution. By this approach, the problem is transformed into a nonlinear algebraic system. Convergence analysis is given and some experiment tests are studied to examine the good accuracy of the numerical algorithm, the proposed technique is compared with some other well known methods.
Journal of Computational and Applied Mathematics, Oct 1, 2011
The paper considers the derivation of families of semi-implicit schemes of weak order N = 3.0 (ge... more The paper considers the derivation of families of semi-implicit schemes of weak order N = 3.0 (general case) and N = 4.0 (additive noise case) for the numerical solution of Itô stochastic differential equations. The degree of implicitness of the schemes depends on the selection of N parameters which vary between 0 and 1 and the families contain as particular cases the 3.0 and 4.0 weak order explicit Taylor schemes. Since the implementation of the multiple integrals that appear in these theoretical schemes is difficult, for the applications they are replaced by simpler random variables, obtaining simplified schemes. In this way, for the multidimensional case with one-dimensional noise, we present an infinite family of semi-implicit simplified schemes of weak order 3.0 and for the multidimensional case with additive one-dimensional noise, we give an infinite family of semi-implicit simplified schemes of weak order 4.0. The mean-square stability of the 3.0 family is analyzed, concluding that, as in the deterministic case, the stability behavior improves when the degree of implicitness grows. Numerical experiments confirming the theoretical results are shown.
An interdisciplinary journal of discontinuity, nonlinearity, and complexity, Dec 1, 2018
Владикавказский математический журнал, 2020
The purpose of this paper is to propose the Chebyshev cardinal functions for solving Volterra sto... more The purpose of this paper is to propose the Chebyshev cardinal functions for solving Volterra stochastic integral equations. The method is based on expanding the required approximate solution as the element of Chebyshev cardinal functions. Though the way, a new operational matrix of integration is derived for the mentioned basis functions. More precisely, the unknown solution is expanded in terms of the Chebyshev cardinal functions including undetermined coefficients. By substituting the mentioned expansion in the original problem, the operational matrix reducing the stochastic integral equation to system of algebraic equations. The convergence and error analysis of the etablished method are investigated in Sobolev space. The method is numerically evaluated by solving test problems caught from the literature by which the computational efficiency of the method is demonstrated. From the computational point of view, the solution obtained by this method is in excellent agreement with th...
Applied Numerical Mathematics, Jun 1, 2013
ABSTRACT As in the deterministic case, the introduction of implicitness in stochastic schemes imp... more ABSTRACT As in the deterministic case, the introduction of implicitness in stochastic schemes improves the stability behavior. In this paper a complete study for the linear MS-stability of the two-parameter family of semi-implicit weak order 2.0 Taylor schemes for scalar stochastic differential equations is given. Figures of the MS-stability regions and numerical examples that confirm the theoretical results are shown.
The purpose of this paper is to propose the Chebyshev cardinal functions for solving Volterra sto... more The purpose of this paper is to propose the Chebyshev cardinal functions for solving Volterra stochastic integral equations. The method is based on expanding the required approximate solution as the element of Chebyshev cardinal functions. Though the way, a new operational matrix of integration is derived for the mentioned basis functions. More precisely, the unknown solution is expanded in terms of the Chebyshev cardinal functions including undetermined coefficients. By substituting the mentioned expansion in the original problem, the operational matrix reducing the stochastic integral equation to system of algebraic equations. The convergence and error analysis of the etablished method are investigated in Sobolev space. The method is numerically evaluated by solving test problems caught from the literature by which the computational efficiency of the method is demonstrated. From the computational point of view, the solution obtained by this method is in excellent agreement with th...
Interpolation by radial basis functions technique have grown significantly in recent years due to... more Interpolation by radial basis functions technique have grown significantly in recent years due to their ability for solving several problems almost impossible to obtain with standard methods. In this paper, an efficient scheme is given to solve nonlinear one-dimensional Fredholm-Volterra-Hammerstein integral equations, this technique is based on hybrid radial basis functions including the multiquadric, the Gaussian and cubic radial basis functions (RBFs). All integrals appeared in the scheme are approximately computed by the Gauss–Legendre integration formula. The new technique can be used for higher dimensional integral equations and does not increase difficulties in computation due to the easy adaption of radial basis functions. The convergence analysis and the accuracy of the proposed approach are given. Numerical examples clearly show the reliability and efficiency of the method.2000-Mathematics subject classification: 65XX, 65Dxx, 65D12.
International Journal of Computational Science and Engineering
International Journal of Mathematics and Computation, Apr 26, 2021
International Journal of Dynamical Systems and Differential Equations, 2021
In this paper, we give a new method for solving stochastic nonlinear Volterra integral equations ... more In this paper, we give a new method for solving stochastic nonlinear Volterra integral equations by using shifted Legendre operational matrix. It is discussed that how the stochastic differential equations (SDE) could numerically be solved as matrix problems. By using this new operational matrix of integration and the so-called collocation method, nonlinear Volterra integral equations is reduced to systems of algebraic equations with unknown Legendre coefficients. Finally, the high accuracy of approximated solutions are illustrated by several experiment.
Mathematics and Computers in Simulation, 2021
Predictor-corrector schemes are designed to be a compromise to retain the stability properties of... more Predictor-corrector schemes are designed to be a compromise to retain the stability properties of the implicit schemes and the computational efficiency of the explicit ones. In this paper a complete analytical study for the linear mean-square stability of the two-parameter family of Euler predictor-corrector schemes for scalar stochastic differential equations is given. The analyzed family is given in terms of two parameters that control the degree of implicitness of the method. For each selection of the parameters the stability region is obtained, letting its comparison. Particular cases of the counter-intuitive fact of losing numerical stability by reducing the step size, is confirmed and proved. Figures of the MS-stability regions and numerical examples that confirm the theoretical results are shown. c
Mathematics and Computers in Simulation, 2019
In this paper, a new computational method based on stochastic operational matrix for integration ... more In this paper, a new computational method based on stochastic operational matrix for integration of Bernoulli polynomials is proposed for solving nonlinear Volterra-Fredholm-Hammerstein stochastic integral equations. By using this new operational matrix of integration and the so-called collocation method, nonlinear Volterra-Fredholm-Hammerstein stochastic integral equation is reduced to nonlinear system of algebraic equations with unknown Bernoulli coefficients. This work is inspired by [9], where the authors study the deterministic integral equations. In order to show the rate of convergence of the suggested approach, we present theorems on convergence analysis and error estimation. Some illustrative error estimations and exapmles are provided and included to demonstrate applicability and accuracy of the technique.
International journal of computing science and mathematics, 2024
Expert Systems with Applications, Feb 29, 2024
There is a growing need of stochastic integral equations (SIEs) to investigate the behavior of co... more There is a growing need of stochastic integral equations (SIEs) to investigate the behavior of complex dynamical systems. Since real-world phenomena frequently dependent on noise sources, modeling them naturally necessitates the use of SIEs. As most SIEs cannot be solved explicitly, thus the behaviors of the studied systems are investigated using approximate solutions of their SIEs. Despite the fact that this problem has been soundly investigated and numerous methods have been presented, the practice demonstrated that obtaining satisfied approximations is not always guaranteed, necessitating the development of new effective techniques. This paper gives a new technique for solving nonlinear Itô-Volterra SIEs by reducing them to linear or nonlinear algebraic systems via the power of a combination of generalized Lagrange functions and Jacobi-Gauss collocation points. The accuracy and reliability of the new technique are evaluated and compared with the existing techniques. Moreover, sufficient conditions to make the estimate error tends to zero are given. The new technique shows surprisingly high efficiency over the existing techniques in terms of computational efficiency and approximation capability. The accuracy of the solution based on the new technique is much higher than that via the existing techniques. The required time of the new technique is much less than that of the existing techniques, where, in some circumstances, the existing techniques take more than 20 times as long as the new technique.
Numerical Algorithms, Dec 28, 2023
The paper introduces an acurrate numerical appraoch based on orthonormal Bernoulli polynomials fo... more The paper introduces an acurrate numerical appraoch based on orthonormal Bernoulli polynomials for solving parabolic partial integro- differential equations (PIDEs). This type of equations arises in physics and engineering. Some operational matrix are given for these polynomials and are also used to obtain the numerical solution. By this approach, the problem is transformed into a nonlinear algebraic system. Convergence analysis is given and some experiment tests are studied to examine the good accuracy of the numerical algorithm, the proposed technique is compared with some other well known methods.
Journal of Computational and Applied Mathematics, Oct 1, 2011
The paper considers the derivation of families of semi-implicit schemes of weak order N = 3.0 (ge... more The paper considers the derivation of families of semi-implicit schemes of weak order N = 3.0 (general case) and N = 4.0 (additive noise case) for the numerical solution of Itô stochastic differential equations. The degree of implicitness of the schemes depends on the selection of N parameters which vary between 0 and 1 and the families contain as particular cases the 3.0 and 4.0 weak order explicit Taylor schemes. Since the implementation of the multiple integrals that appear in these theoretical schemes is difficult, for the applications they are replaced by simpler random variables, obtaining simplified schemes. In this way, for the multidimensional case with one-dimensional noise, we present an infinite family of semi-implicit simplified schemes of weak order 3.0 and for the multidimensional case with additive one-dimensional noise, we give an infinite family of semi-implicit simplified schemes of weak order 4.0. The mean-square stability of the 3.0 family is analyzed, concluding that, as in the deterministic case, the stability behavior improves when the degree of implicitness grows. Numerical experiments confirming the theoretical results are shown.
An interdisciplinary journal of discontinuity, nonlinearity, and complexity, Dec 1, 2018
Владикавказский математический журнал, 2020
The purpose of this paper is to propose the Chebyshev cardinal functions for solving Volterra sto... more The purpose of this paper is to propose the Chebyshev cardinal functions for solving Volterra stochastic integral equations. The method is based on expanding the required approximate solution as the element of Chebyshev cardinal functions. Though the way, a new operational matrix of integration is derived for the mentioned basis functions. More precisely, the unknown solution is expanded in terms of the Chebyshev cardinal functions including undetermined coefficients. By substituting the mentioned expansion in the original problem, the operational matrix reducing the stochastic integral equation to system of algebraic equations. The convergence and error analysis of the etablished method are investigated in Sobolev space. The method is numerically evaluated by solving test problems caught from the literature by which the computational efficiency of the method is demonstrated. From the computational point of view, the solution obtained by this method is in excellent agreement with th...
Applied Numerical Mathematics, Jun 1, 2013
ABSTRACT As in the deterministic case, the introduction of implicitness in stochastic schemes imp... more ABSTRACT As in the deterministic case, the introduction of implicitness in stochastic schemes improves the stability behavior. In this paper a complete study for the linear MS-stability of the two-parameter family of semi-implicit weak order 2.0 Taylor schemes for scalar stochastic differential equations is given. Figures of the MS-stability regions and numerical examples that confirm the theoretical results are shown.
The purpose of this paper is to propose the Chebyshev cardinal functions for solving Volterra sto... more The purpose of this paper is to propose the Chebyshev cardinal functions for solving Volterra stochastic integral equations. The method is based on expanding the required approximate solution as the element of Chebyshev cardinal functions. Though the way, a new operational matrix of integration is derived for the mentioned basis functions. More precisely, the unknown solution is expanded in terms of the Chebyshev cardinal functions including undetermined coefficients. By substituting the mentioned expansion in the original problem, the operational matrix reducing the stochastic integral equation to system of algebraic equations. The convergence and error analysis of the etablished method are investigated in Sobolev space. The method is numerically evaluated by solving test problems caught from the literature by which the computational efficiency of the method is demonstrated. From the computational point of view, the solution obtained by this method is in excellent agreement with th...
Interpolation by radial basis functions technique have grown significantly in recent years due to... more Interpolation by radial basis functions technique have grown significantly in recent years due to their ability for solving several problems almost impossible to obtain with standard methods. In this paper, an efficient scheme is given to solve nonlinear one-dimensional Fredholm-Volterra-Hammerstein integral equations, this technique is based on hybrid radial basis functions including the multiquadric, the Gaussian and cubic radial basis functions (RBFs). All integrals appeared in the scheme are approximately computed by the Gauss–Legendre integration formula. The new technique can be used for higher dimensional integral equations and does not increase difficulties in computation due to the easy adaption of radial basis functions. The convergence analysis and the accuracy of the proposed approach are given. Numerical examples clearly show the reliability and efficiency of the method.2000-Mathematics subject classification: 65XX, 65Dxx, 65D12.
International Journal of Computational Science and Engineering
International Journal of Mathematics and Computation, Apr 26, 2021
International Journal of Dynamical Systems and Differential Equations, 2021
In this paper, we give a new method for solving stochastic nonlinear Volterra integral equations ... more In this paper, we give a new method for solving stochastic nonlinear Volterra integral equations by using shifted Legendre operational matrix. It is discussed that how the stochastic differential equations (SDE) could numerically be solved as matrix problems. By using this new operational matrix of integration and the so-called collocation method, nonlinear Volterra integral equations is reduced to systems of algebraic equations with unknown Legendre coefficients. Finally, the high accuracy of approximated solutions are illustrated by several experiment.
Mathematics and Computers in Simulation, 2021
Predictor-corrector schemes are designed to be a compromise to retain the stability properties of... more Predictor-corrector schemes are designed to be a compromise to retain the stability properties of the implicit schemes and the computational efficiency of the explicit ones. In this paper a complete analytical study for the linear mean-square stability of the two-parameter family of Euler predictor-corrector schemes for scalar stochastic differential equations is given. The analyzed family is given in terms of two parameters that control the degree of implicitness of the method. For each selection of the parameters the stability region is obtained, letting its comparison. Particular cases of the counter-intuitive fact of losing numerical stability by reducing the step size, is confirmed and proved. Figures of the MS-stability regions and numerical examples that confirm the theoretical results are shown. c
Mathematics and Computers in Simulation, 2019
In this paper, a new computational method based on stochastic operational matrix for integration ... more In this paper, a new computational method based on stochastic operational matrix for integration of Bernoulli polynomials is proposed for solving nonlinear Volterra-Fredholm-Hammerstein stochastic integral equations. By using this new operational matrix of integration and the so-called collocation method, nonlinear Volterra-Fredholm-Hammerstein stochastic integral equation is reduced to nonlinear system of algebraic equations with unknown Bernoulli coefficients. This work is inspired by [9], where the authors study the deterministic integral equations. In order to show the rate of convergence of the suggested approach, we present theorems on convergence analysis and error estimation. Some illustrative error estimations and exapmles are provided and included to demonstrate applicability and accuracy of the technique.