A numerical approach based on Bernstein collocation method: Application to differential Lyapunov and Sylvester matrix equations

Sadek, Lakhlifa; Bataineh, A. Sami; Işık, Osman Raşit; Alaoui, Hamad Talibi; Hashim, Ishak

doi:10.1016/j.matcom.2023.05.011

Cited by 10 publications

(2 citation statements)

References 18 publications

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“…The continued exploration and refinement of fractional calculus methodologies promise to unlock new dimensions in our ability to comprehend and engineer complex systems across a multitude of disciplines [8][9][10]. Sadek et al [11] propose a numerical approach based on the Bernstein collocation method, applying it to differential Lyapunov and Sylvester matrix equations.…”

Section: Introductionmentioning

confidence: 99%

Fejér-Quadrature Collocation Algorithm for Solving Fractional Integro-Differential Equations via Fibonacci Polynomials

Youssri,

Atta

2024

Contemp. Math.

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In this article, we introduce a novel spectral algorithm utilizing Fibonacci polynomials to numerically solve both linear and nonlinear integro-differential equations with fractional-order derivatives. Our approach employs a quadrature-collocation method, transforming complex equations and associated conditions into systems of linear or nonlinear algebraic equations. The solutions to these equations, involving unknown coefficients, provide accurate numerical approximations for the original fractional-order equations. To validate the method, we present numerical examples illustrating its robustness and versatility. Comparative analyses with available analytical solutions affirm the reliability and accuracy of our algorithm, establishing its practical utility in addressing fractional-order integro-differential equations. This research contributes to computational mathematics and spectral methods, offering a promising tool for diverse scientific and engineering challenges.

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Section: Introductionmentioning

confidence: 99%

Fejér-Quadrature Collocation Algorithm for Solving Fractional Integro-Differential Equations via Fibonacci Polynomials

Youssri,

Atta

2024

Contemp. Math.

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“…Also, some methods for solving large-scale differential Lyapunov and T-Lyapunov equations were introduced (Sadek and Alaoui, 2021; Sadek and Talibi Alaoui, 2022). In Sadek et al . (2023b), the Bernstein collocation method to construct the solution set of the Sylvester matrix differential equation which involves the Lyapunov matrix differential equation was applied.…”

Section: Introductionmentioning

confidence: 99%

Efficient iterative schemes based on Newton's method and fixed-point iteration for solving nonlinear matrix equation X^p = Q±A(X⁻¹+B)⁻¹A^T

Erfanifar,

Hajarian

2023

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PurposeIn this paper, the authors study the nonlinear matrix equation Xp=Q±A(X-1+B)-1AT, that occurs in many applications such as in filtering, network systems, optimal control and control theory.Design/methodology/approachThe authors present some theoretical results for the existence of the solution of this nonlinear matrix equation. Then the authors propose two iterative schemes without inversion to find the solution to the nonlinear matrix equation based on Newton's method and fixed-point iteration. Also the authors show that the proposed iterative schemes converge to the solution of the nonlinear matrix equation, under situations.Findings The efficiency indices of the proposed schemes are presented, and since the initial guesses of the proposed iterative schemes have a high cost, the authors reduce their cost by changing them. Therefore, compared to the previous scheme, the proposed schemes have superior efficiency indices.Originality/value Finally, the accuracy and effectiveness of the proposed schemes in comparison to an existing scheme are demonstrated by various numerical examples. Moreover, as an application, by using the proposed schemes, the authors can get the optimal controller state feedback of $x(t+1) = A x(t) + C v(t)$.

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The general Bernstein function: Application to χ‐fractional differential equations

Sadek,

Sami Bataineh

2024

Math Methods in App Sciences

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In this paper, we present the general Bernstein functions for the first time. The properties of generalized Bernstein basis functions are given and demonstrated. The classical Bernstein polynomial bases are merely a subset of the general Bernstein functions. Based on the new Bernstein base functions and the collocation method, we present a numerical method for solving linear and nonlinear ‐fractional differential equations ( ‐FDEs) with variable coefficients. The fractional derivative used in this work is the ‐Caputo fractional derivative sense ( ‐CFD). Combining the Bernstein functions basis and the collocation methods yields the approximation solution of nonlinear differential equations. These base functions can be used to solve many problems in applied mathematics, including calculus of variations, differential equations, optimal control, and integral equations. Furthermore, the convergence of the method is rigorously justified and supported by numerical experiments.

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A numerical approach based on Bernstein collocation method: Application to differential Lyapunov and Sylvester matrix equations

Cited by 10 publications

References 18 publications

Fejér-Quadrature Collocation Algorithm for Solving Fractional Integro-Differential Equations via Fibonacci Polynomials

Fejér-Quadrature Collocation Algorithm for Solving Fractional Integro-Differential Equations via Fibonacci Polynomials

Efficient iterative schemes based on Newton's method and fixed-point iteration for solving nonlinear matrix equation X^p = Q±A(X⁻¹+B)⁻¹A^T

The general Bernstein function: Application to χ‐fractional differential equations

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A numerical approach based on Bernstein collocation method: Application to differential Lyapunov and Sylvester matrix equations

Cited by 10 publications

References 18 publications

Fejér-Quadrature Collocation Algorithm for Solving Fractional Integro-Differential Equations via Fibonacci Polynomials

Fejér-Quadrature Collocation Algorithm for Solving Fractional Integro-Differential Equations via Fibonacci Polynomials

Efficient iterative schemes based on Newton's method and fixed-point iteration for solving nonlinear matrix equation Xp = Q±A(X−1+B)−1AT

The general Bernstein function: Application to χ‐fractional differential equations

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Efficient iterative schemes based on Newton's method and fixed-point iteration for solving nonlinear matrix equation X^p = Q±A(X⁻¹+B)⁻¹A^T