Fractional Bernstein operational matrices for solving integro-differential equations involved by Caputo fractional derivative

Alshbool, Mohammed; Mohammad, Mutaz; Işık, Osman Raşit; Hashim, Ishak

doi:10.1016/j.rinam.2022.100258

Cited by 8 publications

(4 citation statements)

References 20 publications

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“…They also studied the solution of nonlinear thin-film flow of 3rd-grade fluid problems with LOM [11]. In [12][13][14][15][16][17][18][19][20], other types of polynomials were used to solve different types of problems.…”

Section: Introductionmentioning

confidence: 99%

Novel Approximate Solutions for Nonlinear Blasius Equations

Mahdi,

AL-Jawary,

Mustafa Turkyilmazoglu

2024

IHJPAS

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The method of operational matrices based on different types of polynomials such as Bernstein, shifted Legendre and Bernoulli polynomials will be presented and implemented to solve the nonlinear Blasius equations approximately. The nonlinear differential equation will be converted into a system of nonlinear algebraic equations that can be solved using Mathematica®12. The efficiency of these methods has been studied by calculating the maximum error remainder ( ), and it was found that their efficiency increases as the polynomial degree (n) increases, since the errors decrease. Moreover, the approximate solutions obtained by the proposed methods are compared with the solution of the 4th order Runge-Kutta method (RK4), which gives very good agreement. In addition, the convergence of the proposed approximate methods is given based on one of the Banach fixed point theorem results.

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Section: Introductionmentioning

confidence: 99%

Novel Approximate Solutions for Nonlinear Blasius Equations

Mahdi,

AL-Jawary,

Mustafa Turkyilmazoglu

2024

IHJPAS

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“…Some of them are Caputo, Atangana-Baleanu, Caputo-Fabrizio, Riesz, Riemann-Liouville, and Hadamard. For example, the authors in [5] introduced the operational matrices of fractional Bernstein functions to solve fractional differential equations (FDEs), and Alshbool et al [6] proposed the concept of operational matrices based on fractional Bernstein functions for solving integro-differential equations under the Caputo operator. The use of new fractional operators in the geometry of realworld models has made significant advancements in this domain [7,8].…”

Section: Introductionmentioning

confidence: 99%

Numerical Investigation of the Fractional Oscillation Equations under the Context of Variable Order Caputo Fractional Derivative via Fractional Order Bernstein Wavelets

et al. 2023

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This article describes an approximation technique based on fractional order Bernstein wavelets for the numerical simulations of fractional oscillation equations under variable order, and the fractional order Bernstein wavelets are derived by means of fractional Bernstein polynomials. The oscillation equation describes electrical circuits and exhibits a wide range of nonlinear dynamical behaviors. The proposed variable order model is of current interest in a lot of application areas in engineering and applied sciences. The purpose of this study is to analyze the behavior of the fractional force-free and forced oscillation equations under the variable-order fractional operator. The basic idea behind using the approximation technique is that it converts the proposed model into non-linear algebraic equations with the help of collocation nodes for easy computation. Different cases of the proposed model are examined under the selected variable order parameters for the first time in order to show the precision and performance of the mentioned scheme. The dynamic behavior and results are presented via tables and graphs to ensure the validity of the mentioned scheme. Further, the behavior of the obtained solutions for the variable order is also depicted. From the calculated results, it is observed that the mentioned scheme is extremely simple and efficient for examining the behavior of nonlinear random (constant or variable) order fractional models occurring in engineering and science.

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“…Considering a broader numerical perspective, Zhang et al's advocacy for the finite-difference method [13] and Tural et al's technique rooted in multi-term variable order FDEs [14] deserve mention. Bernstein polynomial approximations have also been pivotal in various studies [15]- [20] ,highlighting their adaptability and efficacy. In Conclusion, Yarmohammadi et al's introduction of a spectral iterative method [21] formed a significant foundation for our study.…”

Section: Introductionmentioning

confidence: 99%

Solving Nonlinear Multi-Order Fractional Differential Equations Using Bernstein Polynomials

Algazaa,

Saeidian

2023

IEEE Access

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This paper introduces two novel methods for solving multi-order fractional differential equations using Bernstein polynomials. The first method, referred to as the fractional operational matrix of differentiation of Bernstein polynomials, is employed to solve fractional differential equations with high precision. The second method merges the development of the collocation method with Bernstein polynomials, ensuring an even higher accuracy than previously utilized numerical techniques. The Caputo fractional derivative of Bernstein polynomials is calculated using a specialized method, that combines Gauss-Jacobi quadrature rules and three-term recurrence formulas, resulting in a matrix with a notably small condition number. This allows for the solution of the algebraic system through standard linear algebra techniques, and the method provides substantial accuracy,particuarly for problems with smooth solutions. Through a comparison of numerical examples with existing methods and a physical test example predicting the spheres path and behavior within a fluid the practical applicability and accuracy of our novel methods were demonstrated. Utilizing MATLAB 2021, the efficacy of the presented methods was showcased, revealing superior performance in terms of accuracy and reliability in solving fractional differential equations.INDEX TERMS Bernstein polynomials, Collocation method, Fractional differential equations, Numerical methods, Nonlinear differential equations.

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Fractional Bernstein operational matrices for solving integro-differential equations involved by Caputo fractional derivative

Cited by 8 publications

References 20 publications

Novel Approximate Solutions for Nonlinear Blasius Equations

Novel Approximate Solutions for Nonlinear Blasius Equations

Numerical Investigation of the Fractional Oscillation Equations under the Context of Variable Order Caputo Fractional Derivative via Fractional Order Bernstein Wavelets

Solving Nonlinear Multi-Order Fractional Differential Equations Using Bernstein Polynomials

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