Numerical solution of a class of Caputo–Fabrizio derivative problem using Haar wavelet collocation method

Dehda, Bachir; Ahmed, Abdelaziz Azeb; Yazid, Fares; Djeradi, Fatima Siham

doi:10.1007/s12190-023-01859-7

Cited by 3 publications

(5 citation statements)

References 16 publications

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“…In this section, we recall some definitions and properties of fractional derivative and integral operators [7,9].…”

Section: Basic Definitionsmentioning

confidence: 99%

“…[11] For a bounded first derivative function u . Its single definite integral on [a, b] can be evaluated by the following formula: (9) Where:…”

Section: New Approximation To Caputo-fabrizio Fractional Derivativementioning

confidence: 99%

“…In recent years, fractional calculus has received great interest and wide popularity in many scientific fields, especially mathematics, due to the fact that it models many physical [4,10,14,19], engineering [2] and biological phenomena [15,20]. Generally, these models include ordinary fractional differential equations [5], space-time fractional differential equations [3,6] and integral fractional differential equations [16,18] in the one of Riemann Liouville, Caputo or CaputoFabrizio fractional derivative sense [7,9]. Due to the non-singular kernel of the Caputo-Fabrizio fractional derivative, it has been preferred and widely used by many researchers.…”

Section: Introductionmentioning

confidence: 99%

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A new approximation to the first order fractional derivative in the Caputo-Fabrizio sense using Haar Wavelet integration formula

Dehda,

Gao

2024

SEES

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Decades ago, fractional calculus arose to generalize ordinary derivation and integration, and then became a means of modeling and interpreting many phenomena in various fields such as engineering, physics, chemistry, biology and signal processing. The definition of the fractional derivative began with a derivative with a singular kernel, such as the Riemann-Liouville and Caputo derivative. Due to the singularity of the kernel, the definition of Caputo-Fabrizio appeared, which has a non-singular kernel and mathematical properties similar to the derivative of the integer order. This last definition attracted many mathematicians and researchers to use it in modeling phenomena and obtaining historical information about the development of the studied phenomena, but usually the analytical solution does not exist, which necessitated numerical methods to find an approximate solution. These approximate methods depend on finding an approximate formula for the fractional derivative, and then the problem is transformed into a system of algebraic equations that is easy to solve. In fact, all the numerical methods that have been used have a polynomial rate of convergence, which calls for thinking about a new method that is more effective and has a better rate of convergence. For this reason, in this paper, we propose an efficient numerical method to approximate the first order fractional derivative in the Caputo-Fabrizio sense. This method develops a new quadratic formula using Haar wavelet integration method. Error analysis of our proposed method gives an exponential convergence rate of O(2-J ) . To check the effectiveness of the proposed method, we examine some examples with different fractional orders. The quantative results demonstrated the stability and efficiency of the proposed method.

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“…In this section, we recall some definitions and properties of fractional derivative and integral operators [7,9].…”

Section: Basic Definitionsmentioning

confidence: 99%

“…[11] For a bounded first derivative function u . Its single definite integral on [a, b] can be evaluated by the following formula: (9) Where:…”

Section: New Approximation To Caputo-fabrizio Fractional Derivativementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A new approximation to the first order fractional derivative in the Caputo-Fabrizio sense using Haar Wavelet integration formula

Dehda,

Gao

2024

SEES

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“…Due to its property of memory effect, this concept has received a great response in the applied sciences. In this regard, many definitions have been given for both the integral and the fractional derivatives, such as the Riemann-Liouville [1], Caputo [2] and Caputo-Fabrizio fractional integrals and derivatives [1][2][3][4][5]. However, the concepts of Riemann-Liouville and Caputo were used to model the phenomena first, which have singularity in their kernels.…”

Section: Introductionmentioning

confidence: 99%

“…For this reason, many new definitions of integrals and fractional derivatives have been introduced in the literature. For instance, the Caputo-Fabrizio fractional integral and derivative [1] avoid the singularity problem; this property makes it popular in the scientific community. The main problem facing researchers in solving Caputo-Fabrizio fractional differential equations and systems [6] is the difficulty in finding an analytical solution, which leads them to use numerical methods.…”

Section: Introductionmentioning

confidence: 99%

Numerical Approach Based on the Haar Wavelet Collocation Method for Solving a Coupled System with the Caputo–Fabrizio Fractional Derivative

Dehda,

Yazid,

Djeradi

et al. 2024

Symmetry

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In the present paper, we consider an effective computational method to analyze a coupled dynamical system with Caputo–Fabrizio fractional derivative. The method is based on expanding the approximate solution into a symmetry Haar wavelet basis. The Haar wavelet coefficients are obtained by using the collocation points to solve an algebraic system of equations in mathematical physics. The error analysis of this method is characterized by a good convergence rate. Finally, some numerical examples are presented to prove the accuracy and effectiveness of this method.

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A new approximation to the first order fractional derivative in the Caputo-Fabrizio sense using Haar Wavelet integration formula

Dehda,

Gao

2024

Cad. Pedagógico

View full text Add to dashboard Cite

Decades ago, fractional calculus arose to generalize ordinary derivation and integration, and then became a means of modeling and interpreting many phenomena in various fields such as engineering, physics, chemistry, biology and signal processing. The definition of the fractional derivative began with a derivative with a singular kernel, such as the Riemann-Liouville and Caputo derivative. Due to the singularity of the kernel, the definition of Caputo-Fabrizio appeared, which has a non-singular kernel and mathematical properties similar to the derivative of the integer order. This last definition attracted many mathematicians and researchers to use it in modeling phenomena and obtaining historical information about the development of the studied phenomena, but usually the analytical solution does not exist, which necessitated numerical methods to find an approximate solution. These approximate methods depend on finding an approximate formula for the fractional derivative, and then the problem is transformed into a system of algebraic equations that is easy to solve. In fact, all the numerical methods that have been used have a polynomial rate of convergence, which calls for thinking about a new method that is more effective and has a better rate of convergence. For this reason, in this paper, we propose an efficient numerical method to approximate the first order fractional derivative in the Caputo-Fabrizio sense. This method develops a new quadratic formula using Haar wavelet integration method. Error analysis of our proposed method gives an exponential convergence rate of . To check the effectiveness of the proposed method, we examine some examples with different fractional orders. The quantative results demonstrated the stability and efficiency of the proposed method.

show abstract

Numerical solution of a class of Caputo–Fabrizio derivative problem using Haar wavelet collocation method

Cited by 3 publications

References 16 publications

A new approximation to the first order fractional derivative in the Caputo-Fabrizio sense using Haar Wavelet integration formula

A new approximation to the first order fractional derivative in the Caputo-Fabrizio sense using Haar Wavelet integration formula

Numerical Approach Based on the Haar Wavelet Collocation Method for Solving a Coupled System with the Caputo–Fabrizio Fractional Derivative

A new approximation to the first order fractional derivative in the Caputo-Fabrizio sense using Haar Wavelet integration formula

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