A new approach to numerical integration based on Coifman wavelets

Dehda, Bachir

doi:10.17512/jamcm.2019.3.03

Cited by 2 publications

(2 citation statements)

References 8 publications

(11 reference statements)

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“…Let 0 < r < 1 and u Î H 2 (0,b), b > 0 . Then the first order approximation of the Caputo-Fabrizio fractional derivative is given by: Studies in Engineering and Exact Sciences, Curitiba, v.5, n.2, p. 01-17, 2024 (8) Where:…”

Section: New Approximation To Caputo-fabrizio Fractional Derivativementioning

confidence: 99%

“…In this case, we must generalize this method to exploit the abundance of derivation. As an example in this case we can use the approximate integral formula mentioned in the reference [8]. In perspective, we will apply this numerical method to solve ordinary, partial and coupled system of fractional differential equations in the Caputo-Fabrizio sense.…”

Section: Limitation Of the Study And Future Researchmentioning

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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Section: New Approximation To Caputo-fabrizio Fractional Derivativementioning

confidence: 99%

Section: Limitation Of the Study And Future Researchmentioning

confidence: 99%

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

Dehda,

Gao

2024

SEES

View full text Add to dashboard Cite

show abstract

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

Dehda,

Gao

2024

Cad. Pedagógico

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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 . 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

A new approach to numerical integration based on Coifman wavelets

Cited by 2 publications

References 8 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

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

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