A wavelet approach for the variable-order fractional model of ultra-short pulsed laser therapy

Roohi, Reza; Hosseininia, M.; Heydari, Mohammad Hossein

doi:10.1007/s00366-021-01367-x

Cited by 7 publications

(4 citation statements)

References 32 publications

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“…For this reason, numerical methods are applied to solve these problems. Some of numerical methods that have recently been proposed to solve such problems are wavelet approaches [4,5], the Galerkin finite element methods [6,7], finite difference techniques [8,9], meshless schemes [10,11], and spectral methods [12,13].…”

Section: Introductionmentioning

confidence: 99%

A hybrid spectral approach based on 2D cardinal and classical second kind Chebyshev polynomials for time fractional 3D Sobolev equation

Hosseininia,

Heydari,

Razzaghi

2023

Math Methods in App Sciences

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In this work, the Caputo fractional derivative defines the time fractional 3D Sobolev equation. The 2D shifted second kind Chebyshev cardinal polynomials (SSKCCPs) and 2D shifted second kind Chebyshev polynomials (SSKCPs) (as two well‐known classes of basis functions) are utilized to establish a hybrid technique for this new problem. First, the problem solution is approximated simultaneously using the 2D SSKCCPs (relative to ) and 2D SSKCCPs (relative to ). Next, the classical and fractional operational matrices of these polynomials are achieved and applied to make the hybrid algorithm. With a combination of derived operational matrices and the collocation approach, solving the expressed fractional 3D problem turns into solving an equivalent system of algebraic equations. The proposed algorithm's accuracy is checked using four numerical examples.

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Section: Introductionmentioning

confidence: 99%

A hybrid spectral approach based on 2D cardinal and classical second kind Chebyshev polynomials for time fractional 3D Sobolev equation

Hosseininia,

Heydari,

Razzaghi

2023

Math Methods in App Sciences

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“…10 Recently, special consideration has been paid to the application of VO fractional derivatives to model several problems, such as viscoelastic constitutive models, 11 dual phase lag bioheat problems, 12 and ultra-short pulsed laser therapy. 13 Since it is often impossible to solve VO fractional problems, analytically, varied numerical techniques have been used for such problems. Some numerical procedures for solving these class of problems are including the orthonormal Bernoulli polynomials method, 14 the Chebyshev collocation method, 15 the sixth-kind Chebyshev polynomials technique, 16 the meshless approach, 17 the Lagrange polynomials method, 18 the transcendental Bernstein expansion method, 19 and the Lagrangian piecewise interpolation technique.…”

Section: Introductionmentioning

confidence: 99%

“…Such derivatives are called variable‐order (VO) fractional derivatives 10 . Recently, special consideration has been paid to the application of VO fractional derivatives to model several problems, such as viscoelastic constitutive models, 11 dual phase lag bioheat problems, 12 and ultra‐short pulsed laser therapy 13 . Since it is often impossible to solve VO fractional problems, analytically, varied numerical techniques have been used for such problems.…”

Section: Introductionmentioning

confidence: 99%

Third‐kind Chebyshev cardinal functions for variable‐order time fractional RLW‐Burgers equation

Heydari

Razzaghi

2022

Math Methods in App Sciences

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In this article, the Caputo type variable-order fractional differentiation is utilized for defining a novel fractional form of the regularized long wave Burgers problem which arises in several physical applications. The third-kind Chebyshev cardinal functions are generated and some matrix relationships related to them are derived. A numerical procedure dependent on these polynomial cardinal functions is developed for the introduced equation. This scheme converts the primary problem solving into finding an algebraic system solution with low computations using approximating the solution of the equation by the expressed cardinal functions. The accuracy of the presented procedure is investigated in some examples. The presented scheme is straightforward and powerful and can be adopted for many other nonlinear fractional problems.

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“…The remarkable point about such operators is that their memory property is more evident [22]. Some problems that have recently been modeled by such operators can be found in [23,24]. However, similar to constant-order fractional equations, the major challenge in dealing with VO fractional equations is finding their analytical solutions, which is often impossible.…”

Section: Introductionmentioning

confidence: 99%

An accurate approach based on the orthonormal shifted discrete Legendre polynomials for variable-order fractional Sobolev equation

Heydari

Atangana

2021

Adv Differ Equ

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This paper applies the Heydari–Hosseininia nonsingular fractional derivative for defining a variable-order fractional version of the Sobolev equation. The orthonormal shifted discrete Legendre polynomials, as an appropriate family of basis functions, are employed to generate an operational matrix method for this equation. A new fractional operational matrix related to these polynomials is extracted and employed to construct the presented method. Using this approach, an algebraic system of equations is obtained instead of the original variable-order equation. The numerical solution of this system can be found easily. Some numerical examples are provided for verifying the accuracy of the generated approach.

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A wavelet approach for the variable-order fractional model of ultra-short pulsed laser therapy

Cited by 7 publications

References 32 publications

A hybrid spectral approach based on 2D cardinal and classical second kind Chebyshev polynomials for time fractional 3D Sobolev equation

A hybrid spectral approach based on 2D cardinal and classical second kind Chebyshev polynomials for time fractional 3D Sobolev equation

Third‐kind Chebyshev cardinal functions for variable‐order time fractional RLW‐Burgers equation

An accurate approach based on the orthonormal shifted discrete Legendre polynomials for variable-order fractional Sobolev equation

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