New theories and applications of tempered fractional differential equations

Obeidat, Nazek A.; Bentil, Daniel E.

doi:10.1007/s11071-021-06628-4

Cited by 33 publications

(20 citation statements)

References 26 publications

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“…The exact solution of this problem is 𝜐(𝑥) = 𝑒 −𝛼𝑥 (𝑥 + 1) 7+3∕5 . Now, by applying the CPTFDIM at the GL points, we can rewrite Equation (41) as…”

Section: Numerical Testsmentioning

confidence: 99%

“…Many approaches for obtaining exact analytical and approximate solutions to TFDEs have been provided, including the fractional spectral method [61], the fast predictor-corrector technique [19], the tempered fractional Laplace method [3], the fractional Jacobi-predictor-corrector algorithm [33], the fractional natural decomposition method [45,46], the fractional reduced differential transform method [44], and the fractional Fourier transform method [52]. [41] used a flexible tool called the tempered fractional natural transform method to find exact and approximate analytical solutions to tempered fractional linear ordinary and partial differential equations as an alternative to existing techniques, for instance, the Laplace transform and its extension to transforms such as the Sumudu transform and the 𝕁-transform [62]. which is a novel and effective integral transform approach.…”

Section: Introductionmentioning

confidence: 99%

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High-order Chebyshev pseudospectral tempered fractional operational matrices and tempered fractional differential problems

Dahy

ElHawary

Fahim

et al. 2022

Preprint

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This paper focuses on presenting an accurate, efficient, and fast pseudospectral method to solve tempered fractional differential equations (TFDEs) in both spatial and temporal dimensions. We use a tempered integration matrix that can be employed to evaluate 𝑛-fold tempered integrals of a real function 𝑔 for 𝑛 ∈ ℝ + . Also, it may be used to compute any non-integer order 𝜆 < 0 tempered derivatives of 𝑔. We employ the Chebyshev interpolating polynomial for 𝑔 at Gauss-Lobatto (GL) points in the range [−1, 1] and any identically shifted range. The proposed method carries with it a recast of the TFDE into integration formulas to actually take advantage of the adaptation of the integral operators, hence avoiding the ill-conditioning and reduction in the convergence rate of integer differential operators. Via various tempered fractional differential examples, the present technique shows many advantages, for instance, high accuracy, a much higher rate of running, fewer computational hurdles and programming, calculating the tempered-derivative/integral of fractional order and its exceptional accuracy in comparison with other competitive numerical schemes. The study includes the elapsed times taken to construct the collocation matrices and obtain the numerical solutions. Also, numerical examination of the produced condition number 𝜅(𝐴) of the resulted linear systems. The accuracy and efficiency of the proposed method are studied from the standpoint of the 𝐿 2 and 𝐿 ∞ -norms error and fast rate of spectral convergence.

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“…The exact solution of this problem is 𝜐(𝑥) = 𝑒 −𝛼𝑥 (𝑥 + 1) 7+3∕5 . Now, by applying the CPTFDIM at the GL points, we can rewrite Equation (41) as…”

Section: Numerical Testsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

High-order Chebyshev pseudospectral tempered fractional operational matrices and tempered fractional differential problems

Dahy

ElHawary

Fahim

et al. 2022

Preprint

View full text Add to dashboard Cite

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“…This method has been widely used in applied science, and it does not need any transformation or linearization. Recently, Obeidat and Bentil in [19,20] have developed new technique called the tempered fractional natural transform method (TFNTM) where they presented new theories and they proved the existence and uniqueness theorems of the TFNTM and they implemented the method to solve some tempered fractional ODEs and PDEs. Our remit to solve a degenerate parabolic equation arising in the spatial diffusion of biological populations of fractional order with given initial conditions [21][22][23][24][25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

Convergence analysis of the fractional decomposition method with applications to time‐fractional biological population models

Obeidat

Bentil

2022

Numerical Methods Partial

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In this study, we present convergence analysis along with an error estimate for time-fractional biological population equation in terms of the Caputo derivative using a new technique called the fractional decomposition method (FDM). Further, we present exact solutions to four test problems of nonlinear time-fractional biological population models to show the accuracy and efficiency of the FDM. This method based on constructing series solutions in a form of rapidly convergent series with easily computable components and without the need of linearization, discretization and perturbations. The results prove that the FDM is very effective and simple for solving fractional partial differential equations in multi-dimensional spaces, special cases of which we have described in this paper.

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“…Tempered fractional calculus is a type of fractional derivative/integral operator which multiplies an exponential factor to its power law kernel. This type of exponential tempering had been received increasing attention from researchers as having both mathematical and practical advantages [1,2]. Several phenomena were best described by using this tempered fractional derivative/integral operator such as tempered fractional Brownian motion [3], epidemic modelling [4], and diffusion-wave equation [5].…”

Section: Introductionmentioning

confidence: 99%

An Efficient Numerical Scheme for Solving Multiorder Tempered Fractional Differential Equations via Operational Matrix

Owoyemi

Phang

Toh

2022

Journal of Mathematics

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In this paper, we extend the operational matrix method to solve the tempered fractional differential equation, via shifted Legendre polynomial. Although the operational matrix method is widely used in solving various fractional calculus problems, it is yet to apply in solving fractional differential equations defined in the tempered fractional derivatives. We first derive the analytical expression for tempered fractional derivative for x p , hence, using it to derive the new operational matrix of fractional derivative. By using a few terms of shifted Legendre polynomial and via collocation scheme, we were able to obtain a good approximation for the solution of the multiorder tempered fractional differential equation. We illustrate it using some numerical examples.

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New theories and applications of tempered fractional differential equations

Cited by 33 publications

References 26 publications

High-order Chebyshev pseudospectral tempered fractional operational matrices and tempered fractional differential problems

High-order Chebyshev pseudospectral tempered fractional operational matrices and tempered fractional differential problems

Convergence analysis of the fractional decomposition method with applications to time‐fractional biological population models

An Efficient Numerical Scheme for Solving Multiorder Tempered Fractional Differential Equations via Operational Matrix

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