Solving fractal-fractional differential equations using operational matrix of derivatives via Hilfer fractal-fractional derivative sense

Shloof, A. M.; Senu, Norazak; Ahmadian, Ali; Long, Nik Mohd Asri Nik; Salahshour, Soheil

doi:10.1016/j.apnum.2022.02.006

Cited by 15 publications

(7 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Nowadays, the subject of fractional calculus on time scales is very rich and under strong current research [18, 25–30]. Here, we make use of the idea behind a

\psi

‐Hilfer fractional derivative, that is, fractional differentiation of functions with respect to another function, which is a remarkable and relevant idea with a big impact on fractional calculus and its applications, in particular for problems described by fractional differential equations [31]. The study of

\psi

‐Riemann–Liouville fractional integrals with respect to a function

\psi

on time scales was initiated by Mekhalfi and Torres in 2017, where they introduced some generalized fractional operators on time scales of a function with respect to another function and carried out some applications to dynamic equations [32].…”

Section: Introductionmentioning

confidence: 99%

Existence, uniqueness, and controllability for Hilfer differential equations on times scales

Sousa

Oliveira

Frederico

et al. 2023

Math Methods in App Sciences

View full text Add to dashboard Cite

We introduce a new version of ψ$$ \psi $$‐Hilfer fractional derivative, on an arbitrary time scale. The fundamental properties of the new operator are investigated, and in particular, we prove an integration by parts formula. Using the Laplace transform and the obtained integration by parts formula, we then propose a ψ$$ \psi $$‐Riemann–Liouville fractional integral on times scales. The applicability of the new operators is illustrated by considering a fractional initial value problem on an arbitrary time scale, for which we prove existence, uniqueness, and controllability of solutions in a suitable Banach space. The obtained results are interesting and nontrivial even for the following particular choices: (i) of the time scale, (ii) of the order of differentiation, and/or (iii) function ψ$$ \psi $$, opening new directions of investigation. Finally, we end the article with comments and future work.

show abstract

“…Nowadays, the subject of fractional calculus on time scales is very rich and under strong current research [18, 25–30]. Here, we make use of the idea behind a

\psi

\psi

‐Riemann–Liouville fractional integrals with respect to a function

\psi

Section: Introductionmentioning

confidence: 99%

Existence, uniqueness, and controllability for Hilfer differential equations on times scales

Sousa

Oliveira

Frederico

et al. 2023

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…Because the majority of FDEs are complicated to solve analytically, numerical solutions are in high demand. A variety of numerical approaches are present in the literature for solving ordinary and partial FDEs numerically, but the OM approach coupled with the Tau method and the collocation method is commonly utilized [21][22][23][24][25][26][27][28][29][30]. This method works by converting the FDEs into an algebraic equation system that can be solved with any computer programmer.…”

Section: Introductionmentioning

confidence: 99%

A novel fractal-fractional analysis of the stellar helium burning network using extended operational matrix method

Shloof¹,

Senu²,

Ahmadian³

et al. 2023

Phys. Scr.

Self Cite

View full text Add to dashboard Cite

The second stage, in which the star uses the nuclear fuel in its interior, represents the helium burning phase. At that stage, three elements are synthesised, which are carbon, oxygen and neon. The purpose of this paper is to establish a numerical solution for the helium burning system (HBN) fractal fractional diﬀerential equations (FFDEs). The extended operative matrix method (OM) is employed in the solution of a system of diﬀerential equations. and the product abundances, namely helium, carbon, oxygen and neon. The product abundances of the four elements were obtained in a form of divergent series. This divergent series are then accelerated using Euler-Abell transformation (EUAT) and Pade approximation (EUAT-PA) to obtain more reliable results. Nine fractal-fractional (FF) gas models are calculated and the inﬂuence of fractal-fractional parameters on product abundances is discussed. The ﬁndings show that modeling nuclear burning networks with the OM fractal-fractional derivative produces excellent results, establishing it as an accurate, resilient, and trustworthy approach, and the fractional HB models can have a considerable impact on stellar model calculations.

show abstract

“…Thus, we have to perform substantial numerical calculations to solve them. A variety of well-known algorithms can solve FDEs, such as operational matrix, [22][23][24][25][26] Adomian decomposition, 27 Harr wavelet Method, 28 Homotopy perturbation, 29 variational iteration, 30 neural networks (NNs), [31][32][33] and finite difference. 34,35 In comparison to classic numerical approaches, the approximate calculation of ANN appears to be less sensitive to the spatial dimension.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, we have to perform substantial numerical calculations to solve them. A variety of well‐known algorithms can solve FDEs, such as operational matrix, 22‐26 Adomian decomposition, 27 Harr wavelet Method, 28 Homotopy perturbation, 29 variational iteration, 30 neural networks (NNs), 31‐33 and finite difference 34,35 …”

Section: Introductionmentioning

confidence: 99%

A highly accurate artificial neural networks scheme for solving higher multi‐order fractal‐fractional differential equations based on generalized Caputo derivative

Shloof

Ahmadian

Senu

et al. 2023

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

Artificial neural networks have great potential for learning and stability in the face of tiny input data changes. As a result, artificial intelligence techniques and modeling tools have a growing variety of applications. To estimate a solution for fractal‐fractional differential equations (FFDEs) of high‐order linear (HOL) with variable coefficients, an iterative methodology based on a mix of a power series method and a neural network approach was applied in this study. In the algorithm's equation, an appropriate truncated series of the solution functions was replaced. To tackle the issue, this study uses a series expansion of an unidentified function, where this function is approximated using a neural architecture. Some examples were presented to illustrate the efficiency and usefulness of this technique to prove the concept's applicability. The proposed methodology was found to be very accurate when compared to other available traditional procedures. To determine the approximate solution to FFDEs‐HOL, the suggested technique is simple, highly efficient, and resilient.

show abstract

Solving fractal-fractional differential equations using operational matrix of derivatives via Hilfer fractal-fractional derivative sense

Cited by 15 publications

References 43 publications

Existence, uniqueness, and controllability for Hilfer differential equations on times scales

Existence, uniqueness, and controllability for Hilfer differential equations on times scales

A novel fractal-fractional analysis of the stellar helium burning network using extended operational matrix method

A highly accurate artificial neural networks scheme for solving higher multi‐order fractal‐fractional differential equations based on generalized Caputo derivative

Contact Info

Product

Resources

About