Numerical Solutions for Multi-Term Fractional Order Differential Equations with Fractional Taylor Operational Matrix of Fractional Integration

Avcı, İbrahim; Mahmudov, Nazım I.

doi:10.3390/math8010096

Cited by 13 publications

(6 citation statements)

References 62 publications

(80 reference statements)

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“…-In the topic of numerical approximation, we hope to approximate the functions by solving fractional differential equations, following numerical methods such as (Avci and Mahmudov 2020), and to approximate the operators by Bernstein-polynomial techniques, following (Kürt et al 2020). -In applications, we shall demonstrate how the operators can be used in bioengineering applications, and how the functions arise naturally from fractional differential systems.…”

Section: Discussion and Further Workmentioning

confidence: 99%

A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators

2020

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We define an analogue of the classical Mittag-Leffler function which is applied to two variables, and establish its basic properties. Using a corresponding single-variable function with fractional powers, we define an associated fractional integral operator which has many interesting properties. The motivation for these definitions is twofold: firstly, their link with some fundamental fractional differential equations involving two independent fractional orders, and secondly, the fact that they emerge naturally from certain applications in bioengineering. Keywords Mittag-Leffler functions • Fractional integrals • Fractional derivatives • Fractional differential equations • Bivariate Mittag-Leffler functions ρ α,β (x) and E α,β,γ (x), the "two-parameter" and "three-parameter" Mittag-Leffler functions, being defined by power series similar to the Communicated by Roberto Garrappa.

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Section: Discussion and Further Workmentioning

confidence: 99%

A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators

2020

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“…In this section, our aim is to use a new method called the FTOMM (see ref. [62,63]) method (fractional Taylor operational matrix method), to solve the probability-based model of the SARS-CoV-2 virus (95) in the Caputo settings.…”

Section: Model Dynamics In the Caputo Sensementioning

confidence: 99%

A Study on the Fractal-Fractional Epidemic Probability-Based Model of SARS-CoV-2 Virus along with the Taylor Operational Matrix Method for Its Caputo Version

Rezapour

Etemad

Avcı

et al. 2022

Journal of Function Spaces

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SARS-CoV-2 is a strain of the large coronavirus family that has led to COVID-19 disease. The virus has been one of the deadliest known viruses in the world to date. Rapid mutations and the creation of new strains cause researchers to focus on the dynamic behaviors of the virus and to analyze it accurately through clinical research and mathematical models. In this paper, from the point of view of mathematical modeling, we intend to focus on the dynamic behavior of the system and examine its analytical and numerical aspects in two different structures. In other words, by recalling newly formulated hybrid fractional-fractal operators, we present a fractal-fractional probability-based model of SARS-CoV-2 virus for the first time and extract its equivalent compact fractal-fractional IVP to investigate its existence and stability criteria. A type of special admissible contractions will help us in this regard. Moreover, based on the source data, we simulate our system according to algorithms derived by Adams-Bashforth method and explain the effects of variation of the dimension of fractal and fractional order on dynamics of solutions. Finally, we transform our fractal-fractional model into a Caputo probability-based model of SARS-CoV-2 to derive solutions via the operational matrix method under Taylor’s basis. The numerical simulations show close behaviors for both of models.

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“…Because of this quality, these derivatives are excellent for simulating more physical phenomena [2,20,21] and can describe the physical meaning of many problems [22]. In this article, we study the multi-term fractional-order differential equations (Equations ( 1)-( 3)), which have useful qualities as well and can be used to represent complex multi-rate physical processes in a variety of ways; for examples, see [23][24][25][26][27][28]. The Lane-Emden and Bratu equations [29][30][31] are noteworthy examples of a smaller class of multi-term fractional differential equations.…”

Section: Introductionmentioning

confidence: 99%

Semi-Analytical Solutions for Some Types of Nonlinear Fractional-Order Differential Equations Based on Third-Kind Chebyshev Polynomials

El-Sayed,

Boulaaras,

AbaOud

2023

Fractal Fract

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Approximate solutions for a family of nonlinear fractional-order differential equations are introduced in this work. The fractional-order operator of the derivative are provided in the Caputo sense. The third-kind Chebyshev polynomials are discussed briefly, then operational matrices of fractional and integer-order derivatives for third-kind Chebyshev polynomials are constructed. These obtained matrices are a critical component of the proposed strategy. The created matrices are used in the context of approximation theory to solve the stated problem. The fundamental advantage of this method is that it converts the nonlinear fractional-order problem into a system of algebraic equations that can be numerically solved. The error bound for the suggested technique is computed, and numerical experiments are presented to verify and support the accuracy and efficiency of the proposed method for solving the class of nonlinear multi-term fractional-order differential equations.

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Numerical Solutions for Multi-Term Fractional Order Differential Equations with Fractional Taylor Operational Matrix of Fractional Integration

Cited by 13 publications

References 62 publications

A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators

A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators

A Study on the Fractal-Fractional Epidemic Probability-Based Model of SARS-CoV-2 Virus along with the Taylor Operational Matrix Method for Its Caputo Version

Semi-Analytical Solutions for Some Types of Nonlinear Fractional-Order Differential Equations Based on Third-Kind Chebyshev Polynomials

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