Introduction to Fractional Differential Equations

Milici, Constantin; Drăgănescu, Gheorghe; Machado, J.A.T.

doi:10.1007/978-3-030-00895-6

Cited by 100 publications

(89 citation statements)

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“…where D (1) h is the operational matrix of the first derivative of h,M (x) with dimension of (M + 1) × (M + 1).…”

Section: Differentiation Matricesmentioning

confidence: 99%

“…Fractional derivative operators are found to be more real in modelling a variety of engineering processes, physical behaviours, biological models and financial applications, such as viscoelastic materials, anomalous diffusion and non-exponential relaxation patterns, among others (see, e.g. [1][2][3][4][5][6][7][8][9] and the references therein). Regarding their importance, there are difficulties in getting the exact solutions of fractional differential equations (FDEs) due to the property of non-locality of the fractional derivative operators.…”

Section: Introductionmentioning

confidence: 99%

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Application of a Legendre collocation method to the space–time variable fractional-order advection–dispersion equation

Mallawi

Alzaidy

Hafez

2019

Journal of Taibah University for Science

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In this paper, an efficient spectral collocation method is presented for solving the space-time variable-order fractional advection-dispersion equations (ST-VO-FADE). The proposed method is based on the Legendre collocation spectral procedure together with the Legendre operational matrices for fractional derivatives, described in the sense of Riemann-Liouville and Caputo. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations, which greatly simplifies the solution process. The validity of the method is demonstrated by solving two numerical examples. Finally, comparisons between the algorithm derived in this paper and the existing algorithms are given, which show that our numerical schemes exhibit better performances than the existing ones. ARTICLE HISTORY

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“…where D (1) h is the operational matrix of the first derivative of h,M (x) with dimension of (M + 1) × (M + 1).…”

Section: Differentiation Matricesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Application of a Legendre collocation method to the space–time variable fractional-order advection–dispersion equation

Mallawi

Alzaidy

Hafez

2019

Journal of Taibah University for Science

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“…In 2015, Caputo and Fabrizio discovered a new operator of arbitrary order, namely, Caputo‐Fabrizio (CF) operator with arbitrary order and enforced to the several linear and nonlinear physical problems . In 2016, Atangana and Baleanu introduced another nonsingular derivative based on Miitag‐Leffler kernel and applied to the many problems . The parabolic heat equation was first developed and introduced Joseph Fourier in 1822.…”

Section: Introductionmentioning

confidence: 99%

An analysis for heat equations arises in diffusion process using new Yang‐Abdel‐Aty‐Cattani fractional operator

Kumar

Ghosh

Samet

et al. 2020

Math Methods in App Sciences

179

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The heat equation is parabolic partial differential equation and occurs in the characterization of diffusion progress. In the present work, a new fractional operator based on the Rabotnov fractional-exponential kernel is considered. Next, we conferred some fascinating and original properties of nominated new fractional derivative with some integral transform operators where all results are significant. The fundamental target of the proposed work is to solve the multi dimensional heat equations of arbitrary order by using analytical approach homotopy perturbation transform method and residual power series method, where new fractional operator has been taken in new Yang-Abdel-Aty-Cattani (YAC) sense. The obtained results indicate that solution converges to the original solution in language of generalized Mittag-Leffler function. Three numerical examples are discussed to draw an effective attention to reveal the proficiency and adaptability of the recommended methods on new YAC operator. KEYWORDS HPTM and RPSM, multidimensional heat equations, new fractional derivative, new YAC operator, Prabhakar or generalized Mittag-Leffler function MSC CLASSIFICATION 26A33; 34A08; 34A34; 60G22 Math Meth Appl Sci. 2020;43:6062-6080. wileyonlinelibrary.com/journal/mma

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“…The Caputo fractional derivatives are based on integral expressions and gamma functions which are nonlocal. Fractional theory and its applications are mentioned many papers and monographs, we refer [1,12,15,17,24,26,27,28,29,33].…”

Section: Introductionmentioning

confidence: 99%