Euler-Type Fractional Differential Equations

Kilbas, Anatoly A.; Rivero, Margarita; Trujillo, Juan J.

doi:10.1142/9789812835635_0029

Cited by 799 publications

(1,503 citation statements)

References 3 publications

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“…We define the function spaces

{A}&#x0005E;l\left[a,b\right]

A^{l} false[a, b false] = ({}, y : false[a, b false] \to ℝ false| 𝔻_{h}^{l - 1} y false(x false) \in A C false[a, b false]),

where

𝔻_{h} : = x \frac{d}{d x}

and

AC

represents the space of absolutely continuous functions 13 …”

Section: Preliminariesmentioning

confidence: 99%

Numerical solutions of Hadamard fractional differential equations by generalized Legendre functions

Istafa

Rehman

2022

Math Methods in App Sciences

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This paper aims to develop a numerical method for the solutions of Hadamard fractional differential equations. We introduced Hadamard fractional Legendre functions by modifying classical Legendre polynomials and used them to approximate the numerical solutions of linear and nonlinear Hadamard fractional differential equations. The solution is approached by approximating the appropriate terms in Hadamard fractional differential equations using Hadamard fractional Legendre functions and converting the problem to a system of algebraic equations. Quasilinearization technique is employed to linearize the nonlinear Hadamard fractional differential equations. An upper bound for approximation error is derived. This method provides reasonably accurate results for a relatively smaller order of Hadamard fractional Legendre functions.

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“…We define the function spaces

{A}&#x0005E;l\left[a,b\right]

A^{l} false[a, b false] = ({}, y : false[a, b false] \to ℝ false| 𝔻_{h}^{l - 1} y false(x false) \in A C false[a, b false]),

where

𝔻_{h} : = x \frac{d}{d x}

and

AC

represents the space of absolutely continuous functions 13 …”

Section: Preliminariesmentioning

confidence: 99%

Numerical solutions of Hadamard fractional differential equations by generalized Legendre functions

Istafa

Rehman

2022

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…In this article, the Riemann-Liouville definition and the Caputo definition are frequently applied for fractional integral and fractional derivative, respectively. For more details about applications of fractional calculus and extra information see [2,3,7,8,13,14,[20][21][22][23][24]28].…”

Section: Fractional Derivatives and Integralsmentioning

confidence: 99%

On two‐dimensional weakly singular fractional partial integro‐differential equations and dual Bernstein polynomials

Sayevand

Mirzaee

Masti

2022

Numerical Methods Partial

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This article presents an efficient numerical scheme based on some properties of dual Bernstein operational matrices for a class of linear and nonlinear two‐dimensional weakly singular fractional partial integro‐differential equations (WSFPID). A novel property of the suggested scheme is the conversion of the WSFPID into an algebraic system of equations. The corresponding linear and nonlinear system of equations are solved by well‐known Newton–Raphson scheme. Convergence and an upper error bound, of the scheme are also analyzed. Finally, by providing several examples and reviewing the numerical results and comparing them with other available methods, we show that the mentioned method has acceptable accuracy and efficiency. The main important applications of the proposed scheme is that it can be applied on linear as well as nonlinear problems and can be applied on higher order partial differential equations too. Also, with a small number of bases, we can find a reasonable approximate solution.

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“…In almost all branches of science and engineering -fluid mechanics [5], system control [12,20,23], biology [17], finance [9,22,27], and various other engineering fields [11,13,18,19,26] -researchers have started modeling real life phenomenon by fractional differential equations (FDE). Moreover, The variable order FDEs (VOFDE) depending on their non-stationary power-law kernel can describe the memory and hereditary properties of many physical phenomena and processes.…”

Section: Introductionmentioning

confidence: 99%

Learning parameters of a system of variable order fractional differential equations

Singh

Mehra

Gulyani

2021

Numerical Methods Partial

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We introduce a machine learning framework that uses the differential evolution algorithm in combination with Adam-Bashforth-Moulton method to learn the parameters in a system of variable order fractional differential equations. In this work, we present out developments with regards to taking care of a class of problem: data-driven discovery of system of variable order fractional differential equations. The main advantage of the proposed framework is that it works even if data corresponding to only one of the variables in the system of equations is given. We illustrate the working of our framework on several Examples including modeling the 2014-15 Ebola outbreak in Africa via fractional SEIR (susceptible, exposed, infected, removed) model.

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Euler-Type Fractional Differential Equations

Cited by 799 publications

References 3 publications

Numerical solutions of Hadamard fractional differential equations by generalized Legendre functions

Numerical solutions of Hadamard fractional differential equations by generalized Legendre functions

On two‐dimensional weakly singular fractional partial integro‐differential equations and dual Bernstein polynomials

Learning parameters of a system of variable order fractional differential equations

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