A novel operational matrix method for solving singularly perturbed boundary value problems of fractional multi-order

Sayevand, Khosro; Pichaghchi, K.

doi:10.1080/00207160.2017.1296574

Cited by 15 publications

(4 citation statements)

References 28 publications

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“…Among the various definitions for the derived from fractional derivative, the formulations of Riemann-Liouville, Grünwald-Letnikov, and Caputo are mostly used. 23 In the follow-up, we adopt the Riemann-Liouville integral and Caputo definition of order 15 : Definition 2.4. The fractional Riemann-Liouville integral for > 0 is defined as…”

Section: Fractional Calculusmentioning

confidence: 99%

On dual Bernstein polynomials and stochastic fractional integro‐differential equations

Sayevand

Machado

Masti

2020

Math Methods in App Sciences

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In recent years, random functional or stochastic equations have been reported in a large class of problems. In many cases, an exact analytical solution of such equations is not available and, therefore, is of great importance to obtain their numerical approximation. This study presents a numerical technique based on Bernstein operational matrices for a family of stochastic fractional integro-differential equations (SFIDE) by means of the trapezoidal rule. A relevant feature of this method is the conversion of the SFIDE into a linear system of algebraic equations that can be analyzed by numerical methods. An upper error bound, the convergence, and error analysis of the scheme are investigated. Three examples illustrate the accuracy and performance of the technique.

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Section: Fractional Calculusmentioning

confidence: 99%

On dual Bernstein polynomials and stochastic fractional integro‐differential equations

Sayevand

Machado

Masti

2020

Math Methods in App Sciences

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“…The properties and capabilities of B‐spline curves make them widely used in computer graphics, computer‐aided design, geometric modeling, and so on 37 . Numerous problems of FDEs and fractional partial differential equations are solved with the help of B‐spline polynomials 38‐43 …”

Section: Introductionmentioning

confidence: 99%

Optimal control problems with Atangana‐Baleanu fractional derivative

Tajadodi

Khan

Gómez‐Aguilar

et al. 2020

Optim Control Appl Methods

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SummaryIn this paper, we study fractional‐order optimal control problems (FOCPs) involving the Atangana‐Baleanu fractional derivative. A computational method based on B‐spline polynomials and their operational matrix of Atangana‐Baleanu fractional integration is proposed for the numerical solution of this class of problems. With this numerical technique, the FOCPs are reduced to a system of equations which are solved for the unknown parameters with the help of Mathematica® software. Our results show the applicability and usefulness of the numerical technique.

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“…Atangana and Goufo 29 extended the matched asymptotic method for fractional‐order boundary layers problems. Sayevand and Pichaghchi 30 had given an algorithm to solve the singularly perturbed boundary value problem of fractional order in ODEs. They defined the local fractional derivative and extended the matched asymptotic expansion method based on the properties of a local fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of time‐fractional singularly perturbed convection–diffusion problems with a delay in time

Kumar

Chakravarthy

Vigo‐Aguiar

2020

Math Methods in App Sciences

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This work is concerned with the development of a stable finite difference method (SFDM) for time‐fractional singularly perturbed convection–diffusion problems with a delay in time. The fractional derivative is considered in the Caputo sense. The SFDM is constructed based on the stability of the analytical solution. Unlike the other classical numerical methods for singular perturbation problems, this method works nicely on a uniform mesh. The method is easily adaptable on Shishkin mesh and also on any graded mesh in space. Error estimates are presented to show the convergence of the numerical scheme. To support the theory, numerical results are presented in tables for different values of the fractional derivative parameter and perturbation parameter.

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A novel operational matrix method for solving singularly perturbed boundary value problems of fractional multi-order

Cited by 15 publications

References 28 publications

On dual Bernstein polynomials and stochastic fractional integro‐differential equations

On dual Bernstein polynomials and stochastic fractional integro‐differential equations

Optimal control problems with Atangana‐Baleanu fractional derivative

Numerical solution of time‐fractional singularly perturbed convection–diffusion problems with a delay in time

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