Exact Solution of Two-Term Nonlinear Fractional Differential Equation with Sequential Riemann-Liouville Derivatives

Błasik, Marek; Klimek, Małgorzata

doi:10.1007/978-3-319-00933-9_14

Cited by 5 publications

(4 citation statements)

References 12 publications

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“…The integrand in formulae (1-3) is a product of two functions: integrated function (22)(23)(24) and kernel of RiemannLiouville/Caputo formulae.…”

Section: Numerical Accuracy Analysismentioning

confidence: 99%

“…For the purpose of the following comparison analysis, there are calculated derivatives and integrals of selected functions (22)(23)(24) in the (0, 1) range of the fractional orders 0, 1 with step 0.1. Additionally derivatives and integrals of "difficult" orders as for example 0.001 and 0.999 are calculated to test the developed numerical integration methods abilities in the most difficult conditions, in which available methods of numerical integration deliver results usually burdened with 100-200% of relative error.…”

Section: Error Definitionmentioning

confidence: 99%

“…The authors also plans to apply the methods to solve diffusion-wave equation with higher accuracy, inspired by existing works [22][23][24].…”

Section: Dw Brzeziński and P Ostalczykmentioning

confidence: 99%

See 2 more Smart Citations

High-accuracy numerical integration methods for fractional order derivatives and integrals computations

Brzeziński¹,

Ostalczyk²

2014

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. In this paper the authors present highly accurate and remarkably efficient computational methods for fractional order derivatives and integrals applying Riemann-Liouville and Caputo formulae: the Gauss-Jacobi Quadrature with adopted weight function, the Double Exponential Formula, applying two arbitrary precision and exact rounding mathematical libraries (GNU GMP and GNU MPFR). Example fractional order derivatives and integrals of some elementary functions are calculated. Resulting accuracy is compared with accuracy achieved by applying widely known methods of numerical integration. Finally, presented methods are applied to solve Abel's Integral equation (in Appendix).Key words: accuracy of numerical calculations, fractional order derivatives and integrals, double exponential formula, gauss-jacobi quadrature with adopted weight function, arbitrary precision, numerical integration, abel's integral equation.

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“…The integrand in formulae (1-3) is a product of two functions: integrated function (22)(23)(24) and kernel of RiemannLiouville/Caputo formulae.…”

Section: Numerical Accuracy Analysismentioning

confidence: 99%

Section: Error Definitionmentioning

confidence: 99%

See 1 more Smart Citation

High-accuracy numerical integration methods for fractional order derivatives and integrals computations

Brzeziński¹,

Ostalczyk²

2014

Bulletin of the Polish Academy of Sciences Technical Sciences

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“…Recently, papers have been published that deal with the existence and multiplicity of the solution of nonlinear initial fractional differential equation by the use of techniques of nonlinear analysis, see [15][16][17]. However, most of the papers offer the problem using the standard Riemann-Liouville differentiation, see [18,19]. However, Our aim is to study the existence and the uniqueness of the solution for a class of fractional boundary value problems.…”

Section: Introductionmentioning

confidence: 99%

Analytical and numerical study for a fractional boundary value problem with a conformable fractional derivative of Caputo and its fractional integral

Bekkouche¹,

Guebbai²

2020

J. Appl. Math. Comput. Mech.

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We study the existence and uniqueness of the solution of a fractional boundary value problem with conformable fractional derivation of the Caputo type, which increases the interest of this study. In order to study this problem we have introduced a new definition of fractional integral as an inverse of the conformable fractional derivative of Caputo, therefore, the proofs are based upon the reduction of the problem to a equivalent linear Volterra-Fredholm integral equations of the second kind, and we have built the minimum conditions to obtain the existence and uniqueness of this solution. The analytical study is followed by a complete numerical study.

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On the solvability fractional of a boundary value problem with new fractional integral

Bekkouche

Guebbai²,

Kurulay

2020

J. Appl. Math. Comput.

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Exact Solution of Two-Term Nonlinear Fractional Differential Equation with Sequential Riemann-Liouville Derivatives

Cited by 5 publications

References 12 publications

High-accuracy numerical integration methods for fractional order derivatives and integrals computations

High-accuracy numerical integration methods for fractional order derivatives and integrals computations

Analytical and numerical study for a fractional boundary value problem with a conformable fractional derivative of Caputo and its fractional integral

On the solvability fractional of a boundary value problem with new fractional integral

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