Existence Results for Solutions of Nonlinear Fractional Differential Equations

Yakar, Ali; Köksal, Mehmet Emir

doi:10.1155/2012/267108

Cited by 19 publications

(11 citation statements)

References 37 publications

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“…It is only a few decades ago, it was realized that the derivatives of arbitrary order provide an excellent framework for modeling the real world problems in a variety of disciplines. There has been a growing interest in this new area to study the concept of fractional differential equations and fractional dynamical systems [11][12][13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Initial time difference quasilinearization for Caputo Fractional Differential Equations

Yakar

2012

Adv Differ Equ

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This paper deals with an application of the method of quasilinearization by not demanding the Hölder continuity assumption of functions involved and by choosing upper and lower solutions with initial time difference for nonlinear Caputo fractional differential equations. Thus, we construct monotone flows that are generated by solutions of linear fractional differential equations which converge uniformly and quadratically to the unique solution of the problem. Also, necessary comparison result concerning lower and upper solutions are proved without using Hölder continuity. Mathematics subject classification: 34A12, 34A45, 34C11.

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Section: Introductionmentioning

confidence: 99%

Initial time difference quasilinearization for Caputo Fractional Differential Equations

Yakar

2012

Adv Differ Equ

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“…Some fundamental descriptions and applications of fractional calculus are given in [5] and [6]. Existence and uniqueness of the solutions are also studied in [7].…”

Section: Introductionmentioning

confidence: 99%

New Analytical Solutions of Fractional Symmetric Regularized-Long-Wave Equation

Şenol

2020

Rev. Mex. Fís.

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In this study, new extended direct algebraic method is successfully implemented to acquire new exact wave solution sets for symmetric regularized-long-wave (SRLW) equation which arise in long water flow models. By the help of Mathematica symbolic calculation package, the method produced a great number of analytical solutions. We also presented a few graphical illustrations for some surfaces. The fractional derivatives are considered in the conformable sense. All of the solutions were checked by substitution to ensure the reliability of the method. Obtained results confirm that the method is straightforward, powerful and effective method to attain exact solutions for nonlinear fractional differential equations. Therefore, the method is a good candidate to take part in the existing literature.

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“…That is why, it is owing to the fact that each of fractional calculus and impulsive theory serves very practical instruments for mathematical modeling of many concepts in different branches of science and engineering [1][2][3][4][5][6][7]. See [8][9][10][11][12][13][14][15][16][17][18][19][20][21] for some recent works on fractional differential equations and inclusions, and see [22][23][24][25][26][27][28][29][30][31] for the ones on impulsive fractional differential equations and inclusions.…”

Section: Introductionmentioning

confidence: 99%