Analytical solutions of some nonlinear fractional‐order differential equations by different methods

Odabaşı, Meryem; Pınar, Zehra; Koçak, Hüseyin

doi:10.1002/mma.6313

Cited by 11 publications

(4 citation statements)

References 28 publications

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“…Some authors [11][12][13][14][15][16] have studied fractional differential equations by different methods. Besides these, to investigate analytical solutions of PDEs, Liu proposed an approach called trial equation method which aims to reduce the examined equation to the solvable differential equations.…”

Section: Conformable Fractional Derivative and Trial Solution Algorithmmentioning

confidence: 99%

Exact analytical solutions of the fractional biological population model, fractional EW and modified EW equations

Odabaşı

2020

Int. J. Optim. Control, Theor. Appl. (IJOCTA)

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In this paper, exact analytical solutions of the biological population model, the EW and the modified EW equations with a conformable derivative operator have been examined by means of the trial solution algorithm and the complete discrimination system. Dark, bright and singular traveling wave solutions of the equations have been obtained by algorithm. Also, revealed singular periodic solutions have been listed. All solutions were verified by substituting them into their corresponding equation via Mathematica package program.

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Section: Conformable Fractional Derivative and Trial Solution Algorithmmentioning

confidence: 99%

Exact analytical solutions of the fractional biological population model, fractional EW and modified EW equations

Odabaşı

2020

Int. J. Optim. Control, Theor. Appl. (IJOCTA)

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“…In later years, it was the subject of major different publications by a lot of remarkable scientists. Odabasi et al have developed the unique methods by applying for some nonlinear fractional order differential equations [11]. By Belgacem et al, Natural transform have been introduced to the literature [12].…”

Section: Introductionmentioning

confidence: 99%

Representation of the solution of the M-Sturm-Liouville problem with natural transform

Bas,

Karaoglan

2023

International Journal of Mathematics and Computer in Engineering

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In this article, we develop the natural transform in terms of the M-derivative improving the basic notions for a new interesting version of the fractional transform. The properties and relations of certain functions for the natural transform of the M-derivative are introduced. The natural transform with the M-derivative is the more general version of the natural transform for the conformable operator. Furthermore, this method is successfully applied to find the general solutions of some fractional differential equations with M-derivative. We propose a significant spectral data with boundary conditions under M-Sturm-Liouville problem. We offer the representation of the solution for the M-Sturm-Liouville problem, depending on both the boundary and initial conditions. Our main aim is to extract the solution representation of the M-Sturm-Liouville problem by using the natural transform and observe the problem by supporting the spectral structure of the M-Sturm- Liouville problem with graphs. Finally, these results show that our new approach is simple, effective and accurate.

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“…For other recent developments in the literature on theoretical and numerical studies of FDEs, see for example [10][11][12] for the inverse problem associated with FDEs, ref. [13] for Hermite-Hadamard-type inequalities, [14][15][16] for numerical methods of FDEs, and [17][18][19] for analytical and numerical solutions of some FDEs.…”

Section: Introductionmentioning

confidence: 99%

On the Stability and Numerical Scheme of Fractional Differential Equations with Application to Biology

Hattaf

2022

Computation

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The fractional differential equations involving different types of fractional derivatives are currently used in many fields of science and engineering. Therefore, the first purpose of this study is to investigate the qualitative properties including the stability, asymptotic stability, as well as Mittag–Leffler stability of solutions of fractional differential equations with the new generalized Hattaf fractional derivative, which encompasses the popular forms of fractional derivatives with non-singular kernels. These qualitative properties are obtained by constructing a suitable Lyapunov function. Furthermore, the second aim is to develop a new numerical method in order to approximate the solutions of such types of equations. The developed method recovers the classical Euler numerical scheme for ordinary differential equations. Finally, the obtained analytical and numerical results are applied to a biological nonlinear system arising from epidemiology.

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Analytical solutions of some nonlinear fractional‐order differential equations by different methods

Cited by 11 publications

References 28 publications

Exact analytical solutions of the fractional biological population model, fractional EW and modified EW equations

Exact analytical solutions of the fractional biological population model, fractional EW and modified EW equations

Representation of the solution of the M-Sturm-Liouville problem with natural transform

On the Stability and Numerical Scheme of Fractional Differential Equations with Application to Biology

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