Series solutions for the Laguerre and Lane-Emden fractional differential equations in the sense of conformable fractional derivative

Al–Zhour, Zeyad; Al-Mutairi, Nouf; Alrawajeh, Fatimah Abdulrahman; Alkhasawneh, Raed

doi:10.1016/j.aej.2019.11.012

Cited by 28 publications

(12 citation statements)

References 32 publications

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“…Furthermore, many research studies have been conducted on the theoretical and practical elements of conformable differential equations shortly after the proposition of this new definition [5,7,12,[18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. Conformable derivative has also been applied in modeling and investigating phenomena in applied sciences and engineering [12] such as the nonlinear Boussinesq equation's travelling wave solutions [36], the coupled nonlinear Schrödinger equations [34] and regularized long wave Burgers equation [35] deterministic and stochastics forms, the approximate long water wave equation's exact solutions [37], the (1 + 3)-Zakharov-Kuznetsov equation with power-law nonlinearity analytical and numerical solutions [38], the (2 + 1)-dimensional Zoomeron equation [39,40] and 3 rd -order modified KdV equation analytical solutions [39], and the exact solutions for Whitham-Broer-Kaup equation's three various models in shallow water [41].…”

Section: Introductionmentioning

confidence: 99%

Generalized Conformable Mean Value Theorems with Applications to Multivariable Calculus

Martínez

Kaabar

et al. 2021

Journal of Mathematics

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The conformable derivative and its properties have been recently introduced. In this research work, we propose and prove some new results on the conformable calculus. By using the definitions and results on conformable derivatives of higher order, we generalize the theorems of the mean value which follow the same argument as in the classical calculus. The value of conformable Taylor remainder is obtained through the generalized conformable theorem of the mean value. Finally, we introduce the conformable version of two interesting results of classical multivariable calculus via the conformable formula of finite increments.

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Section: Introductionmentioning

confidence: 99%

Generalized Conformable Mean Value Theorems with Applications to Multivariable Calculus

Martínez

Kaabar

et al. 2021

Journal of Mathematics

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“…[58][59][60] One of the approaches that makes it possible to establish the exact solutions of CFDEs is a modified version of the Kudryashov method, 34,[61][62][63][64] which has proved its merits and robustness in the solution of CFDEs. Till now, classical analytical-approximate methods 65,66,80 for some ordinary and partial CFDEs have been developed for Adomian decomposition method, 29,67 Sumudu transform method, 68 invariant subspace method, 53 first integral method, [69][70][71][72] homotopy analysis method, [73][74][75] residual power series method, 74,76 and other methods. [77][78][79] In recent years, many researchers have focused their attention on certain generalizations of Sturm-Liouville problems.…”

Section: Introductionmentioning

confidence: 99%

“…Some analytical and numerical methods have attracted great interest and became an important tool for differential equations with CFDs, (see previous studies 52‐80 ). Ünal et al 52 have presented a method based on the well‐known differential transform technique; it is suitable for finding the numerical solution of conformable fractional ordinary differential equations.…”

Section: Introductionmentioning

confidence: 99%

Two classes of conformable fractional Sturm‐Liouville problems: Theory and applications

Mortazaasl

Akbarfam

2020

Math Methods in App Sciences

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In this paper, we investigate in more detail some useful theorems related to conformable fractional derivative (CFD) and integral and introduce two classes of conformable fractional Sturm-Liouville problems (CFSLPs): namely, regular and singular CFSLPs. For both classes, we study some of the basic properties of the Sturm-Liouville theory. In the class of r-CFSLPs, we discuss two types of CFSLPs which include left-and right-sided CFDs, each of order ∈ (n, n + 1], and prove properties of the eigenvalues and the eigenfunctions in a certain Hilbert space. Also, we apply a fixed-point theorem for proving the existence and uniqueness of the eigenfunctions. As an operator for the class of s-CFSLPs, we first derive two fractional types of the hypergeometric differential equations of order ∈ (0, 1] and obtain their analytical eigensolutions as Gauss hypergeometric functions. Afterwards, we define the conformable fractional Legendre polynomial/functions (CFLP/Fs) as Jacobi polynomial and investigate their basic properties. Moreover, the conformable fractional integral Legendre transforms (CFILTs) based on CFLP/Fs-I and-II are introduced, and using these new transforms, an effective procedure for solving explicitly certain ordinary and partial conformable fractional differential equations (CFDEs) are given. Finally, as a theoretical application, some fractional diffusion equations are solved. KEYWORDS conformable fractional derivatives and integrals, eigenvalues and eigenfunctions, fixed-point theorem, fractional diffusion, fractional Sturm-Liouville operators, transform methods JEL CLASSIFICATION

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“…In [8], the conformable version of Euler´s Theorem on homogeneous is introduced. Furthermore, in a short time, various research studies have been conducted on the theory and applications of fractional differential equations in the context of this newly introduced fractional derivative, [9][10][11][12][13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Note on the Conformable Boundary Value Problems: Sturm’s Theorems and Green’s Function

Martínez

Kaabar³

et al. 2020

Preprint

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Recently, the conformable derivative and its properties have been introduced. In this paper, we propose and prove some new results on conformable Boundary Value Problems. First, we introduce a conformable version of classical Sturm´s separation, and comparison theorems. For a conformable Sturm-Liouville problem, Green's function is constructed, and its properties are also studied. In addition, we propose the applicability of the Green´s Function in solving conformable inhomogeneous linear differential equations with homogeneous boundary conditions, whose associated homogeneous boundary value problem has only trivial solution. Finally, we prove the generalized Hyers-Ulam stability of the conformable inhomogeneous boundary value problem.

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Series solutions for the Laguerre and Lane-Emden fractional differential equations in the sense of conformable fractional derivative

Cited by 28 publications

References 32 publications

Generalized Conformable Mean Value Theorems with Applications to Multivariable Calculus

Generalized Conformable Mean Value Theorems with Applications to Multivariable Calculus

Two classes of conformable fractional Sturm‐Liouville problems: Theory and applications

Note on the Conformable Boundary Value Problems: Sturm’s Theorems and Green’s Function

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