Solution of singular one-dimensional Boussinesq equation by using double conformable Laplace decomposition method

Eltayeb, Hassan; Bachar, Imed; Gad-Allah, Musa R.

doi:10.1186/s13662-019-2230-1

Cited by 17 publications

(12 citation statements)

References 16 publications

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“…So that, utilizing (58) with = 0 and = − 1, we achieve (55). Moreover, from the properties of the Jacobi polynomials, we have…”

Section: Theorem 49 Fractional Legendre Polynomials Of the First Kindmentioning

confidence: 94%

“…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%

“…The single Laplace transform method was first introduced and used in Hashemi 53 . Also in Özkan and Kurt, 54 the idea was extended to the conformable double Laplace transform, and then, the new conformable fractional double Laplace transform decomposition method recommended for developing the solutions of linear and nonlinear singular one‐dimensional conformable Pseudohyperbolic and Boussinesq equations, see other works 55,56 for more detail. Analytical solutions of the conformable nonlinear time‐fractional wave equations by the Jacobi elliptic function expansion method have been investigated in Tasbozan et al 57 Furthermore, Painlevé Burgers, Nizhnik‐Novikov‐Veselov, and Klein‐Gordon equations and fractional DSW system with the CFD have been solved by using the expansion method in other works 58‐60 .…”

Section: Introductionmentioning

confidence: 99%

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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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“…So that, utilizing (58) with = 0 and = − 1, we achieve (55). Moreover, from the properties of the Jacobi polynomials, we have…”

Section: Theorem 49 Fractional Legendre Polynomials Of the First Kindmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Mortazaasl

Akbarfam

2020

Math Methods in App Sciences

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“…is new fractional derivative quickly becomes the subject of many contributions in several areas of science [11][12][13][14][15][16][17][18][19][20][21][22]. Motivated by the better effect of the fractional derivative and simple properties of the conformable fractional derivative, we consider model (1) in the framework of conformable fractional calculus.…”

Section: Introductionmentioning

confidence: 99%

Nonlocal Conformable-Fractional Differential Equations with a Measure of Noncompactness in Banach Spaces

Bouaouid

Hannabou

Hilal

2020

Journal of Mathematics

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is paper deals with the existence of mild solutions for the following Cauchy problem:where d α (.)/dt α is the so-called conformable fractional derivative. e linear part A is the infinitesimal generator of a uniformly continuous semigroup (T(t)) t≥0 on a Banach space X, f and g are given functions. e main result is proved by using the Darbo-Sadovskii fixed point theorem without assuming the compactness of the family (T(t)) t>0 and the Lipshitz condition on the nonlocal part g.

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“…Fractional partial differential equations as generalizations of classical partial differential equations, and they have been proposed and applied to many applications in various fields of physical sciences and engineering such as electromagnetic, acoustics, visco-elasticity and electro-chemistry. Recently, the solution of fractional partial differential equations has been obtained through a double Laplace decomposition method by the authors [1][2][3]. The natural transform decomposition method has been successfully used to handle linear and nonlinear problems appearing in physical and engineering disciplines [4,5].…”

Section: Introductionmentioning

confidence: 99%

A note on time-fractional Navier–Stokes equation and multi-Laplace transform decomposition method

2020

Self Cite

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In this study, the double Laplace Adomian decomposition method and the triple Laplace Adomian decomposition method are employed to solve one- and two-dimensional time-fractional Navier–Stokes problems, respectively. In order to examine the applicability of these methods some examples are provided. The presented results confirm that the proposed methods are very effective in the search of exact and approximate solutions for the problems. Numerical simulation is used to sketch the exact and approximate solution.

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Solution of singular one-dimensional Boussinesq equation by using double conformable Laplace decomposition method

Cited by 17 publications

References 16 publications

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

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

Nonlocal Conformable-Fractional Differential Equations with a Measure of Noncompactness in Banach Spaces

A note on time-fractional Navier–Stokes equation and multi-Laplace transform decomposition method

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