The principal purpose of this work is to present an efficient analytical approach based on the q-homotopy analysis transform technique in order to analyze a fractional model of the vibration equation ...for large membranes. The Laplace decomposition technique is also employed to obtain the solution of the fractional-order vibration equation. Numerical examples are provided to examine the efficiency of the proposed schemes and the results derived are demonstrated graphically. Our examples illustrate that the suggested computational approaches are straightforward to perform and computationally very striking.
•Prabhakar and related operators can be expressed as series of Riemann–Liouville operators.•Fundamental properties of Prabhakar operators are recovered from the series formulae.•The product and chain ...rules hold for Prabhakar fractional-calculus operators.•Fractional iteration for these operators is discussed.
We consider an integral transform introduced by Prabhakar, involving generalised multi-parameter Mittag-Leffler functions, which can be used to introduce and investigate several different models of fractional calculus. We derive a new series expression for this transform, in terms of classical Riemann–Liouville fractional integrals, and use it to obtain or verify series formulae in various specific cases corresponding to different fractional-calculus models. We demonstrate the power of our result by applying the series formula to derive analogues of the product and chain rules in more general fractional contexts. We also discuss how the Prabhakar model can be used to explore the idea of fractional iteration in connection with semigroup properties.
In this paper, a family of local fractional two-dimensional Burgers-type equations (2DBEs) is investigated. The local fractional Riccati differential equation method is proposed here for the first ...time. The travelling wave transformation of the non-differentiable type is presented. The non-differentiable exact travelling wave solutions for the problems are obtained. The present methodology is shown to provide a useful approach to solve the local fractional nonlinear partial differential equations (LFNPDEs) in mathematical physics.
This article investigates a family of approximate solutions for the fractional model (in the Liouville-Caputo sense) of the Ebola virus via an accurate numerical procedure (Chebyshev spectral ...collocation method). We reduce the proposed epidemiological model to a system of algebraic equations with the help of the properties of the Chebyshev polynomials of the third kind. Some theorems about the convergence analysis and the existence-uniqueness solution are stated. Finally, some numerical simulations are presented for different values of the fractional-order and the other parameters involved in the coefficients. We also note that we can apply the proposed method to solve other models.
Recently, the authors introduced some generalizations of the Apostol–Bernoulli polynomials and the Apostol–Euler polynomials (see Q.-M. Luo, H.M. Srivastava, J. Math. Anal. Appl. 308 (2005) 290–302 ...and Q.-M. Luo, Taiwanese J. Math. 10 (2006) 917–925). The main object of this paper is to investigate an analogous generalization of the Genocchi polynomials of higher order, that is, the so-called Apostol–Genocchi polynomials of higher order. For these generalized Apostol–Genocchi polynomials, we establish several elementary properties, provide some explicit relationships with the Apostol–Bernoulli polynomials and the Apostol–Euler polynomials, and derive various explicit series representations in terms of the Gaussian hypergeometric function and the Hurwitz (or generalized) zeta function. We also deduce their special cases and applications which are shown here to lead to the corresponding results for the Genocchi and Euler polynomials of higher order. By introducing an analogue of the Stirling numbers of the second kind, that is, the so-called λ-Stirling numbers of the second kind, we derive some basic properties and formulas and consider some interesting applications to the family of the Apostol type polynomials. Furthermore, we also correct an error in a previous paper Q.-M. Luo, H.M. Srivastava, Comput. Math. Appl. 51 (2006) 631–642 and pose two open problems on the subject of our investigation.
In this paper, we introduce and investigate a fractional calculus with an integral operator which contains the following family of generalized Mittag–Leffler functions:
E
α
,
β
γ
,
κ
(
z
)
=
∑
n
=
0
...∞
(
γ
)
κ
n
Γ
(
α
n
+
β
)
z
n
n
!
(
z
,
β
,
γ
∈
C
;
R
(
α
)
>
max
{
0
,
R
(
κ
)
-
1
}
;
R
(
κ
)
>
0
)
in its kernel,
(
λ
)
ν
being the familiar Pochhammer symbol. A number of corollaries and consequences of the main results presented here are also considered.
In the present paper, we introduce and investigate two interesting subclasses of normalized analytic and univalent functions in the open unit disk
U
≔
{
z
:
z
∈
C
and
|
z
|
<
1
}
,
whose inverse has ...univalently analytic continuation to
U
. Among other results, bounds for the Taylor–Maclaurin coefficients
|
a
2
|
and
|
a
3
|
are found in our investigation.
The main purpose of this work is to study the dynamics of a fractional-order Covid-19 model. An efficient computational method, which is based on the discretization of the domain and memory ...principle, is proposed to solve this fractional-order corona model numerically and the stability of the proposed method is also discussed. Efficiency of the proposed method is shown by listing the CPU time. It is shown that this method will work also for long-time behaviour. Numerical results and illustrative graphical simulation are given. The proposed discretization technique involves low computational cost.
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