Representation of solutions for Sturm–Liouville eigenvalue problems with generalized fractional derivative

2020 ◽  
Vol 30 (3) ◽  
pp. 033137 ◽  
Author(s):  
Ramazan Ozarslan ◽  
Erdal Bas ◽  
Dumitru Baleanu
Keyword(s):  
Fractional Derivative ◽  
Eigenvalue Problems ◽  
Representation Of Solutions ◽  
Sturm Liouville ◽  
Generalized Fractional Derivative

scholarly journals Delta Fractional Sturm--Liouville Problems: From Discrete to Continuous

2021 ◽  
Author(s):  
Ramazan Ozarslan ◽  
Erdal Bas
Keyword(s):  
Initial Value Problems ◽  
Eigenvalue Problems ◽  
Point Of View ◽  
Difference Operators ◽  
Initial Value ◽  
Fractional Difference ◽  
Differential Problem ◽  
Representation Of Solutions ◽  
Sturm Liouville ◽  
Z Transformation

In this study, we consider delta fractional Sturm--Liouville (DFSL) initial value problems in the sense of delta Caputo and delta Riemann-Liouville (R--L) operators. One of the properties of delta fractional difference operators which makes it different from nabla counterpart is to shift its domain. This feature makes it more complex than the nabla fractional operator. We obtain sum representation of solutions for DFSL initial value problems with the help of $\mathcal{Z}-$ transformation. Moreover, we get analytical solutions of homogeneous DFSL problem within Riemann-Liouville (R--L) and Caputo sense, discrete Sturm--Liouville (DSL) problem, continuous fractional Sturm--Liouville (FSL) problem in the sense of R--L and Caputo operators, and continuous Sturm--Liouville (SL) differential problem. From this point of view, we compare all the solutions with each other. Consequently, we show that all results for these four eigenvalue problems are compatible with each other and approach to each other while the orders tends to one, i.e. $\Delta^{\mu }\left( \Delta x\left( t-\mu \right) \right)\cong D_{0^{+}}^{\mu }\left( x^{\prime }\left( t\right) \right)\cong \Delta^2x(n-1) \cong x^{\prime \prime }\left( t\right) =\lambda x\left( t\right),\ \mu\rightarrow1 $ . We support our results comparatively by tables and simulations in detail.

scholarly journals The Principle of Differential Subordination and Its Application to Analytic and p-Valent Functions Defined by a Generalized Fractional Differintegral Operator

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1083 ◽  
Author(s):  
Nak Eun Cho ◽  
Mohamed Kamal Aouf ◽  
Rekha Srivastava
Keyword(s):  
Fractional Derivative ◽  
Integral Operators ◽  
Differential Subordination ◽  
Open Unit ◽  
Open Unit Disk ◽  
The Family ◽  
Sandwich Type ◽  
Fractional Differintegral ◽  
Generalized Fractional Derivative ◽  
Fractional Differintegral Operator

A useful family of fractional derivative and integral operators plays a crucial role on the study of mathematics and applied science. In this paper, we introduce an operator defined on the family of analytic functions in the open unit disk by using the generalized fractional derivative and integral operator with convolution. For this operator, we study the subordination-preserving properties and their dual problems. Differential sandwich-type results for this operator are also investigated.

scholarly journals Homotopy Perturbation ρ-Laplace Transform Method and Its Application to the Fractional Diffusion Equation and the Fractional Diffusion-Reaction Equation

2019 ◽  
Vol 3 (2) ◽  
pp. 14 ◽  
Author(s):  
Ndolane Sene ◽  
Aliou Niang Fall
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Diffusion Reaction ◽  
Reaction Equation ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Generalized Fractional Derivative

In this paper, the approximate solutions of the fractional diffusion equations described by the fractional derivative operator were investigated. The homotopy perturbation Laplace transform method of getting the approximate solution was proposed. The Caputo generalized fractional derivative was used. The effects of the orders α and ρ in the diffusion processes was addressed. The graphical representations of the approximate solutions of the fractional diffusion equation and the fractional diffusion-reaction equation both described by the Caputo generalized fractional derivative were provided.

scholarly journals Stability Results for Implicit Fractional Pantograph Differential Equations via ϕ-Hilfer Fractional Derivative with a Nonlocal Riemann-Liouville Fractional Integral Condition

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 94 ◽  
Author(s):  
Idris Ahmed ◽  
Poom Kumam ◽  
Kamal Shah ◽  
Piyachat Borisut ◽  
Kanokwan Sitthithakerngkiet ◽  
...  
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Fixed Point Theorems ◽  
Integral Condition ◽  
Hilfer Fractional Derivative ◽  
Fractional Differential ◽  
Generalized Fractional Derivative ◽  
Stability Results

This paper presents a class of implicit pantograph fractional differential equation with more general Riemann-Liouville fractional integral condition. A certain class of generalized fractional derivative is used to set the problem. The existence and uniqueness of the problem is obtained using Schaefer’s and Banach fixed point theorems. In addition, the Ulam-Hyers and generalized Ulam-Hyers stability of the problem are established. Finally, some examples are given to illustrative the results.

scholarly journals ANALYSIS OF DENGUE FEVER OUTBREAK BY GENERALIZED FRACTIONAL DERIVATIVE

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040038
Author(s):  
PAUL BOSCH ◽  
J. F. GÓMEZ-AGUILAR ◽  
JOSÉ M. RODRÍGUEZ ◽  
JOSÉ M. SIGARRETA
Keyword(s):  
Fractional Derivative ◽  
Dengue Fever ◽  
Gaussian Model ◽  
Integral Operators ◽  
Real Data ◽  
Practical Application ◽  
Gaussian Models ◽  
Fractional Differential ◽  
Generalized Differential ◽  
Generalized Fractional Derivative

In this paper, we use the generalized fractional derivative in order to study the fractional differential equation associated with a fractional Gaussian model. Moreover, we propose new properties of generalized differential and integral operators. As a practical application, we estimate the order of the derivative of the fractional Gaussian models by solving an inverse problem involving real data on the dengue fever outbreak.

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