An inverse problem for a multi‐term fractional differential equation with two‐parameter fractional derivatives in time and Bessel operator

2021 ◽  
Author(s):  
Arifa Samreen ◽  
Salman A. Malik
Keyword(s):  
Differential Equation ◽  
Inverse Problem ◽  
Fractional Differential Equation ◽  
Fractional Derivatives ◽  
Bessel Operator ◽  
Fractional Differential ◽  
Two Parameter

Operational method for solving fractional differential equations with the left-and right-hand sided Erdélyi-Kober fractional derivatives

2020 ◽  
Vol 23 (1) ◽  
pp. 103-125 ◽  
Author(s):  
Latif A-M. Hanna ◽  
Maryam Al-Kandari ◽  
Yuri Luchko
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Fractional Derivatives ◽  
Fractional Integrals ◽  
Operational Method ◽  
Initial Value ◽  
Right Hand ◽  
Fractional Differential ◽  
Left And Right ◽  
Fractional Integrals And Derivatives

AbstractIn this paper, we first provide a survey of some basic properties of the left-and right-hand sided Erdélyi-Kober fractional integrals and derivatives and introduce their compositions in form of the composed Erdélyi-Kober operators. Then we derive a convolutional representation for the composed Erdélyi-Kober fractional integral in terms of its convolution in the Dimovski sense. For this convolution, we also determine the divisors of zero. These both results are then used for construction of an operational method for solving an initial value problem for a fractional differential equation with the left-and right-hand sided Erdélyi-Kober fractional derivatives defined on the positive semi-axis. Its solution is obtained in terms of the four-parameters Wright function of the second kind. The same operational method can be employed for other fractional differential equation with the left-and right-hand sided Erdélyi-Kober fractional derivatives.

scholarly journals Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales

2016 ◽  
Vol 2016 ◽  
pp. 1-21 ◽  
Author(s):  
Yanning Wang ◽  
Jianwen Zhou ◽  
Yongkun Li
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Fractional Differential Equation ◽  
Time Scales ◽  
Fractional Derivatives ◽  
Point Theory ◽  
Existence Results ◽  
Recent Approach ◽  
Equation Boundary ◽  
Fractional Differential

Using conformable fractional calculus on time scales, we first introduce fractional Sobolev spaces on time scales, characterize them, and define weak conformable fractional derivatives. Second, we prove the equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, uniform convexity, and compactness of some imbeddings, which can be regarded as a novelty item. Then, as an application, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for ap-Laplacian conformable fractional differential equation boundary value problem on time scaleT:  Tα(Tαup-2Tα(u))(t)=∇F(σ(t),u(σ(t))),Δ-a.e.  t∈a,bTκ2,u(a)-u(b)=0,Tα(u)(a)-Tα(u)(b)=0,whereTα(u)(t)denotes the conformable fractional derivative ofuof orderαatt,σis the forward jump operator,a,b∈T,  0<a<b,  p>1, andF:[0,T]T×RN→R. By establishing a proper variational setting, we obtain three existence results. Finally, we present two examples to illustrate the feasibility and effectiveness of the existence results.

Inverse problem for a multi-term fractional differential equation

2020 ◽  
Vol 23 (3) ◽  
pp. 799-821
Author(s):  
Muhammad Ali ◽  
Sara Aziz ◽  
Salman A. Malik
Keyword(s):  
Differential Equation ◽  
Inverse Problem ◽  
Fractional Differential Equation ◽  
Local Existence ◽  
Operational Calculus ◽  
Banach Fixed Point Theorem ◽  
Local Boundary ◽  
Fractional Differential ◽  
Problem Stability ◽  
Orthogonal Set

AbstractInverse problem for a family of multi-term time fractional differential equation with non-local boundary conditions is studied. The spectral operator of the considered problem is non-self-adjoint and a bi-orthogonal set of functions is used to construct the solution. The operational calculus approach has been used to obtain the solution of the multi-term time fractional differential equations. Integral type over-determination condition is considered for unique solvability of the inverse problem. Different estimates of multinomial Mittag-Leffler functions alongside Banach fixed point theorem are used to prove the unique local existence of the solution of the inverse problem. Stability of the solution of the inverse problem on the given datum is established.

Well-Posedness of Fractional Differential Equations

2017 ◽  
Author(s):  
Changpin Li ◽  
Li Ma
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Well Posedness ◽  
Fractional Differential ◽  
Well Posed

Generally speaking, definite conditions of fractional differential equations with Riemann-Liouville, Riesz or Hadamard fractional derivatives are quite different from those of classic differential equations. In this paper, we propose the well-posed conditions for fractional differential equation involving right Riemann-Liouville, Riesz and Hadamard fractional derivatives.

scholarly journals Positive Solutions for a Higher-Order Semipositone Nonlocal Fractional Differential Equation with Singularities on Both Time and Space Variable

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Kemei Zhang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fractional Differential Equation ◽  
Point Theorem ◽  
Fractional Derivatives ◽  
Space Variable ◽  
Higher Order ◽  
Existence Results ◽  
Fractional Differential

In this paper, we consider the following higher-order semipositone nonlocal Riemann-Liouville fractional differential equation D0+αx(t)+f(t,x(t),D0+βx(t))+e(t)=0,  0<t<1,D0+βx(0)=D0+β+1x(0)=⋯=D0+n+β-2x(0)=0, and D0+βx(1)=∑i=1m-2ηiD0+βx(ξi), where D0+α and D0+β are the standard Riemann-Liouville fractional derivatives. The existence results of positive solution are given by Guo-krasnosel’skii fixed point theorem and Schauder’s fixed point theorem.

Attractivity for differential equations of fractional order and ψ-Hilfer type

2020 ◽  
Vol 23 (4) ◽  
pp. 1188-1207
Author(s):  
J. Vanterler da C. Sousa ◽  
Mouffak Benchohra ◽  
Gaston M. N’Guérékata
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fractional Derivatives ◽  
Broad Class ◽  
Hilfer Fractional Derivative ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Fractional Differential

AbstractThis paper investigates the overall solution attractivity of the fractional differential equation involving the ψ-Hilfer fractional derivative and using the Krasnoselskii’s fixed point theorem. We highlight some particular cases of the results presented here, especially involving the Riemann-Liouville, thus illustrating the broad class of fractional derivatives to which these results can be applied.

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