scholarly journals Coupled system of a fractional order differential equations with weighted initial conditions

2019 ◽  
Vol 17 (1) ◽  
pp. 1737-1749
Author(s):  
Ahmed M. A. El-Sayed ◽  
Sheren A. Abd El-Salam
Keyword(s):  
Fractional Order ◽  
Continuous Dependence ◽  
Initial Conditions ◽  
Coupled System ◽  
Cauchy Type ◽  
Type Problem ◽  
Integrable Solution ◽  
Fractional Order Differential Equations ◽  
Cauchy Type Problem ◽  
Uniqueness Of The Solution

Abstract Here, a coupled system of nonlinear weighted Cauchy-type problem of a diffre-integral equations of fractional order will be considered. We study the existence of at least one integrable solution of this system by using Schauder fixed point Theorem. The continuous dependence of the uniqueness of the solution is proved.

scholarly journals Existence and Uniqueness of Solution for a Class of Nonlinear Fractional Order Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Azizollah Babakhani ◽  
Dumitru Baleanu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Uniqueness Of Solution ◽  
Fractional Order Differential Equations ◽  
Banach Contraction ◽  
Uniqueness Of The Solution

We discuss the existence and uniqueness of solution to nonlinear fractional order ordinary differential equations(Dα-ρtDβ)x(t)=f(t,x(t),Dγx(t)),t∈(0,1)with boundary conditionsx(0)=x0,  x(1)=x1or satisfying the initial conditionsx(0)=0,  x′(0)=1, whereDαdenotes Caputo fractional derivative,ρis constant,1<α<2,and0<β+γ≤α. Schauder's fixed-point theorem was used to establish the existence of the solution. Banach contraction principle was used to show the uniqueness of the solution under certain conditions onf.

scholarly journals On the Operator Method for Solving Linear Integro-Differential Equations with Fractional Conformable Derivatives

2021 ◽  
Vol 5 (3) ◽  
pp. 109
Author(s):  
Batirkhan Kh. Turmetov ◽  
Kairat I. Usmanov ◽  
Kulzina Zh. Nazarova
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Fractional Order ◽  
Operator Method ◽  
Explicit Solutions ◽  
Cauchy Type ◽  
Type Problem ◽  
Volterra Type ◽  
Cauchy Type Problem ◽  
Conformable Derivatives

The methods for constructing solutions to integro-differential equations of the Volterra type are considered. The equations are related to fractional conformable derivatives. Explicit solutions of homogeneous and inhomogeneous equations are constructed, and a Cauchy-type problem is studied. It should be noted that the considered method is based on the construction of normalized systems of functions with respect to a differential operator of fractional order.

scholarly journals On the Operator Method for Solving Linear Integro-Differential Equations with Fractional Conformable Derivatives

2021 ◽  
Author(s):  
Batirkhan kh. Turmetov ◽  
Kairat I. Usmanov ◽  
Kulzina Zh. Nazarova
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Fractional Order ◽  
Operator Method ◽  
Explicit Solutions ◽  
Cauchy Type ◽  
Type Problem ◽  
Volterra Type ◽  
Cauchy Type Problem ◽  
Conformable Derivatives

The methods for constructing solutions to integro-differential equations of the Volterra type are considered. The equations are related to fractional conformable derivatives. Explicit solutions of homogeneous and inhomogeneous equations are constructed and a Cauchy-type problem is studied. It should be noted that the considered method is based on the construction of normalized systems of functions with respect to a differential operator of fractional order.

scholarly journals On the Unique Solvability of Incomplete Cauchy Type Problems for a Class of Multi-Term Equations with the Riemann–Liouville Derivatives

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 75
Author(s):  
Vladimir E. Fedorov ◽  
Wei-Shih Du ◽  
Mikhail M. Turov
Keyword(s):  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Initial Boundary ◽  
Cauchy Type ◽  
Initial Boundary Value Problems ◽  
Type Problem ◽  
Spatial Variables ◽  
Closed Operators ◽  
Cauchy Type Problem ◽  
Initial Boundary Value

Incomplete Cauchy-type problems are considered for linear multi-term equations solved with respect to the highest derivative in Banach spaces with fractional Riemann–Liouville derivatives and with linear closed operators at them. Some new existence and uniqueness theorems for solutions are presented explicitly and the analyticity of the solutions of the homogeneous equations are also shown. The asymmetry of the Cauchy-type problem under study is expressed in the presence of a so-called defect, which shows the number of lower-order initial conditions that should not be set when setting the problem. As applications, our abstract results are used in the study of a class of initial-boundary value problems for multi-term equations with Riemann–Liouville derivatives in time and with polynomials of a self-adjoint elliptic differential operator with respect to spatial variables.

Expansions with Poisson kernels and related topics

2010 ◽  
Vol 53 (1) ◽  
pp. 153-173 ◽  
Author(s):  
Cristina Giannotti ◽  
Paolo Manselli
Keyword(s):  
Unit Disc ◽  
Poisson Kernel ◽  
Cauchy Type ◽  
Two Dimensional ◽  
Type Problem ◽  
Special Sequence ◽  
Poisson Kernels ◽  
Cauchy Type Problem ◽  
In Series

AbstractLet P(r, θ) be the two-dimensional Poisson kernel in the unit disc D. It is proved that there exists a special sequence {ak} of points of D which is non-tangentially dense for ∂D and such that any function on ∂D can be expanded in series of P(|ak|, (·)–arg ak) with coefficients depending continuously on f in various classes of functions. The result is used to solve a Cauchy-type problem for Δu = μ, where μ is a measure supported on {ak}.

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