Applications of Erdélyi-Kober fractional integral for solving time-fractional Tricomi-Keldysh type equation

2020 ◽  
Vol 23 (5) ◽  
pp. 1381-1400 ◽  
Author(s):  
Kangqun Zhang
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Integral Operator ◽  
Nonlinear Equations ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Formal Solution ◽  
Type Equation ◽  
Multiplier Theorem ◽  
Problem Of Time

Abstract In this paper we consider Cauchy problem of time-fractional Tricomi-Keldysh type equation. Based on the theory of a Erdélyi-Kober fractional integral operator, the formal solution of the inhomogeneous differential equation involving hyper-Bessel operator is presented with Mittag-Leffler function, then nonlinear equations are considered by applying Gronwall-type inequalities. At last, we establish the existence and uniqueness of L p -solution of time-fractional Tricomi-Keldysh type equation by use of Mikhlin multiplier theorem.

scholarly journals Fractional Operators in p -adic Variable Exponent Lebesgue Spaces and Application to p -adic Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Leonardo Fabio Chacón-Cortés ◽  
Humberto Rafeiro
Keyword(s):  
Cauchy Problem ◽  
Integral Operator ◽  
Existence And Uniqueness ◽  
Variable Exponent ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Fractional Operators ◽  
Lebesgue Spaces ◽  
Variable Exponent Lebesgue Spaces ◽  
Uniqueness Of The Solution

In this paper, we prove the boundedness of the fractional maximal and the fractional integral operator in the p -adic variable exponent Lebesgue spaces. As an application, we show the existence and uniqueness of the solution for a nonhomogeneous Cauchy problem in the p -adic variable exponent Lebesgue spaces.

scholarly journals Existence, Uniqueness, and Eq–Ulam-Type Stability of Fuzzy Fractional Differential Equation

2021 ◽  
Vol 5 (3) ◽  
pp. 66
Author(s):  
Azmat Ullah Khan Niazi ◽  
Jiawei He ◽  
Ramsha Shafqat ◽  
Bilal Ahmed
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
The Cauchy Problem ◽  
Fractional Differential ◽  
Ulam Type Stability ◽  
Fuzzy Fractional Differential Equations ◽  
Fuzzy Fractional Differential Equation

This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy fractional differential equation with Caputo derivative of order q∈(1,2], 0cD0+qu(t)=λu(t)⊕f(t,u(t))⊕B(t)C(t),t∈[0,T] with initial conditions u(0)=u0,u′(0)=u1. Moreover, by using direct analytic methods, the Eq–Ulam-type results are also presented. In addition, several examples are given which show the applicability of fuzzy fractional differential equations.

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 On the Theory of ψ-Hilfer Nonlocal Cauchy Problem

2021 ◽  
pp. 161-177
Author(s):  
Mohammed A. Almalahi ◽  
Satish K. Panchal
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Fixed Point ◽  
Constant Coefficient ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Representation Formula ◽  
Qualitative Properties ◽  
Fractional Differential ◽  
Nonlocal Cauchy Problem

In this paper, we derive the representation formula of the solution for ψ-Hilfer fractional differential equation with constant coefficient in the form of Mittag-Leffler function by using Picard’s successive approximation. Moreover, by using some properties of Mittag-Leffler function and fixed point theorems such as Banach and Schaefer, we introduce new results of some qualitative properties of solution such as existence and uniqueness. The generalized Gronwall inequality lemma is used in analyze Eα -Ulam-Hyers stability. Finally, one example to illustrate the obtained results

scholarly journals Cauchy Problem for a Stochastic Fractional Differential Equation with Caputo-Itô Derivative

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1479
Author(s):  
Jorge Sanchez-Ortiz ◽  
Omar U. Lopez-Cresencio ◽  
Francisco J. Ariza-Hernandez ◽  
Martin P. Arciga-Alejandre
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Contraction Mapping ◽  
Stochastic Fractional Differential Equation ◽  
Fractional Differential

In this note, we define an operator on a space of Itô processes, which we call Caputo-Itô derivative, then we considerer a Cauchy problem for a stochastic fractional differential equation with this derivative. We demonstrate the existence and uniqueness by a contraction mapping argument and some examples are given.

LP-solutions for fractional integral equations

2014 ◽  
Vol 17 (1) ◽  
Author(s):  
Sadia Arshad ◽  
Vasile Lupulescu ◽  
Donal O’Regan
Keyword(s):  
Integral Operator ◽  
Integral Equations ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Existence Of Solutions ◽  
Type Condition ◽  
Existence And Uniqueness Results ◽  
Global Existence Of Solutions ◽  
Uniqueness Results ◽  
Lp Solutions

AbstractIn this article, we examine L p-solutions of fractional integral equations in Banach spaces involving the Riemann-Liouville integral operator. Using a compactness type condition, we obtain local and global existence of solutions. Also other types of existence and uniqueness results are established. At the end, an application is given to illustrate the main result.

Cauchy problem for an ordinary fractional differential equation with variable delay

2021 ◽  
Vol 21 (3) ◽  
pp. 16-20
Author(s):  
M.G. Mazhgikhova ◽  
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Uniqueness Theorem ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Initial Problem ◽  
Variable Delay ◽  
Existence And Uniqueness Theorem ◽  
Method Of Steps ◽  
Fractional Differential

For a fractional differential equation with variable delay, a solution to the initial problem is obtained by the method of steps. The existence and uniqueness theorem of the solution is proved.

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