scholarly journals On the solutions of some fractional q-differential equations with the Riemann-Liouville fractional q-derivative

2021 ◽  
Vol 104 (4) ◽  
pp. 130-141
Author(s):  
S. Shaimardan ◽  
◽  
N.S. Tokmagambetov ◽  
◽  
Keyword(s):  
Differential Equations ◽  
Difference Equations ◽  
Numerical Solutions ◽  
Operational Calculus ◽  
Explicit Solutions ◽  
Cauchy Type ◽  
Numerical Treatment ◽  
Type Problem ◽  
Real Numbers ◽  
Cauchy Type Problem

This paper is devoted to explicit and numerical solutions to linear fractional q-difference equations and the Cauchy type problem associated with the Riemann-Liouville fractional q-derivative in q-calculus. The approaches based on the reduction to Volterra q-integral equations, on compositional relations, and on operational calculus are presented to give explicit solutions to linear q-difference equations. For simplicity, we give results involving fractional q-difference equations of real order a > 0 and given real numbers in q-calculus. Numerical treatment of fractional q-difference equations is also investigated. Finally, some examples are provided to illustrate our main results in each subsection.

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 Fractional Cauchy Problem with Caputo Nabla Derivative on Time Scales

2015 ◽  
Vol 2015 ◽  
pp. 1-23 ◽  
Author(s):  
Jiang Zhu ◽  
Ling Wu
Keyword(s):  
Explicit Solutions ◽  
Cauchy Type ◽  
Type Problem ◽  
Initial Value ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Cauchy Type Problem ◽  
Definition Of ◽  
Nabla Derivative ◽  
Fractional Cauchy Problem

The definition of Caputo fractional derivative is given and some of its properties are discussed in detail. After then, the existence of the solution and the dependency of the solution upon the initial value for Cauchy type problem with fractional Caputo nabla derivative are studied. Also the explicit solutions to homogeneous equations and nonhomogeneous equations are derived by using Laplace transform method.

NONLINEAR DIFFERENTIAL EQUATIONS WITH MARCHAUD‐HADAMARD-TYPE FRACTIONAL DERIVATIVE IN THE WEIGHTED SPACE OF SUMMABLE FUNCTIONS

2007 ◽  
Vol 12 (3) ◽  
pp. 343-356 ◽  
Author(s):  
Anatoly Kilbas ◽  
Anatoly Titioura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Nonlinear Differential Equations ◽  
Banach Fixed Point Theorem ◽  
Cauchy Type ◽  
Type Problem ◽  
Cauchy Type Problem ◽  
Linear Differential ◽  
Half Axis

The paper is devoted to the study of the Cauchy‐type problem for the nonlinear differential equation of fractional order 0 < α < 1: containing the Marchaud-Hadamard-type fractional derivative (Dα 0+, μ y)(x), on the half-axis R+ = (0, +oo) in the space Xp,α c,0 (R+) defined for α > 0 by where Xp c, 0 (R+) is the subspace of Xp c (R+) of functions g Xp c (R + ) with compact support on infinity: g(x) = 0 for large enough x > R. The equivalence of this problem and of the nonlinear Volterra integral equation is established. The existence and uniqueness of the solution y(x) of the above Cauchy‐type problem is proved by using the Banach fixed point theorem. Solution in closed form of the above problem for the linear differential equation with f[x, y(x)] = λy(x) + f(x) is constructed. The corresponding assertions for the differential equations with the Marchaud‐Hadamard fractional derivative (Dα 0+ y)(x) are presented. Examples are given.

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}.

scholarly journals Solution of Volterra-type integro-differential equations with a generalized Lauricella confluent hypergeometric function in the kernels

2005 ◽  
Vol 2005 (8) ◽  
pp. 1155-1170 ◽  
Author(s):  
R. K. Saxena ◽  
S. L. Kalla
Keyword(s):  
Hypergeometric Function ◽  
Fractional Derivatives ◽  
Confluent Hypergeometric Function ◽  
Several Complex Variables ◽  
Cauchy Type ◽  
Type Problem ◽  
Volterra Type ◽  
Caputo Fractional Derivatives ◽  
Cauchy Type Problem ◽  
Special Cases

The object of this paper is to solve a fractional integro-differential equation involving a generalized Lauricella confluent hypergeometric function in several complex variables and the free term contains a continuous functionf(τ). The method is based on certain properties of fractional calculus and the classical Laplace transform. A Cauchy-type problem involving the Caputo fractional derivatives and a generalized Volterra integral equation are also considered. Several special cases are mentioned. A number of results given recently by various authors follow as particular cases of formulas established here.

scholarly journals Solvability and Stability of Impulsive Set Dynamic Equations on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-19 ◽  
Author(s):  
Shihuang Hong ◽  
Jing Gao ◽  
Yingzi Peng
Keyword(s):  
Differential Equations ◽  
Time Scales ◽  
Difference Equations ◽  
Time Scale ◽  
Dynamic Equations ◽  
Stability Of Solutions ◽  
Real Numbers ◽  
Generalized Derivative ◽  
Special Cases ◽  
Existence And Stability

A class of new nonlinear impulsive set dynamic equations is considered based on a new generalized derivative of set-valued functions developed on time scales in this paper. Some novel criteria are established for the existence and stability of solutions of such model. The approaches generalize and incorporate as special cases many known results for set (or fuzzy) differential equations and difference equations when the time scale is the set of the real numbers or the integers, respectively. Finally, some examples show the applicability of our results.

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