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.

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.

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 Choice of the regularization parameter for the Cauchy problem for the Laplace equation

2020 ◽  
Vol 30 (10) ◽  
pp. 4475-4492
Author(s):  
Magda Joachimiak
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Chebyshev Polynomials ◽  
Regularization Parameter ◽  
Rectangular Domain ◽  
Heated Wall ◽  
Cauchy Type ◽  
Type Problem ◽  
Content Type ◽  
Cauchy Type Problem

Purpose In this paper, the Cauchy-type problem for the Laplace equation was solved in the rectangular domain with the use of the Chebyshev polynomials. The purpose of this paper is to present an optimal choice of the regularization parameter for the inverse problem, which allows determining the stable distribution of temperature on one of the boundaries of the rectangle domain with the required accuracy. Design/methodology/approach The Cauchy-type problem is ill-posed numerically, therefore, it has been regularized with the use of the modified Tikhonov and Tikhonov–Philips regularization. The influence of the regularization parameter choice on the solution was investigated. To choose the regularization parameter, the Morozov principle, the minimum of energy integral criterion and the L-curve method were applied. Findings Numerical examples for the function with singularities outside the domain were solved in this paper. The values of results change significantly within the calculation domain. Next, results of the sought temperature distributions, obtained with the use of different methods of choosing the regularization parameter, were compared. Methods of choosing the regularization parameter were evaluated by the norm Nmax. Practical implications Calculation model described in this paper can be applied to determine temperature distribution on the boundary of the heated wall of, for instance, a boiler or a body of the turbine, that is, everywhere the temperature measurement is impossible to be performed on a part of the boundary. Originality/value The paper presents a new method for solving the inverse Cauchy problem with the use of the Chebyshev polynomials. The choice of the regularization parameter was analyzed to obtain a solution with the lowest possible sensitivity to input data disturbances.

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.

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