The Weighted Cauchy Type Problem and Cauchy Problem for the Linear Barbashin Integro-Differential Equations with Fractional Partial Derivative

2016 ◽  
Vol 219 (1) ◽  
pp. 125-132
Author(s):  
A. S. Kalitvin ◽  
V. A. Kalitvin
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Partial Derivative ◽  
Cauchy Type ◽  
Type Problem ◽  
Cauchy Type Problem

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 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.

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.

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.

Application of measure of noncompactness to a Cauchy problem for fractional differential equations in Banach spaces

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Asadollah Aghajani ◽  
Ehsan Pourhadi ◽  
Juan Trujillo
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Fixed Point ◽  
Measure Of Noncompactness ◽  
Existence Of Solutions ◽  
Cauchy Type ◽  
Type Problem ◽  
Nonlinear Fractional Differential Equation ◽  
Cauchy Type Problem ◽  
Fractional Differential

AbstractThis paper is devoted to study the existence of solutions of a Cauchy type problem for a nonlinear fractional differential equation, via the techniques of measure of noncompactness. The investigation is based on a new fixed point result which is a generalization of the well known Darbo’s fixed point theorem. The main result is less restrictive than those given in the literature. Some illustrative examples are given.

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