scholarly journals Cauchy Problem for a Linear System of Ordinary Differential Equations of the Fractional Order

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1475
Author(s):  
Murat Mamchuev
Keyword(s):  
Linear System ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Existence And Uniqueness ◽  
Differentiation Operator ◽  
Initial Problem ◽  
Fractional Differentiation ◽  
The Matrix ◽  
Constant Coefficients ◽  
Fractional Differentiation Operator

We investigate the initial problem for a linear system of ordinary differential equations with constant coefficients and with the Dzhrbashyan–Nersesyan fractional differentiation operator. The existence and uniqueness theorems of the solution of the boundary value problem under the study are proved. The solution is constructed explicitly in terms of the Mittag–Leffler function of the matrix argument. The Dzhrbashyan–Nersesyan operator is a generalization of the Riemann–Liouville, Caputo and Miller–Ross fractional differentiation operators. The obtained results as particular cases contain the results related to the study of initial problems for the systems of ordinary differential equations with Riemann–Liouville, Caputo and Miller–Ross derivatives and the investigated initial problem that generalizes them.

scholarly journals Linearization of Two Second-Order Ordinary Differential Equations via Fiber Preserving Point Transformations

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
Sakka Sookmee ◽  
Sergey V. Meleshko
Keyword(s):  
Linear System ◽  
General Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Necessary Conditions ◽  
Second Order ◽  
Point Transformations ◽  
Constant Coefficients ◽  
Linearizable System

The necessary form of a linearizable system of two second-order ordinary differential equations y1″=f1(x,y1,y2,y1′,y2′), y2″=f2(x,y1,y2,y1′,y2′) is obtained. Some other necessary conditions were also found. The main problem studied in the paper is to obtain criteria for a system to be equivalent to a linear system with constant coefficients under fiber preserving transformations. A linear system with constant coefficients is chosen because of its simplicity in finding the general solution. Examples demonstrating the procedure of using the linearization theorems are presented.

A nonlocal problem for a differential operator of even order with involution

2020 ◽  
Vol 26 (2) ◽  
pp. 297-307
Author(s):  
Petro I. Kalenyuk ◽  
Yaroslav O. Baranetskij ◽  
Lubov I. Kolyasa
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Riesz Basis ◽  
Existence And Uniqueness ◽  
Spectral Properties ◽  
Nonlocal Problem ◽  
Even Order

AbstractWe study a nonlocal problem for ordinary differential equations of {2n}-order with involution. Spectral properties of the operator of this problem are analyzed and conditions for the existence and uniqueness of its solution are established. It is also proved that the system of eigenfunctions of the analyzed problem forms a Riesz basis.

scholarly journals Weak Estimates of Singular Integrals with Variable Kernel and Fractional Differentiation on Morrey-Herz Spaces

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Yanqi Yang ◽  
Shuangping Tao
Keyword(s):  
Integral Operator ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differentiation Operator ◽  
Fractional Differentiation ◽  
Herz Spaces ◽  
Variable Kernel ◽  
Harmonic Decomposition ◽  
Spherical Harmonic Decomposition ◽  
Fractional Differentiation Operator

Let T be the singular integral operator with variable kernel defined by Tf(x)=p.v.∫Rn(Ω(x,x-y)/x-yn)f(y)dy and let Dγ  (0≤γ≤1) be the fractional differentiation operator. Let T⁎and T♯ be the adjoint of T and the pseudoadjoint of T, respectively. In this paper, the authors prove that TDγ-DγT and (T⁎-T♯)Dγ are bounded, respectively, from Morrey-Herz spaces MK˙p,1α,λ(Rn) to the weak Morrey-Herz spaces WMK˙p,1α,λ(Rn) by using the spherical harmonic decomposition. Furthermore, several norm inequalities for the product T1T2 and the pseudoproduct T1∘T2 are also given.

scholarly journals Existence and Uniqueness of Periodic Solutions for Second Order Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Yuanhong Wei
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Function Space ◽  
Point Theorem ◽  
Existence And Uniqueness ◽  
Second Order ◽  
Second Order Differential Equations

We study some second order ordinary differential equations. We establish the existence and uniqueness in some appropriate function space. By using Schauder’s fixed-point theorem, new results on the existence and uniqueness of periodic solutions are obtained.

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