On the nonexistence conditions of solution of two-point in time problem for nonhomogeneous PDE

2021 ◽  
Vol 71 (5) ◽  
pp. 1125-1132
Author(s):  
Zinovii Nytrebych ◽  
Oksana Malanchuk
Keyword(s):  
Infinite Order ◽  
Entire Functions ◽  
Second Order ◽  
Time Variable ◽  
Spatial Variables ◽  
Characteristic Determinant ◽  
Time Problem ◽  
Second Order In Time

Abstract We investigate the problem with local homogeneous two-point conditions with respect to time for nonhomogeneous PDE of second order in time variable and generally infinite order in spatial variables in the case when the characteristic determinant of the problem identically equals zero. We establish the nonexistence conditions of solution of this problem in the class of entire functions.

The conditions of existence of a solution of the degenerate two-point in time problem for PDE

2019 ◽  
Vol 12 (03) ◽  
pp. 1950037 ◽  
Author(s):  
Zinovii Nytrebych ◽  
Oksana Malanchuk
Keyword(s):  
Infinite Order ◽  
Entire Functions ◽  
Second Order ◽  
Existence Of A Solution ◽  
Degenerate Problem ◽  
Spatial Variables ◽  
Characteristic Determinant ◽  
Time Problem ◽  
Second Order In Time

The problem with local nonhomogeneous two-point in time conditions for homogeneous PDE of the second order in time and, generally, infinite order in spatial variables is investigated. This problem is degenerated namely its characteristic determinant is identically zero. The condition of existence of a solution of the degenerate problem is established. Also, we proposed the differential-symbol method of constructing the solution of the problem in the classes of entire functions. Some examples of solving the degenerate two-point in time problems are presented.

scholarly journals Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables

2017 ◽  
Vol 15 (1) ◽  
pp. 101-110 ◽  
Author(s):  
Oksana Malanchuk ◽  
Zinoviy Nytrebych
Keyword(s):  
Trivial Solution ◽  
Infinite Order ◽  
Homogeneous Problem ◽  
Second Order ◽  
Time Variable ◽  
Point Problem ◽  
Nontrivial Solutions ◽  
Spatial Variables ◽  
Opposite Case ◽  
Second Order In Time

Abstract We prove that homogeneous problem for PDE of second order in time variable, and generally infinite order in spatial variables with local two-point conditions with respect to time variable, has only trivial solution in the case when the characteristic determinant of the problem is nonzero. In another, opposite case, we prove the existence of nontrivial solutions of the problem, and we propose a differential-symbol method of constructing them.

scholarly journals The differential-symbol method of constructing the quasi-polynomial solutions of two-point problem

2019 ◽  
Vol 52 (1) ◽  
pp. 88-96 ◽  
Author(s):  
Zinovii Nytrebych ◽  
Oksana Malanchuk
Keyword(s):  
Infinite Order ◽  
Second Order ◽  
Spatial Variable ◽  
Point Problem ◽  
Existence Of A Solution ◽  
Polynomial Solutions ◽  
Characteristic Determinant ◽  
Right Hand ◽  
Second Order In Time ◽  
The Right

Abstract The solvability of the problem with local nonhomogeneous two-point in time conditions for a homogeneous PDE of the second order in time and infinite order in spatial variable in the case when the set of zeroes of the characteristic determinant is not empty and does not coincide with C is investigated. The existence of a solution of the problem in which the right-hand sides of the two-point conditions are quasi-polynomials is proved. We propose the differential-symbol method of constructing the solutions of the problem.

On entire functions of infinite order

1963 ◽  
Vol 14 (1) ◽  
pp. 323-327 ◽  
Author(s):  
S. M. Shah
Keyword(s):  
Infinite Order ◽  
Entire Functions

scholarly journals On the Growth of Solutions of Some Second Order Linear Differential Equations with Entire Coefficients

2013 ◽  
Vol 21 (2) ◽  
pp. 35-52
Author(s):  
Benharrat Belaïdi ◽  
Habib Habib
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Infinite Order ◽  
Entire Functions ◽  
Complex Numbers ◽  
Order Of Growth ◽  
Hyper Order ◽  
Linear Differential ◽  
Growth Of Solutions

Abstract In this paper, we investigate the order and the hyper-order of growth of solutions of the linear differential equation where n≥2 is an integer, Aj (z) (≢0) (j = 1,2) are entire functions with max {σ A(j) : (j = 1,2} < 1, Q (z) = qmzm + ... + q1z + q0 is a nonoonstant polynomial and a1, a2 are complex numbers. Under some conditions, we prove that every solution f (z) ≢ 0 of the above equation is of infinite order and hyper-order 1.

scholarly journals An Efficient Compact Difference Method for Temporal Fractional Subdiffusion Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Lei Ren ◽  
Lei Liu
Keyword(s):  
Finite Difference ◽  
Fourth Order ◽  
Difference Method ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Approximation Formula ◽  
Discrete Energy ◽  
Compact Finite Difference ◽  
Second Order In Time

In this paper, a high-order compact finite difference method is proposed for a class of temporal fractional subdiffusion equation. A numerical scheme for the equation has been derived to obtain 2-α in time and fourth-order in space. We improve the results by constructing a compact scheme of second-order in time while keeping fourth-order in space. Based on the L2-1σ approximation formula and a fourth-order compact finite difference approximation, the stability of the constructed scheme and its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Applications using two model problems demonstrate the theoretical results.

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