On solvability and unique solvability of two-point singular boundary value problems

“…where * is given in (18). Since lim →T − α ( ) = (T ) = 0 and lim →T − β ( ) = 2S, Lemma 4.1 guarantees that for each ∈ N there exists a solution ∈ AC 1 [ε T ] of equation (42) satisfying the inequality…”

Section: Lemma 42mentioning

confidence: 98%

Positive and maximal positive solutions of singular mixed boundary value problem

Agarwal¹,

O’Regan

Staněk

2009

Open Mathematics

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“…In the case of singular secondary ODEs, solvability is still the subject of mathematical studies. Kiguradze et al [24][25][26] published several related studies that thoroughly examine the difficulties of these mathematical problems to be solved.…”

Section: Introductionmentioning

confidence: 99%

General Exact Schemes for Second-Order Linear Differential Equations Using the Concept of Local Green Functions

Vizvari

Klincsik

Ódry

et al. 2023

Axioms

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In this paper, we introduce a special system of linear equations with a symmetric, tridiagonal matrix, whose solution vector contains the values of the analytical solution of the original ordinary differential equation (ODE) in grid points. Further, we present the derivation of an exact scheme for an arbitrary mesh grid and prove that its application can completely avoid other errors in discretization and numerical methods. The presented method is constructed on the basis of special local green functions, whose special properties provide the possibility to invert the differential operator of the ODE. Thus, the newly obtained results provide a general, exact solution method for the second-order ODE, which is also effective for obtaining the arbitrary grid, Dirichlet, and/or Neumann boundary conditions. Both the results obtained and the short case study confirm that the use of the exact scheme is efficient and straightforward even for ODEs with discontinuity functions.

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“…It arises naturally in studying many physical, economic, demographic and other processes. This problem was investigated in many papers, in particular, the linear case in [1,2,35], the nonlinear case in [18,20,40], the degenerate case in [34,36] etc. Some efficient methods of constructing the approximate solutions were elaborated in [11,52].…”

mentioning

confidence: 99%

Two-point boundary value problem for a partial differential equation in spaces of periodic functions

et al. 2020

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We investigate the two-point in time boundary value problem for the partial differential equations of the second-order with one spatial variable and constant coefficients. The problem is considered in in the spaces of functions which Fourier coefficients are characterized by exponential behavior on the Cartesian product of the time interval and spatial domain $\mathbb{R}/2\pi\mathbb{Z}$. The correct solvability of the problem is established, the formulas for solutions are presented, the kernel is described and the smoothness of the solution is established in the spaces of functions that are periodic in one spatial variable. We have established the conditions which are close to the necessary conditions of solvability of the problem in scale of spaces of functions with exponentially increasing (or decreasing) Fourier coefficients.We also found the asymptotic estimates demonstrating the absence of the problem of small denominators, which arises of many spatial variables and makes the boundary value problem incorrect. We have established sufficient conditions of the finite-dimensionality of the kernel of the problem and found upper bounds for its dimension. The results are obtained under the condition of minimum smoothness on the right-hand sides of two-point conditions, which is close to the necessary condition.

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On solvability and unique solvability of two-point singular boundary value problems

Cited by 10 publications

References 13 publications

Positive and maximal positive solutions of singular mixed boundary value problem

Positive and maximal positive solutions of singular mixed boundary value problem

General Exact Schemes for Second-Order Linear Differential Equations Using the Concept of Local Green Functions

Two-point boundary value problem for a partial differential equation in spaces of periodic functions

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