Remarks on Boundary Value Problems and FOURIER Method for right invertible Operators

1976 ◽  
Vol 72 (1) ◽  
pp. 109-117
Author(s):  
D. Przeworska-Rolewicz
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Fourier Method

On the well-posedness of boundary value problems for nonclassical differential equations of higher order

2017 ◽  
Vol 10 (03) ◽  
pp. 1750059
Author(s):  
N. R. Pinigina
Keyword(s):  
Boundary Value Problems ◽  
Spectral Parameter ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Fourier Method ◽  
Countable System ◽  
Priori Estimates ◽  
Complex Eigenvalues ◽  
Even Order

This paper investigates a high even-order nonclassical differential equation with a spectral parameter. We proved that this equation has a countable system of nontrivial solutions if spectral parameter is negative. We consider two cases, one where the spectral parameter is equal to eigenvalues and one where the spectral parameter is not equal to eigenvalues. In both cases, we proved the existence of regular solutions of boundary value problems for this equation. To do this, we combined the Fourier method and the method of a priori estimates. Moreover, we found some conditions for unsolvability of boundary value problems. In addition, for adjoint problems, we proved that there is no complex eigenvalues.

scholarly journals Solving the Boundary Value Problems for Differential Equations with Fractional Derivatives by the Method of Separation of Variables

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1877
Author(s):  
Temirkhan Aleroev
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Fractional Derivatives ◽  
Separation Of Variables ◽  
Boundary Value ◽  
Fourier Method ◽  
Associated Functions ◽  
Sturm Liouville ◽  
Eigenfunctions And Associated Functions ◽  
Adjoint Operators

This paper is devoted to solving boundary value problems for differential equations with fractional derivatives by the Fourier method. The necessary information is given (in particular, theorems on the completeness of the eigenfunctions and associated functions, multiplicity of eigenvalues, and questions of the localization of root functions and eigenvalues are discussed) from the spectral theory of non-self-adjoint operators generated by differential equations with fractional derivatives and boundary conditions of the Sturm–Liouville type, obtained by the author during implementation of the method of separation of variables (Fourier). Solutions of boundary value problems for a fractional diffusion equation and wave equation with a fractional derivative are presented with respect to a spatial variable.

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