Application of the Fourier method to the solution of boundary value problems for functions analytic in disk bidomains

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.

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

Mathematische Nachrichten ◽

10.1002/mana.19760720109 ◽

1976 ◽

Vol 72 (1) ◽

pp. 109-117

Author(s):

D. Przeworska-Rolewicz

Keyword(s):

Boundary Value Problems ◽

Boundary Value ◽

Fourier Method

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

Mathematics ◽

10.3390/math8111877 ◽

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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