On a Green’s function and eigenvalues of a second-order differential operator with involution and Neumann boundary conditions

2016 ◽  
Author(s):  
Abdisalam A. Sarsenbi
Keyword(s):  
Differential Operator ◽  
Boundary Conditions ◽  
Green’S Function ◽  
Green's Function ◽  
Neumann Boundary ◽  
Second Order ◽  
Neumann Boundary Conditions ◽  
Order Differential Operator

scholarly journals The ill-posed problem for the heat transfer equation with involution

2019 ◽  
Vol 21 (1) ◽  
pp. 48-59
Author(s):  
Abdisalam A. Sarsenbi
Keyword(s):  
Heat Transfer ◽  
Differential Operator ◽  
Green’S Function ◽  
Mixed Problem ◽  
Green's Function ◽  
Uniform Estimate ◽  
Fourier Method ◽  
Second Order ◽  
Order Differential Operator ◽  
Ill Posed

A mixed problem for an equation of heat transfer with involution is considered. The uniqueness of the problem's solution is proved. The ill-posedness of the mixed problem with Dirichlet-type boundary conditions for this equation is shown. By application of Fourier method, we obtain a spectral problem for a second-order differential operator with involution with an infinite number of positive and negative eigenvalues. The Green function of obtained second-order differential operator with involution is constructed. Uniform estimate of the Green's function is established for sufficiently large values of the spectral parameter. The existence of the Green's function of a second-order differential operator with involution and with variable coefficient is proved. By estimation of the Green's function completeness of the eigenfunctions's system for operator discussed is proved. In the class of polynomials the existence of a solution of this ill-posed problem is proved.

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