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.

scholarly journals Green's Function and Convergence of Fourier Series for Elliptic Differential Operators with Potential from Kato Space

2010 ◽  
Vol 2010 ◽  
pp. 1-18 ◽  
Author(s):  
Valery Serov
Keyword(s):  
Fourier Series ◽  
Differential Operator ◽  
Green’S Function ◽  
Green's Function ◽  
Differential Operators ◽  
Fourier Method ◽  
Smooth Coefficients ◽  
Equations Of Mathematical Physics ◽  
Adjoint Extension ◽  
Bounded Smooth

We consider the Friedrichs self-adjoint extension for a differential operatorAof the formA=A0+q(x)⋅, which is defined on a bounded domainΩ⊂ℝn,n≥1(forn=1we assume thatΩ=(a,b)is a finite interval). HereA0=A0(x,D)is a formally self-adjoint and a uniformly elliptic differential operator of order2mwith bounded smooth coefficients and a potentialq(x)is a real-valued integrable function satisfying the generalized Kato condition. Under these assumptions for the coefficients ofAand for positiveλlarge enough we obtain the existence of Green's function for the operatorA+λIand its estimates up to the boundary ofΩ. These estimates allow us to prove the absolute and uniform convergence up to the boundary ofΩof Fourier series in eigenfunctions of this operator. In particular, these results can be applied for the basis of the Fourier method which is usually used in practice for solving some equations of mathematical physics.

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