Abstract
In the present paper, we find out the eigenvalues and the corresponding eigenfunctions of the modified Frankl problem with a nonlocal parity condition, the completeness and the basis property in the elliptic part of the third kind of a domain in L<sup>2</sup>(0, π/2). We also consider a new boundaries condition and analyze the orthogonal basis of the eigenfunctions depending on parameters of the problem.
Highlights
The classical Frankl problem was considered in [3]
We consider boundaries conditions of the third kind on the intervals (-1,0) and (0,1) of the axis OY for which the derivatives of functions with respect to x on these intervals are related by linear dependence
We show that if the dependence coefficient exceeds -1, the systems of eigenfunctions of the problem forms a Riesz basis in the elliptic part of the domain
Summary
The classical Frankl problem was considered in [3]. On the solution of the Frankl prolem in a special domain in [12]. About spectrum of the gasedynamic problem of Frankl for the model equation of mixed type in [10]. About construction of the gasedynamic problem of Frankl in [11]. Basis property of eigen -functions of the generalized problem of Frankl with a nonlocal parity condition and with the discontinuity of the gradient of solution in [9]. We consider boundaries conditions of the third kind on the intervals (-1,0) and (0,1) of the axis OY for which the derivatives of functions with respect to x on these intervals are related by linear dependence. We show that if the dependence coefficient exceeds -1 (the coefficient cannot be zero, since, otherwise, the problem will degenerate), the systems of eigenfunctions of the problem forms a Riesz basis in the elliptic part of the domain
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