The problem of finding eigenvalues and eigenfunctions of boundary value problems for an equation of mixed type
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Ibrokhimjon Tojaliyevich Tojiboev
Abstract
In this work, we investigate the eigenvalue problem for an elliptic-hyperbolic equation with a non-smooth line of type changing within a special domain. The boundary conditions considered include Frankl’s condition, which is crucial for such problems involving mixed type equations. We derive the eigenvalues and corresponding eigenfunctions of the problem, paying special attention to the structural complexities introduced by the discontinuity in the equation’s type across the domain. Furthermore, we rigorously prove that the system of eigenfunctions obtained from the problem is not complete in the L2 space, meaning that not every square-integrable function in the domain can be represented as a series expansion in terms of these eigenfunctions. This non-completeness is demonstrated by constructing a specific function orthogonal to the entire system of eigenfunctions. We utilize advanced techniques, including the transformation to polar coordinates and variable substitution, to express integrals involving Bessel functions, thereby providing a clear framework for proving the non-completeness. Our findings are applicable to both symmetric and anti-symmetric cases of the problem, with boundary conditions that either involve equal or opposite values at points reflected across a line of symmetry. This study highlights significant challenges in dealing with mixed-type equations and contributes to the broader understanding of how eigenvalue problems with Frankl’s conditions behave in non-standard geometries, particularly in domains with discontinuities.
(Communicated by Andras Ronto)
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Artikel in diesem Heft
- Generalized Sasaki mappings in d0-Algebras
- On a theorem of Nathanson on Diophantine approximation
- Constructing infinite families of number fields with given indices from quintinomials
- Partitions into two Lehmer numbers in ℤq
- Fundamental systems of solutions of some linear differential equations of higher order
- On k-Circulant matrices involving the Lucas numbers of even index
- Explicit formulae for the Drazin inverse of the sum of two matrices
- On nonoscillation of fractional order functional differential equations with forcing term and distributed delays
- Novel generalized tempered fractional integral inequalities for convexity property and applications
- Convergence of α-Bernstein-Durrmeyer operators about a collection of measures
- The problem of finding eigenvalues and eigenfunctions of boundary value problems for an equation of mixed type
- Orbital Hausdorff dependence and stability of the solution to differential equations with variable structure and non-instantaneous impulses
- Improvements on the Leighton oscillation theorem for second-order dynamic equations
- Topogenous orders on forms
- Comparison of topologies on fundamental groups with subgroup topology viewpoint
- An elementary proof of the generalized Itô formula
- Asymptotic normality for kernel-based test of conditional mean independence in Hilbert space
- Advancing reliability and medical data analysis through novel statistical distribution exploration
- Historical notes on the 75th volume of Mathematica Slovaca - Authors of the first issue from 1951
