Uniqueness and non-uniqueness of solutions of the boundary value problems of the heat equation

Meiramkul M. Amangaliyeva, Muvasharkhan Jenaliyev, M.T. Kosmakova, M.I. Ramazanov

2015

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DOI 10.1063/1.4930454
Data confidence 70 medium
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Abstract

The article addresses the singular Volterra integral equation of the second kind which has the “incompressible” kernel. It is shown that the corresponding homogeneous equation on |λ| > exp{|arg λ|}, arg λ ∈ [−π, π] has a continuous spectrum, and the multiplicity of the characteristic numbers grows with increasing |λ|. The equation is reduced to Abel equation by the regularization method. The eigenfunctions of the equation are found in an explicit form. We prove the solvability theorem of the inhomogeneous equation in a case when the right-hand side of the equation belongs to a certain class.