About Dirichlet boundary value problem for the heat equation in the infinite angular domain

Muvasharkhan He Jenaliyev,Meiramkul Amangaliyeva,Minzilya Kosmakova,Murat Ramazanov

doi:10.1186/s13661-014-0213-4

Muvasharkhan He Jenaliyev, Meiramkul Amangaliyeva + Show 2 more

Open Access

https://doi.org/10.1186/s13661-014-0213-4

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Abstract

Abstract In this paper it is established that in an infinite angular domain for Dirichlet problem of the heat conduction equation the unique (up to a constant factor) non-trivial solution exists, which does not belong to the class of summable functions with the found weight. It is shown that for the adjoint boundary value problem the unique (up to a constant factor) non-trivial solution exists, which belongs to the class of essentially bounded functions with the weight found in the work. It is proved that the operator of a boundary value problem of heat conductivity in an infinite angular domain in a class of growing functions is Noetherian with an index which is equal to minus one. MSC: 35D05, 35K20, 45D05.

Highlights

Different kinds of processes of mass and heat transfer lead to solving boundary value problems for parabolic equations in a domain with a moving in time boundary
Consideration of a wide range of issues of mathematical physics [, ], in particular, the solving of boundary value problems in the heat equation degenerating domains leads to the need to study the singular integral equations of Volterra type when the norm of a integral operator is equal to unit
7 Statement of the adjoint boundary value problem L∗ In the domain G = {(x; t) : –∞ < t

Summary

Introduction

Different kinds of processes of mass and heat transfer lead to solving boundary value problems for parabolic equations in a domain with a moving in time boundary (non-cylindrical domain). 4 Statement of the boundary value problem L In the domain G = {(x; t) : –∞ < t < , < x < –t} it is required to find a solution to the heat conduction equation ut(x, t) = a uxx(x, t), ( )

Results

Conclusion