scholarly journals On a boundary value problem for the heat equation and a singular integral equation associated with it

2021 ◽  
Vol 399 ◽  
pp. 126009
Author(s):  
Meiramkul Amangaliyeva ◽  
Muvasharkhan Jenaliyev ◽  
Sagyndyk Iskakov ◽  
Murat Ramazanov
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Singular Integral ◽  
Boundary Value

Analysis of the Relaxation of Wall Electrons in a Plasma

1975 ◽  
Vol 30 (8) ◽  
pp. 937-946
Author(s):  
G. Ecker ◽  
K.-U. Riemann ◽  
A. Scholz
Keyword(s):  
Integral Equation ◽  
Kinetic Equation ◽  
Singular Integral Equation ◽  
Boundary Value Problem ◽  
Singular Integral ◽  
Boundary Value ◽  
Iteration Process ◽  
Standard Technique ◽  
Inelastic Collisions ◽  
Zeroth Order

Abstract A procedure is developed which renders the simultaneous description of angular and energy relaxation of wall electrons in a plasma possible. The plasma is assumed to be field free, allowance is made for elastic and inelastic collisions with neutrals, as well as for electron-electron encounters. The procedure is based on an expansion in terms of the eigenfunctions of a suitable part of the kinetic equation. In the zeroth order, the problem is reduced to a boundary value problem of the Sturm type, and subsequent solution of a singular integral equation by standard technique. To account for the rest of the kinetic equation, we construct a Green′s function which forms the basis for an iteration process. The application of the procedure is illustrated with an example.

scholarly journals Reduction of Boundary Value Problem to Possio Integral Equation in Theoretical Aeroelasticity

2008 ◽  
Vol 2008 ◽  
pp. 1-27 ◽  
Author(s):  
A. V. Balakrishnan ◽  
M. A. Shubov
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Boundary Value Problem ◽  
Singular Integral ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Normal Velocity ◽  
Initial Boundary Value ◽  
Slender Wing

The present paper is the first in a series of works devoted to the solvability of the Possio singular integral equation. This equation relates the pressure distribution over a typical section of a slender wing in subsonic compressible air flow to the normal velocity of the points of a wing (downwash). In spite of the importance of the Possio equation, the question of the existence of its solution has not been settled yet. We provide a rigorous reduction of the initial boundary value problem involving a partial differential equation for the velocity potential and highly nonstandard boundary conditions to a singular integral equation, the Possio equation. The question of its solvability will be addressed in our forthcoming work.

scholarly journals To the solution of the Solonnikov-Fasano problem with boundary moving on arbitrary law x = γ(t).

2021 ◽  
Vol 101 (1) ◽  
pp. 37-49
Author(s):  
M.T. Jenaliyev ◽  
◽  
M.I. Ramazanov ◽  
A.O. Tanin ◽  
◽  
...  
Keyword(s):  
Integral Equation ◽  
Integral Operator ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Volterra Integral Equation ◽  
Boundary Value ◽  
Nonzero Solution ◽  
Initial Moment ◽  
Homogeneous Integral Equation

In this paper we study the solvability of the boundary value problem for the heat equation in a domain that degenerates into a point at the initial moment of time. In this case, the boundary changing with time moves according to an arbitrary law x = γ(t). Using the generalized heat potentials, the problem under study is reduced to a pseudo-Volterra integral equation such that the norm of the integral operator is equal to one and it is shown that the corresponding homogeneous integral equation has a nonzero solution.

scholarly journals To solving the fractionally loaded heat equation

2021 ◽  
Vol 101 (1) ◽  
pp. 65-77
Author(s):  
M.T. Kosmakova ◽  
◽  
S.A. Iskakov ◽  
L.Zh. Kasymova ◽  
◽  
...  
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Fractional Derivative ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Continuous Functions ◽  
Uniqueness Of Solutions ◽  
Loaded Term ◽  
Differential Part

In this paper we consider a boundary value problem for a fractionally loaded heat equation in the class of continuous functions. Research methods are based on an approach to the study of boundary value problems, based on their reduction to integral equations. The problem is reduced to a Volterra integral equation of the second kind by inverting the differential part. We also carried out a study the limit cases for the fractional derivative order of the term with a load in the heat equation of the boundary value problem. It is shown that the existence and uniqueness of solutions to the integral equation depends on the order of the fractional derivative in the loaded term.

scholarly journals The solution of a singular integral equation with some applications in potential theory

1998 ◽  
Vol 21 (1) ◽  
pp. 189-196
Author(s):  
N. T. Shawagfeh
Keyword(s):  
Integral Equation ◽  
Analytical Solution ◽  
Singular Integral Equation ◽  
Boundary Value Problems ◽  
Potential Theory ◽  
Singular Integral ◽  
Boundary Value ◽  
Region Exterior

An analytical solution is derived for a singular integral equation which governs some twodimensional potential boundary value problems in a region exterior ton-infinite co-axial circular strips. An application in electrostatics is discussed.

scholarly journals Method of singular integral equations in liquid vibration problems for coaxial shells

2019 ◽  
Keyword(s):  
Integral Equation ◽  
Free Surface ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Integral Equations ◽  
Laplace Equation ◽  
Singular Integral ◽  
Boundary Value ◽  
Singular Equation ◽  
One Dimensional

The paper deals with the problem of free vibrations of an ideal incompressible fluid in coaxial shells of revolution. It is assumed that the motion of the fluid is irrotational that allows us to introduce the velocity potential. In these suppositions the potential is satisfied to Laplace equation. The boundary conditions are formulated on the wetted surfaces of the shells and on the free liquid surface. The non-penetration conditions are applied to the wetted surfaces. On the free surface we consider dynamical and kinematical boundary conditions. The dynamical condition consists in equality of the liquid pressure on the free surface to the atmospheric one. The kinematic condition requires that total time derivative of the free surface elevation will be equal to zero at any instant. Regarding the potential of velocities, a boundary value problem is formulated that is further reduced to the eigenvalue problem. To solve the boundary value problem for the Laplace equation, the boundary element method is used in a direct formulation. The axial symmetric form of the shells allows us to reduce the obtained system of singular equations to one-dimensional equations. The kernels in singular operators of obtained integral equations are expressed on terms of elliptical integrals of the first and second kinds, and have the logarithmic singularities. The special numerical technique is elaborated to treat with such kind integral equations. The resulting one-dimensional singular equation is solved by the method of discrete singularities. The integration region contains the free surface of the fluid that in the case of coaxial shells is a ring. So, the possibility of using the boundary integral equation approach coupled with application of the discrete singularities method is established to solution of the singular integral equation with incoherent boundaries. A numerical study has been carried out that made it possible to determine the frequencies and modes of the liquid sloshing in the shells for different ratios of the inner and outer radii of cylindrical coaxial shells. The obtained modes of natural vibrations will be used for numerical simulation of forced liquid vibrations in the tanks and reservoirs.

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