Boundary value problems for a spectrally loaded heat operator with load line approaching the time axis at zero or infinity

Solving a nonhomogeneous integral equation with the variable lower limit An nonhomogeneous integral equation with a singular kernel is considered. A feature of the equation under study is the incompressibility of the integral operator. In the study of the equation, an auxiliary simpler equation is used with the right-hand side equal to 1. The incompressibility of the integral operator for the equation under study is shown. Using the relations for an independent variable, the equation is equivalently reduced to a certain simplified equation. With the help of replacements for independent variables, the equation is reduced to an integral equation with a difference kernel. By applying the Laplace transform, the obtained equation is reduced to an ordinary first-order differential equation (linear). Its solution is found. By using the inverse Laplace transform, a solution of the auxiliary integral equation is obtained in the form of a convergent series in some domain. The solution of the initial equation with an arbitrary right-hand side is written through the solution of the auxiliary equation.

Section: Resultsmentioning

confidence: 99%

Solving a nonhomogeneous integral equation with the variable lower limit

Kosmakova¹,

Akhmanova²,

Tuleutaeva³

et al. 2019

“…Siberian Mathematical Journal, 52, 1, 1-10. DOI: 10.1134 /S0037446606010010 5 Amangaliyeva, M.M., Akhmanova, D.M., Dzhenaliev, M.T., & Ramazanov, M.I. (2011).…”

Section: Referencesmentioning

confidence: 99%

Untitled

2019

On strongly loaded heat equations The article is devoted to the research of boundary value problems for the spectrum-loaded operator of heat conduction with the moving point of loading to the temporary axle in zero or on infinity. For strongly loaded parabolic 2k-order equations the adjoint boundary value problems, when order of loaded term is greater then one of differential part of equation, is studied. In this article we continue a investigation of the boundary value problems for spectrally loaded parabolic equations in unbounded domains.The boundary value problem for the spectral-loaded equation of thermal conductivity, which on the one hand is quite close to the problems with the load containing the second derivative of the spatial variable, and is of independent interest on the other hand in this work, is considered.

“…After substituting (6) into 5, we can write out the partial solution of the nonhomogeneous equation 4aŝ…”

Section: Reducing the Equation (1) To A Differential Equation In Imagesmentioning

confidence: 99%

“…In this connection, the weight classes of the solution existence differ from the class of the solution existence for the equation considered in this work. We also note that boundary value problems for a spectrally loaded parabolic equation reduce to this kind of singular integral equations, when the load line moves according to the law x = t [5][6][7][8][9][10] and problems for essentially loaded equation of heat conduction [11][12][13][14][15].…”

Section: Reducing the Equation (1) To A Differential Equation In Imagesmentioning

confidence: 99%

On a pseudo-Volterra nonhomogeneous integral equation

Kosmakova¹,

Akhmanova²,

Tuleutaeva³

et al. 2019

In this paper the issues of the solvability of a pseudo-Volterra nonhomogeneous integral equation of the second kind are studied. The solution to the corresponding homogeneous equation and the classes of the uniqueness of the solution are found in [1]. By replacing the right-hand side and the unknown function, the integral equation is reduced to an integral equation, the kernel of which is not compressible. Using the Laplace transform, the obtained equation is reduced to an ordinary first-order differential equation (linear). Its solution is found. By using the solution of the homogeneous equation the form of a particular solution of the nonhomogeneous differential equation is defined (by the variation method of an arbitrary constant). By using the inverse Laplace transform, a particular solution of the pseudo-Volterra nonhomogeneous integral equation under study is obtained. The case of an nonhomogeneous integral equation with the value of the parameter k = 1 is considered and studied. Classes for the right side and the solution of the integral equation are indicated.