Simulation of the processes of heat conductivity in three-layer inhomogeneous environment with soft borders by means of hybrid differential Legendre, Bessel and Fourier operators

The processes of heat conductivity occurring in the environment, attracted attention of the scientists throughout the history of mankind. The simplest mathematical model of this process is the differential equation of heat conductivity of parabolic type with the initial condition and boundary conditions.

Practice requirements led to generalizations of this equation. But in all cases the processes of heat conductivity were studied when the border of the environment was rigid with respect to the reflection of waves. The picture of heat distribution changes, if the limit of the environment is soft with respect to the reflection of waves. Mathematically, it means the presence of derivative in terms of the time variable in boundary operators and differential operators of conjugation.

In the second half of the twentieth century there was a wide spread method of piecewise constant coefficients, which led to differential equations with singular coefficients, such as delta-function and its derivatives. But it was impossible to obtain integral image of accurate analytical solution of these problems. These difficulties can be avoided to model the process of heat conductivity by means of hybrid differential operators.

The work is devoted to the simulation of the nonstationary processes of heat conductivity by means of hybrid differential Legendre – (Bessel, Fourier) operators, when the border of the environment is soft with respect to the reflection of waves.

The first and second paragraphs are devoted to the simulation of the processes of heat conductivity in inhomogeneous environment with soft borders by means of hybrid differential Legendre – Bessel – Fourier and Legendre – Fourier – Bessel operators on polar axis, respectively. The third and fourth paragraphs are devoted to the simulation of the processes of heat conductivity in inhomogeneous environment with soft borders by means of hybrid differential Legendre – Fourier – Bessel and Legendre – Bessel – ­­Fourier operators on polar axis , respectively. In the fifth paragraph there are presented the basic requirements for safety.

Consider the details in the first paragraph, which is called «Simulation of the processes of heat conductivity in inhomogeneous environment with soft borders by means of hybrid differential Legendre – Bessel – Fourier operator on the polar axis»

The consideration of the problem of construction of a limited in solution of separative system of differential equations of parabolic type conductivity:

(1)

with initial conditions

(2)

and conjugation conditions

(3)

In equalities there are involved Legendre differential operators , Bessel , Fourier and generalized differential operators of conjugation

.

We obtain the solution of problem (1)-(3) by the Laplace method of integral transformation on t in the assumption that the desired and specified functions are images on the Laplace variable t ().

In the image by Laplace we obtain a boundary problem: build a limited on set solution of separate system of ordinary differential equations of Legendre, Bessel and Fourier for modified functions

(4)

with conjugation conditions:

(5)

Fundamental system of solutions for Legendre differential equations is formed by generalized connected Legendre functions of the first kind and second kind ; fundamental system of solutions for Bessel differential equations is formed by Bessel functions of imaginary argument of the first kind and second kind ; fundamental system of solutions for Fourier differential equations is formed by functions and .

The presence of a fundamental system of solutions allows us to build a solution of the boundary problem (4),(5) by Cauchy functions:

(9)

In the equalities (9) there are involved -Cauchy functions, which are found by using their properties:

, (10)

Now we have to find the values A₁,A₂,B₂,B₃. To find them we use the conjugation conditions (5), giving heterogeneous algebraic system of four equations which is solved by the Cramer rule:

(21)

Then define the main boundary problem solutions (4),(5), that is the Green function generated by the inhomogeneity of conjugation conditions and the impact function generated by the inhomogeneity of the system (4).

The unique solvability of the boundary problem (4),(5), the substitution of defined values A₁,A₂,B₂,B₃ in equalities (9) and returning to the original offer the only solution of the parabolic problem (1)-(3):

The results can be formulated as the following theorem:

Theorem: If are the an original by Laplace, – are twice continuously differentiable for variable r and they satisfy the homogeneous conditions of conjugation, then the parabolic problem (1)-(3) has a solution and if the condition of explicit solvability of the boundary problem is performed then the solution is unique.

Дата добавления: 2015-11-14