Numerous research works are devoted to study Cauchy mixed problem for model heat equations because of its theoretical and practical importance. Among them we can notice monographers Vladimirov (1988), Ladyzhenskaya (1973), and Tikhonov and Samarskyi (1980) which demonstrate main research methods, such as Fourier method, integral equations method, and the method of a priori estimates. But at the same time to represent the solution of Cauchy mixed problem in integral form by given and known functions has not been achieved up to now. This paper completes this omission for the one-dimensional heat equation.

Partial differential equations of parabolic type are widely represented in the study of heat conductivity and diffusion process. Numerous research works are devoted to study Cauchy mixed problem for model heat equations because of its theoretical and practical importance. Among them we can notice monographers [

Exterior potential method as a special continuation of a solution for all half-space is widely used under the solution of Cauchy mixed problem. Our idea is based on a representation possibility of general solution only in the form of volume potential excluding surface integrals. Thus the system of integral equations obtained by this method allows us to construct the solution in quadrature.

Consider the following problem in a plane domain

Our goal is to construct a classical solution of the problem (

It should be noticed that the heat potential

It is easy to verify that the first term in representation (

Our aim is to choose unknown functions

We will seek functions

We introduce the following notations:

The main result of this paper is as follows.

Let

To prove the theorem, the main role plays the following.

Let unknown functions be given by the formulae (

By substituting function (

After putting

Now we will construct the solution of the system (

Since Laplace inverse transformation of function

Then from (

To complete the proof of the theorem we represent functions

As long as

Therefore, the following relations hold:

We define function

From here and relation (

We obtain formula (

From (

Since

Consequently, representation of the solution