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ISRN Mathematical Analysis
Volume 2013 (2013), Article ID 640985, 5 pages
http://dx.doi.org/10.1155/2013/640985
Research Article

Method of External Potential in Solution of Cauchy Mixed Problem for the Heat Equation

Institute of Mathematics and Mathematical Modeling, MES RK, Shevchenko Street, 28, Almaty 050010, Kazakhstan

Received 30 May 2013; Accepted 18 July 2013

Academic Editors: S. Filippas and K. Lurie

Copyright © 2013 T. Sh. Kalmenov and N. E. Tokmagambetov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. Introduction

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 [13] which demonstrate main research methods, such as Fourier method, integral equations method, and 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.

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.

2. Material and Methods

Consider the following problem in a plane domain .

Cauchy Mixed Problem. To find a regular solution of the following equation in with the initial condition and boundary conditions

Our goal is to construct a classical solution of the problem (1)–(3) in a quadrature. We will seek a solution in the form of sum of three volume potentials: where Here is a fundamental solution of the heat equation (1) and is Heaviside theta-function.

It should be noticed that the heat potential satisfies the following boundary condition: where is a boundary of the domain . Note that in works [4, 5] differential operators with nonlocal boundary conditions are investigated as above.

It is easy to verify that the first term in representation (4), , is a solution of nonhomogeneous equation (1) and the second and third ones, , are solutions of homogeneous equation. Consequently, representation (4) gives a solution of non-homogeneous equation (1) satisfying initial condition (2) for arbitrary , .

Our aim is to choose unknown functions and such that the solution will satisfy boundary condition (3).

We will seek functions , in the forms Let be periodical function with period defined on the interval by the formula . This function can be represented by the formula out of the interval [0, 2] .

We introduce the following notations:

3. Results and Discussion

The main result of this paper is as follows.

Theorem 1. Let , and then the solution of the problem (1)–(3) is represented by formula (4), where and are given by formulae (8):

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

Lemma 2. Let unknown functions be given by the formulae (8). Then and satisfy the system of equations where , are defined by formula (10).

Proof. By substituting function (4) in boundary condition (3) and taking account of (10), we get a system of interval equations with respect to unknown functions and : Integrating by part it is easy to verify correctness of the following equations:
After putting in integrals (13), our system can be transformed to By differentiating these relations we obtain the system (12). The lemma is proved.
Now we will construct the solution of the system (12) by using the Laplace transformation properties: The system (12) can be transformed in the next form: where , are images of the Laplace transform. After solving this system with respect to and we find
Since Laplace inverse transformation of function has no table form, so by using the expansions and table values of the Laplace inverse transformation,
Then from (18) we obtain where
To complete the proof of the theorem we represent functions and by integrals of known functions.
As long as , is a tempered distribution [6, page 112], and then we have
Therefore, the following relations hold:
We define function on the interval and then continue it periodically on all axes . It is shown in [6, page 113] where is regarded in the general function sense.
From here and relation (25) we conclude that Analogically, the relation follows from (26).
We obtain formula (9) after substituting the last relations in (22) and (23).
From (23) we have , and from (22) .
Since, , then , . Therefore it is not difficult to establish .
Consequently, representation of the solution in the form (4) is established. The theorem is fully proved.

References

  1. V. S. Vladimirov, Equations of Mathematical Physics, “Nauka”, Moscow, Russia, 1988. View at Zentralblatt MATH · View at MathSciNet
  2. O. A. Ladyzhenskaya, Boundary Value Problems for the Parabolic Equations, Nauka, 1973.
  3. A. N. Tikhonov and A. A. Samarskyi, Equations of Mathematical Physics, Nauka, 1980.
  4. T. Sh. Kal'menov and N. E. Tokmagambetov, “On a boundary condition for a spatially one-dimensional generalized volume heat potential,” Izvestiya Natsional noĭ Akademii Nauk Respubliki Kazakhstan, no. 5, pp. 3–9, 2010. View at MathSciNet
  5. D. Suragan and N. Tokmagambetov, “On transparent boundary conditions for the high-order heat equation,” Siberian Electronic Mathematical Reports, vol. 10, pp. 141–149, 2013.
  6. V. A. Ditkin and A. P. Prudnikov, Integral Transformations and Operator Calculus, Nauka, 1984.