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

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 [1–3] 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