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A new method is suggested for obtaining the exact and numerical solutions of the initial-boundary value problem for a nonlinear parabolic type equation in the domain with the free boundary. With this aim, a special auxiliary problem having some advantages over the main problem and being equivalent to the main problem in a definite sense is introduced. The auxiliary problem allows us to obtain the weak solution in a class of discontinuous functions. Moreover, on the basis of the auxiliary problem a higher-resolution numerical method is developed so that the solution accurately describes all physical properties of the problem. In order to extract the significance of the numerical solutions obtained by using the suggested auxiliary problem, some computer experiments are carried out.

It is known that many practical problems such as distribution of heat waves, melting glaciers, and filtration of a gas in a porous medium, and so forth, are described by nonlinear equations of the parabolic type.

In [

These problems are also called free boundary problems. Therefore, it is necessary to obtain the moving unknown boundary together with the solution of a differential problem. Its nature raises several difficulties for finding analytical as well as numerical solutions of this problem.

The questions of the existence and uniqueness of the solutions of the free boundary problems are studied in [

In the literature there are some numerical algorithms (homogeneous schemes) which are approximated by finite differences of the differential problem without taking into account the properties occurring in the exact solution [

We consider the equation

for

It is easily shown that the problem (

the function

when

for

for

Therefore when

A nonnegative function

Because the function

Now, we show that the solution defined by the formula (

According to the definition of the weak solution, we can write

In order to find the weak solution of the problem (

If the function

The auxiliary problem has the following advantages.

The function

The function

In the process of finding the solution

The graphs of the function structured by the formulas (

(a) The exact solution of the main problem. (b) The exact solution of the auxiliary problem.

(a) The function

In this section we investigate an algorithm for finding a numerical solution of the problem (

Now, we will develop a numerical algorithm as follows. Since the function

It can be easily shown that

In this section we will investigate some properties of the numerical solution and of the question of convergence of the numerical solution to the weak exact solution. Suppose that

At first, we show that the finite difference scheme (

The solution of problem (

At first, let us write (

Assume that

If

From Theorem

Now, we will prove convergence of the

In order to extract the significance of the suggested method, the numerical solution obtained using the proposed auxiliary problem is compared with the exact solution of the problem (

(a) Numerical solution of the problem (

Comparing Figures

Thus, the numerical experiments carried out show that the suggested numerical algorithms are efficient and economical from a computer point of view. The proposed algorithms permit us to develop the higher-resolution methods where the obtained solution correctly describes all physical features of the problem, even if the differentiability order of the solution of the problem is less than the order of differentiability which is required by the equation from the solution.

The finite differences scheme (

The new method is suggested for obtaining the regular weak solution for the free boundary problem of the nonlinear parabolic type equation.

The auxiliary problem which has some advantages over the main problem is introduced and it permits us to find the exact solution with singular properties.

The auxiliary problem introduced previously allows us to develop the higher-resolution method where the obtained solution correctly describes all physical features of the problem, even if the differentiability order of the solution of the problem is less than the order of differentiability which is required by the equation from the solution.