The Analytical and a Higher-Accuracy Numerical Solution of a Free Boundary Problem in a Class of Discontinuous Functions
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 , at first the effect of localization of the solution of the equation describing the motion of perfect gas in a porous medium is observed and the solution in the traveling wave form is structured. Then, the mentioned properties of the solution for the nonlinear parabolic type equation are studied in [2, 3], and so forth.
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 [4, 5]. In , Oleĭnik introduced the notion of a generalized solution of the Stefan problem whose uniqueness and existence were guaranteed in the class of measurable bounded functions. In , Kamenomostskaya considered the classical quasilinear heat conduction equation and constructed the generalized solution by the use of an explicit difference scheme.
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 [6–8].
2. Traveling Wave Solution of the Main Problem
We consider the equation with following initial and boundary conditions, where . Here, and are real known constants. In order to study the properties of the exact solution of the problem (2.1)–(2.3) for the sake of simplicity the case is considered. It is clear that the function satisfies the following conditions:(i),(ii), for and ,(iii)for , have alternative signs on the domain when .
It is easily shown that the problem (2.1)–(2.3) has the solution in the traveling wave form as Via simple calculation we get the following:(1)the function and are continuous in , but and do not exist when ;(2)when , and are finite;(3)for , all derivatives , , and exist;(4)for the and exist, but does not exist.
Therefore when , we must only consider the weak solution for the problem (2.1)–(2.3). As it is seen from the formula (2.4), when , the solution is equal to zero at . Here, is the speed of the front of the traveling wave and Taking into account (2.5), we get It is clear that when , (2.1) degenerates to the first-order equation. The weak solution is defined as follows.
Definition 2.1. A nonnegative function which is satisfying the initial condition (2.2) and boundary conditions (2.3) and (2.5) is called a weak solution of the problem (2.1)–(2.3), and (2.5), if the following integral relation holds for every test functions from and .
Because the function and are continuous in the domain , the integral involving (2.7) exists in the Riemann sense.
According to the definition of the weak solution, we can write Changing the order of integration in the first integral and then applying the integration by parts to the inner integrals of the first and second terms in the last expression with respect to and , respectively, we have If we reapply integration by parts to the inner integrals in the first term and second term by and , respectively, we prove that the solution in the form (2.4) satisfies the integral relation (2.7).
3. Auxiliary Problem and Its Exact Solution
In order to find the weak solution of the problem (2.1)–(2.3), according to [9, 10] the special auxiliary problem is introduced. Here, the function is any solution of the equation The problem (3.1)–(3.3) has the solution in the traveling wave form . As is seen from (3.5), the differentiability property of the function is more than the differentiability property of the solution . In addition to this, from (3.5) we get
The auxiliary problem has the following advantages.(i)The function is smoother than .(ii)The function is an absolutely continuous function.(iii)In the process of finding the solution , one does not need to use the first and second derivatives of with respect to .
4. Developing a Numerical Algorithm in a Class of Discontinuous Functions
In this section we investigate an algorithm for finding a numerical solution of the problem (2.1)–(2.3). At first, we cover the region by a special grid: where is the step of the grid with respect to variable.
Now, we will develop a numerical algorithm as follows. Since the function is continuous, we can approximate the problem (3.1)–(3.3) by the following finite differences schemes: . Here, and is any grid function of (3.4).
It can be easily shown that and the grid function defined by (4.5) is a solution of the nonlinear system of algebraic equations: . The initial and boundary conditions for (4.6) are Here, and are the approximation values of and at any point of the grid . For the sake of simplicity we introduce the notations , and .
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 , and are the errors of approximations of the functions , and by finite differences, respectively. Then, we can write (3.1) in the following form: or where
At first, we show that the finite difference scheme (4.2) is consistent; that is, approaches zero when . It is known that the suitable characteristic of continuity of any function on the any interval is its modulus continuity: Finally, Therefore, , if .
Proof. At first, let us write (4.2) in the following form:
Assume that is not constant and takes the greatest value at some point of the grid rather than at boundary nodes of . Then, there is such a point that takes the maximal value and even some neighborhood points less than .
If , because the function is monotone, the left part of the relation (4.14) is positive, but the right part is negative. Hence we arrive to inconsistency. We arrive to the same inconsistency if . Similarly, we can prove that does not take a minimal value at the inner nodes of the grid .
From Theorem 4.1, it follows that the solution of the problem (4.2)–(4.4) converges to the solution of problem (3.1)–(3.3) pointwise, that is, It can be easily seen that the solution of the mentioned problem is continuously dependent on initial data.
Now, we will prove convergence of the to the solution of the main problem. To this end, by subtracting (4.2) from (4.9) and taking (4.5) into account, we have or where . By multiplying the last equation with and summing with respect to and , we get Here, the symbol denotes the differences analogy of the scalar product of the functions of and in sense: With some algebra we have It follows that the numerical solution converges in mean to .
5. Numerical Experiments
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 (2.1)–(2.3) on an equal footing. With this aim, firstly, computer experiments are performed on the algorithm (4.2)–(4.4) with the values , , and 6. The graphs of the obtained numerical solutions are presented in Figure 3(a). The numerical solutions of the main problem (2.1)–(2.3) obtained by using the algorithm (4.2)–(4.4) are shown in Figure 3(b). The numerical solution of the main problem obtained using the algorithm (4.6)–(4.7) is demonstrated in Figure 2(b).
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 (4.2)–(4.4) has the first-order approximation with respect to . But using the different versions of Runge-Kutta's methods we can increase the order of the algorithms mentioned previously.
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.
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