Volume 2012, Article ID 791026,10pages doi:10.1155/2012/791026

*Research Article*

**The Analytical and a Higher-Accuracy Numerical** **Solution of a Free Boundary Problem in a Class of** **Discontinuous Functions**

**Bahaddin Sinsoysal**

*Department of Mathematics and Computing, Beykent University, 34396 Istanbul, Turkey*

Correspondence should be addressed to Bahaddin Sinsoysal,bsinsoysal@beykent.edu.tr Received 8 July 2011; Revised 10 September 2011; Accepted 8 October 2011

Academic Editor: Ezzat G. Bakhoum

Copyrightq2012 Bahaddin Sinsoysal. 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.

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.

**1. Introduction**

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.

In1, 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 in2,3, 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 5, Ole˘ınik 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 4, 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 algorithmshomogeneous schemeswhich are approximated by finite diﬀerences of the diﬀerential problem without taking into account the properties occurring in the exact solution6–8.

**2. Traveling Wave Solution of the Main Problem**

We consider the equation

*∂u*

*∂t* *∂*^{2}*ϕu*

*∂x*^{2} *,* in*R*^{2} 2.1

with following initial

*ux,*0 *u*_{0}x 0, in*I* 0,∞ 2.2

and boundary

*u0, t u*_{1}t *μ*_{0}*t*^{n}*,* *t >*0 2.3

conditions, where *R*^{2} *I* ×0, T. Here, *μ*_{0} and *n* are real known constants. In order to
study the properties of the exact solution of the problem2.1–2.3for the sake of simplicity
the case *ϕu * *u** ^{σ}* is considered. It is clear that the function

*ϕu*satisfies the following conditions:

i*ϕu*∈*C*^{2}R^{2},

ii*ϕ*^{}u≥0, for*u*≥0 and*σ*≥2,

iiifor*σ*≥2,*ϕ*^{}uhave alternative signs on the domain when*ux, t/*0.

It is easily shown that the problem2.1–2.3has the solution in the traveling wave form as

*ux, t *

⎧⎪

⎨

⎪⎩

*Dσ*−1
*σ*

_{1/σ−1}

Dt−*x*^{1/σ−1}*,* 0*< x < Dt,*

0, *x*≥*Dt.*

2.4

Via simple calculation we get the following:

1the function*ux, t*and*wx, t *−∂u* ^{σ}*x, t/∂x

*Dux, t*are continuous in

*D*

*{x, t|0≤*

_{t}*x*≤

*Dt,*0≤

*t*≤

*T}, butu*

*t*and

*u*

*x*do not exist when

*σ >*2;

2when*σ*2,*u**t*and*u**x*are finite;

3for 1*< σ <*3/2, all derivatives*u** _{t}*,

*u*

*, and*

_{x}*u*

*exist;*

_{xx}4for 3/2*< σ <*2 the*u** _{t}*and

*u*

*exist, but*

_{x}*u*

*does not exist.*

_{xx}Therefore when*σ >*2, we must only consider the weak solution for the problem2.1–

2.3. As it is seen from the formula2.4, when*n*1/σ−1, the solution is equal to zero at
*x* *t Dt. Here,D* ±

σ/σ−1μ^{σ−1}_{0} is the speed of the front of the traveling wave
and

*ut, t *0. 2.5

Taking into account2.5, we get
*dx*

*dt* −*∂u/∂t*

*∂u/∂x* *D∂ux, t/∂x*

*∂u/∂x* *D.* 2.6

It is clear that when*u*0,2.1degenerates to the first-order equation. The weak solution is
defined as follows.

*Definition 2.1. A nonnegative function* *ux, t*which is satisfying the initial condition 2.2
and boundary conditions2.3and2.5is called a weak solution of the problem2.1–2.3,
and2.5, if the following integral relation

*D**t*

*ux, t∂fx, t*

*∂t* −*∂u** ^{σ}*x, t

*∂x*

*∂f*x, t

*∂x*

*dx dt*0 2.7

holds for every test functions*f*x, tfrom*C** ^{o}*1,1D

*t*and

*fx, T*0.

Because the function*ux, t*and*wx, t *−∂u* ^{σ}*x, t/∂xare continuous in the domain

*D*

*, the integral involving2.7exists in the Riemann sense.*

_{t}Now, we show that the solution defined by the formula 2.4 satisfies the integral
equality2.7; that is,*ux, t*is the weak solution of the problem2.1–2.3.

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

0 ^{T}

0
*Dt*
0

*ux, t∂fx, t*

*∂t* *Dux, t∂f*x, t

*∂x*

*dx dt*

^{T}

0
*Dt*
0

*ux, t∂f*x, t

*∂t* *dx dt* *D*

*T*
0

*Dt*
0

*ux, t∂fx, t*

*∂x* *dx dt.*

2.8

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
*t*and*x, respectively, we have*

*DT*
0

*T*
*x/D*

*ux, t∂fx, t*

*∂t* *dt dx* *D*

*T*
0

*Dt*
0

*ux, t∂fx, t*

*∂x* *dx dt*0. 2.9

If we reapply integration by parts to the inner integrals in the first term and second term by*t*
and*x, respectively, we prove that the solution in the form*2.4satisfies the integral relation
2.7.

**3. Auxiliary Problem and Its Exact Solution**

In order to find the weak solution of the problem2.1–2.3, according to9,10the special auxiliary problem

*∂v*

*∂t* *∂*

*∂x*
*∂v*

*∂x*
_{σ}

*,* 3.1

*vx,*0 *v*_{0}x, 3.2

*u0, t * *∂v0, t*

*∂x* *μ*_{0}*t** ^{n}* 3.3

is introduced. Here, the function*v*_{0}xis any solution of the equation

*dv*0x

*dx* 0. 3.4

The problem3.1–3.3has the solution

*vx, t *

⎧⎪

⎨

⎪⎩

−D^{1/σ−1}
*σ*−1

*σ*

_{σ/σ−1}

Dt−*x*^{σ/σ−1}*, x < Dt,*

0, *x*≥*Dt*

3.5

in the traveling wave form 9. As is seen from 3.5, the diﬀerentiability property of the
function*vx, t*is more than the diﬀerentiability property of the solution*ux, t. In addition*
to this, from3.5we get

*ux, t * *∂vx, t*

*∂x* *.* 3.6

* Theorem 3.1. If the functionvx, tis a soft solution of the problem*3.1–3.3, then the function

*ux, tobtained by*3.6

*is a weak solution of the main problem*2.1–2.3

*in sense of*2.7.

The auxiliary problem has the following advantages.

iThe function*vx, t*is smoother than*ux, t.*

iiThe function*vx, t*is an absolutely continuous function.

iiiIn the process of finding the solution*ux, t, one does not need to use the first and*
second derivatives of*ux, t*with respect to*x.*

The graphs of the function structured by the formulas2.4,3.5, and3.6are shown in the Figures1a,1b, and2a, respectively.

0 2 4 6 8 10 0

0.5 1 1.5 2 2.5

*T*=2
*T*=4

*T*=6

*x*

*u*(*x*,*t*)

a

0 2 4 6 8 10

−12

−10

−8

−6

−4

−2 0

*v*(*x*,*t*)

*T*=2
*T*=4

*T*=6

*x*
b

**Figure 1:**aThe exact solution of the main problem.bThe exact solution of the auxiliary problem.

0 2 4 6 8 10

0 0.5 1 1.5 2 2.5

*x*
*u*(*x,**t*)=*v**x*(*x,**t*)

a

0 2 4 6 8 10

0 0.5 1 1.5 2 2.5

*T*=2
*T*=4
*U**i,k*

*x**i*

*T*=6

b

**Figure 2:**aThe function*ux, t * *∂vx, t/∂x.*bNumerical solution obtained by using the classical
algorithm4.6–4.7.

**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*D** _{t}*by a special grid:

*ω** _{τ}* {x

*i*

*, t*

*|*

_{k}*x*

_{i}*Dt*

_{i}*, t*

_{k}*kτ, i, k*0,1,2, . . .}, 4.1

where*τ*is the step of the grid with respect to*t*variable.

Now, we will develop a numerical algorithm as follows. Since the function*∂u*^{σ}*/∂x*is
continuous, we can approximate the problem3.1–3.3by the following finite diﬀerences
schemes:

*V*_{i,k 1}*V*_{i,k}*τ*
*h*^{σ 1}

V*i 1,k 1*−*V*_{i,k 1}* ^{σ}*−V

*i,k 1*−

*V*

_{i−1,k 1}

^{σ}*,* 4.2

*V*_{i,0}*v*_{0}x*i*, 4.3

*V*_{0,k 1}*V*_{1,k 1}−*hμ*0*t*^{n}* _{k 1}* 4.4

i0,1,2, . . .;*k*0,1,2, . . .. Here,*hDτ*and*v*_{0}x*i*is any grid function of3.4.

It can be easily shown that

*U*_{i,k 1}*V** _{i 1,k 1}*−

*V*

_{i,k 1}*h* *,* 4.5

and the grid function*U** _{i,k 1}*defined by4.5is a solution of the nonlinear system of algebraic
equations:

*U*_{i,k 1}*U*_{i,k}*τ*
*h*^{2}

*U*^{σ}* _{i 1,k 1}*−2U

_{i,k 1}

^{σ}*U*

^{σ}

_{i−1,k 1}4.6

i1,2, . . .;*k*0,1,2, . . .. The initial and boundary conditions for4.6are

*U** _{i,0}*0 i0,1,2, . . .,

*U*_{0,k 1}*μ*_{0}*t*^{n}* _{k 1}* k0,1,2, . . .. 4.7

Here,*V**i,k*and*U**i,k* are the approximation values of*vx, t*and*ux, t*at any pointx*i**, t**k*of
the grid*ω** _{τ}*. For the sake of simplicity we introduce the notations

*V*

_{i,k}*V*,

*V*

_{i,k 1}*V,*and

*V*

_{i±1,k 1}*V*

_{±}.

**4.1. Convergence**

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
*ε*_{i,k}*, δ** _{i,k}*, and

*η*

*are the errors of approximations of the functions*

_{i,k}*∂v/∂x, ∂v/∂t, and∂ϕ/∂x*by finite diﬀerences, respectively. Then, we can write3.1in the following form:

*v**t* *δ*_{i,k 1}*ϕ**x**v**x* *ε*_{i,k 1}*η** _{i,k 1}* 4.8

or

*v*_{t}*ϕ*_{x}*v*_{x}*γ*_{i,k 1}*,* 4.9

where

*γ*_{i,k 1}*δ*_{i,k 1}*ϕ*_{xx}*y*_{x}

*ε*_{i,k 1}*η*_{i,k 1}*.* 4.10

At first, we show that the finite diﬀerence scheme 4.2 is consistent; that is, *γ** _{i,k 1}*
approaches zero when

*τ*→ 0. It is known that the suitable characteristic of continuity of any function

*f*xon the any intervala, bis its modulus continuity:

*ω*
*δ, f*

≡*π*
*f*

sup

|t−x|<δ

*ft*−*fx,*
*ε*_{i,k 1}*∂vx**i**, t*_{k 1}

*∂x* −*v*_{x}*∂vx**i**, t*_{k 1}

*∂x* −*∂v*

*x*^{∗}_{i}*, t*_{k 1}

*∂x*
*ux**i**, t** _{k 1}*−

*u*

*x*_{i}^{∗}*, t*_{k 1}

*πu*−→0, *x*^{∗}* _{i}* ∈x

*i*−

*h, x*

*,*

_{i}*δ*_{i,k 1}*∂vx**i**, t*_{k 1}

*∂t* −*v**t* *∂vx**i**, t*_{k 1}

*∂t* −*∂v*
*x*_{i}*, t*^{∗}_{k 1}

*∂t*
*∂ϕux**i**, t*_{k 1}

*∂x* −*∂ϕ*

*u*

*x*_{i}*, t*^{∗}_{k 1}

*∂x* *π*

*∂ϕu*

*∂x*

−→0, *t*^{∗}* _{k 1}*∈t

*k 1*

*, t*

_{k 1}*τ*. 4.11

Finally,

*η*_{i,k 1}*∂ϕu*

*∂x* −*ϕ*_{x}*u * *∂ϕux**i**, t*_{k 1}

*∂x* −*∂ϕ*

*u*

*x*^{∗∗}_{i}*, t*_{k 1}

*∂x* −→0, *x*^{∗∗}* _{i}* ∈x

*i*−

*h, x*

*. 4.12*

_{i}Therefore,*γ** _{i,k 1}* → 0, if

*τ*→ 0.

**Theorem 4.1**Maximum Principle. The solution of problem4.2–4.4*takes its maximum (or*
*minimum) value on the boundary of the domain of definition of the solution, that is,*

0≤*V** _{i,k 1}* ≤

*M*max

*i,k* {|u0|,|u1|}. 4.13

*Proof. At first, let us write*4.2in the following form:

*V** _{i,k 1}*−

*V*

*i,k*

*τ* 1

*h*

*V** _{i 1,k 1}*−

*V*

_{i,k 1}*h*

*σ*

−

*V** _{i,k 1 1}*−

*V*

_{i−1,k 1}*h*

*σ*

1
*h*

*U*^{σ}* _{i}* −

*U*

^{σ}

_{i−1}−wx*i**, t** _{k}*.

4.14

Assume that*V** _{i,k 1}*is not constant and

*V*

*takes the greatest value at some point of the grid*

_{i,k 1}*ω*

*rather than at boundary nodes of*

_{τ}*γ*

*. Then, there is such a pointx1*

_{τ}*, t*

_{1}∈

*ω*

*that*

_{τ}*V*takes the maximal value and even some neighborhood points

*V*x1

*, t*

_{1}less than

*V*x1

*, t*

_{1}.

If*V**x*1*,t*1 *> V**x*1*,t*1, because the function*ϕu u** ^{σ}*is monotone, the left part of the relation
4.14is positive, but the right part is negative. Hence we arrive to inconsistency. We arrive to
the same inconsistency if

*V*

_{±}x1

*, t*1

*>V*x1

*, t*1. Similarly, we can prove that

*V*does not take a minimal value at the inner nodes of the grid

*ω*

*.*

_{τ}FromTheorem 4.1, it follows that the solution of the problem4.2–4.4converges to the solution of problem3.1–3.3pointwise, that is,

max*i* |v*i,k*−*V** _{i,k}*| −→0. 4.15

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*U** _{i,k}* to the solution of the main problem. To
this end, by subtracting4.2from4.9and taking4.5into account, we have

*R**t*

*ϕ*^{}*uv**x*−*V**x*

*x* *γ**i,k* 4.16

or

*R*_{t}

*ϕ*^{}*uu* *i,k*−*U*_{i,k}

*x* *γ*_{i,k}*,* *u*∈u*i,k**, U** _{i,k}*, 4.17

where*Rv** _{i,k 1}*−

*V*

*. By multiplying the last equation with*

_{i,k 1}*R*and summing with respect to

*i*and

*k, we get*

*R, R* *t*

*L*2ω*τ*
*R,*

*ϕ*^{}*uu**i,k*−*U**i,k*

*x*

*L*2ω*τ*

*R, v* *i,k*

*L*2ω*τ**.* 4.18

Here, the symbolf, gdenotes the diﬀerences analogy of the scalar product of the functions
of*f*and*g*in*L*2ω*τ*sense:

*f, g*

*L*2ω*τ**τh*

*i1*

*k0*

*f*_{i,k}*g*_{i,k}*.* 4.19

With some algebra we have 1 2

*i*

*R*^{2}_{i}

*t**k*

*T*

*u** _{i,k}*−

*U*

_{i,k}*, ϕ*

^{}

*uu*

*i,k*−

*U*

_{i,k}*L*2ω*τ*

≤ 1 2

*i*

*R*^{2}_{i}

*t**k*0 *T*max

*i* *R*^{2}_{i}*γ**i,k*

*L*2ω*τ**.*

4.20

It follows that the numerical solution*U** _{i,k}*converges in mean to

*u*

*.*

_{i,k}0 2 4 6 8 10

−14

−12

−10

−8

−6

−4

−2 0

*V**i,k*

*x**i*

*T*=2
*T*=4

*T*=6

a

0 2 4 6 8 10

0 0.5 1 1.5 2 2.5

*x**i*

*T*=2
*T*=4

*T*=6

*U**i,k***=***V**x*

b

**Figure 3:**aNumerical solution of the problem4.2–4.4.bNumerical solution of the main problem
obtained by using the solution of the problem4.2–4.4.

**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
2.1–2.3on an equal footing. With this aim, firstly, computer experiments are performed
on the algorithm4.2–4.4with the values*σ*3,*T* 2,4, and 6. The graphs of the obtained
numerical solutions are presented in Figure 3a. The numerical solutions of the main
problem2.1–2.3 obtained by using the algorithm 4.2–4.4 are shown in Figure 3b.

The numerical solution of the main problem obtained using the algorithm 4.6–4.7 is demonstrated inFigure 2b.

Comparing Figures1aand3bshows that the numerical and the exact solutions of
the main problem2.1–2.3coincide. Moreover, the graphs of the functions*vx, t*and*V**i,k*

coincide, too.

Thus, the numerical experiments carried out show that the suggested numerical algo- rithms 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 scheme4.2–4.4has the first-order approximation with respect
to*t. But using the diﬀerent versions of Runge-Kutta’s methods we can increase the order of*
the algorithms mentioned previously.

**6. Conclusions**

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 intro- duced and it permits us to find the exact solution with singular properties.

The auxiliary problem introduced previously allows us to develop the higher-reso- lution 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.

**References**

1 G. I. Barenblatt and M. I. Viˇsik, “On finite velocity of propagation in problems of non-stationary
*filtration of a liquid or gas,” Prikladnaja Matematika i Mehanika, vol. 20, pp. 411–417, 1956.*

2 S. N. Antoncev, “On the localization of solutions of non-linear degenerate elliptic and parabolic
*equations,” Soviet Mathematics. Doklady, vol. 24, pp. 420–424, 1981.*

3 L. K. Martinson and K. B. Pavlov, “On the problem of spatial localization of thermal perturbations in
*the theory of non-linear heat conduction,” USSR Computational Math and Math Physics, vol. 12, no. 4,*
pp. 261–268, 1972.

4 *S. L. Kamenomostskaya, “On Stefan’s problem,” Mathematics Sbornik, vol. 53, pp. 489–514, 1961.*

5 O. A. Ole˘ınik, “A method of solution of the general Stefan problem,” Soviet Mathematics. Doklady, vol.

1, pp. 1350–1354, 1960.

6 A. A. Abramov and A. N. Gaipova, “On the numerical solution of some sets of diﬀerential equations
*for problems of Stefan’s type,” USSR Computational Mathematics and Mathematical Physics, vol. 11, no.*

1, pp. 162–170, 1971.

7 *V. F. Baklanovskaya, “The numerical solution of a nonstationary filtration problem,” USSR*
*Computational Mathematics and Mathematical Physics, vol. 1, no. 1, pp. 114–122, 1962.*

8 B. M. Budak and A. B. Uspenskii, “A diﬀerence method with front straightening for solving Stefan-
*type problems,” USSR Computational Mathematics and Mathematical Physics, vol. 9, no. 6, pp. 83–103,*
1969.

9 *M. A. Rasulov, “A numerical method of solving a parabolic equation with degeneration,” Diﬀerential*
*Equations, vol. 18, no. 8, pp. 1418–1427, 1992.*

10 B. Sinsoysal, “A new numerical method for Stefan-type problems in a class of unsmooth functions,”

*International Journal of Contemporary Mathematical Sciences, vol. 5, no. 27, pp. 1323–1335, 2010.*

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