2. Traveling Wave Solution of the Main Problem

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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

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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 ∂²ϕu

∂x² , inR² 2.1

with following initial

ux,0 u₀x 0, inI 0,∞ 2.2

and boundary

u0, t u₁t μ₀tⁿ, t >0 2.3

conditions, where R² I ×0, T. Here, μ₀ 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:

iϕu∈C²R²,

iiϕu≥0, foru≥0 andσ≥2,

iiiforσ≥2,ϕuhave alternative signs on the domain whenux, 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 functionux, tandwx, t −∂u^σx, t/∂xDux, tare continuous in D_t{x, t|0≤x≤Dt, 0≤t≤T}, bututanduxdo not exist whenσ >2;

2whenσ2,utanduxare finite;

3for 1< σ <3/2, all derivativesu_t,u_x, andu_xxexist;

4for 3/2< σ <2 theu_tandu_xexist, butu_xxdoes not exist.

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Therefore whenσ >2, we must only consider the weak solution for the problem2.1–

2.3. As it is seen from the formula2.4, whenn1/σ−1, the solution is equal to zero at x t Dt. Here,D ±

σ/σ−1μ^σ−1₀ 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 whenu0,2.1degenerates to the first-order equation. The weak solution is defined as follows.

Definition 2.1. A nonnegative function ux, twhich 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

Dt

ux, t∂fx, t

∂t −∂u^σx, t

∂x

∂fx, t

∂x

dx dt0 2.7

holds for every test functionsfx, tfromC^o1,1Dtandfx, T 0.

Because the functionux, tandwx, t −∂u^σx, t/∂xare continuous in the domain D_t, the integral involving2.7exists in the Riemann sense.

Now, we show that the solution defined by the formula 2.4 satisfies the integral equality2.7; that is,ux, tis 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∂fx, t

∂x

dx dt

^T

0 Dt 0

ux, t∂fx, 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 tandx, 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 dt0. 2.9

If we reapply integration by parts to the inner integrals in the first term and second term byt andx, respectively, we prove that the solution in the form2.4satisfies the integral relation 2.7.

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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 v₀x, 3.2

u0, t ∂v0, t

∂x μ₀tⁿ 3.3

is introduced. Here, the functionv₀xis any solution of the equation

dv0x

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 functionvx, tis more than the diﬀerentiability property of the solutionux, 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 problem3.1–3.3, then the function ux, tobtained by3.6is a weak solution of the main problem2.1–2.3in sense of 2.7.

The auxiliary problem has the following advantages.

iThe functionvx, tis smoother thanux, t.

iiThe functionvx, tis an absolutely continuous function.

iiiIn the process of finding the solutionux, t, one does not need to use the first and second derivatives ofux, twith respect tox.

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

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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)=vx(x,t)

a

0 2 4 6 8 10

0 0.5 1 1.5 2 2.5

T=2 T=4 Ui,k

xi

T=6

b

Figure 2:aThe functionux, 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 regionD_tby a special grid:

ω_τ {xi, t_k|x_iDt_i, t_kkτ, i, k0,1,2, . . .}, 4.1

whereτis the step of the grid with respect totvariable.

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Now, we will develop a numerical algorithm as follows. Since the function∂u^σ/∂xis 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}

Vi 1,k 1−V_{i,k 1}^σ−Vi,k 1−V_{i−1,k 1}^σ

, 4.2

V_i,0 v₀xi, 4.3

V_{0,k 1}V_{1,k 1}−hμ0tⁿ_{k 1} 4.4

i0,1,2, . . .;k0,1,2, . . .. Here,hDτandv₀xiis 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 functionU_{i,k 1}defined by4.5is a solution of the nonlinear system of algebraic equations:

U_{i,k 1}U_i,k τ h²

U^σ_{i 1,k 1}−2U_{i,k 1}^σ U^σ_{i−1,k 1}

4.6

i1,2, . . .;k0,1,2, . . .. The initial and boundary conditions for4.6are

U_i,00 i0,1,2, . . .,

U_{0,k 1}μ₀tⁿ_{k 1} k0,1,2, . . .. 4.7

Here,Vi,kandUi,k are the approximation values ofvx, tandux, tat any pointxi, tkof the gridω_τ. For the sake of simplicity we introduce the notationsV_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, andη_i,kare the errors of approximations of the functions∂v/∂x, ∂v/∂t, and∂ϕ/∂x by finite diﬀerences, respectively. Then, we can write3.1in the following form:

vt δ_{i,k 1}ϕxvx ε_{i,k 1} η_{i,k 1} 4.8

or

v_tϕ_xv_x γ_{i,k 1}, 4.9

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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 whenτ → 0. It is known that the suitable characteristic of continuity of any functionfxon the any intervala, bis its modulus continuity:

ω δ, f

≡π f

sup

|t−x|<δ

ft−fx, ε_{i,k 1} ∂vxi, t_{k 1}

∂x −v_x ∂vxi, t_{k 1}

∂x −∂v

x^∗_i, t_{k 1}

∂x uxi, t_{k 1}−u

x_i^∗, t_{k 1}

πu−→0, x^∗_i ∈xi−h, x_i,

δ_{i,k 1} ∂vxi, t_{k 1}

∂t −vt ∂vxi, t_{k 1}

∂t −∂v x_i, t^∗_{k 1}

∂t ∂ϕuxi, t_{k 1}

∂x −∂ϕ

u

x_i, t^∗_{k 1}

∂x π

∂ϕu

∂x

−→0, t^∗_{k 1}∈tk 1, t_{k 1} τ. 4.11

Finally,

η_{i,k 1} ∂ϕu

∂x −ϕ_xu ∂ϕuxi, t_{k 1}

∂x −∂ϕ

u

x^∗∗_i , t_{k 1}

∂x −→0, x^∗∗_i ∈xi−h, x_i. 4.12

Therefore,γ_{i,k 1} → 0, ifτ → 0.

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

0≤V_{i,k 1} ≤Mmax

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

Proof. At first, let us write4.2in the following form:

V_{i,k 1}−Vi,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

−wxi, t_k.

4.14

Assume thatV_{i,k 1}is not constant andV_{i,k 1}takes the greatest value at some point of the gridω_τrather than at boundary nodes ofγ_τ. Then, there is such a pointx1, t₁∈ω_τ that V takes the maximal value and even some neighborhood pointsVx1, t₁less thanVx1, t₁.

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IfVx1,t1 > Vx1,t1, because the functionϕu u^σis 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 ifV_±x1, t1>Vx1, t1. Similarly, we can prove thatV does not take a minimal value at the inner nodes of the gridω_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,

maxi |vi,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 theU_i,k to the solution of the main problem. To this end, by subtracting4.2from4.9and taking4.5into account, we have

Rt

ϕuvx−Vx

x γi,k 4.16

or

R_t

ϕuu i,k−U_i,k

x γ_i,k, u∈ui,k, U_i,k, 4.17

whereRv_{i,k 1}−V_{i,k 1}. By multiplying the last equation withRand summing with respect toiandk, we get

R, R t

L2ωτ R,

ϕuui,k−Ui,k

x

L2ωτ

R, v i,k

L2ωτ. 4.18

Here, the symbolf, gdenotes the diﬀerences analogy of the scalar product of the functions offandginL2ωτsense:

f, g

L2ωττh

i1

k0

f_i,k g_i,k. 4.19

With some algebra we have 1 2

i

R²_i

tk

T

u_i,k−U_i,k, ϕuui,k−U_i,k

L2ωτ

≤ 1 2

i

R²_i

tk0 Tmax

i R²_iγi,k

L2ωτ.

4.20

It follows that the numerical solutionU_i,kconverges in mean tou_i,k.

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0 2 4 6 8 10

−14

−12

−10

−8

−6

−4

−2 0

Vi,k

xi

T=2 T=4

T=6

a

0 2 4 6 8 10

0 0.5 1 1.5 2 2.5

xi

T=2 T=4

T=6

Ui,k=Vx

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 valuesσ3,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 functionsvx, tandVi,k

coincide, too.

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 scheme4.2–4.4has the first-order approximation with respect tot. 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 introduced and it permits us to find the exact solution with singular properties.

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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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