A numerical method to approximate the solutions of non-linearabsorption diffusion equations

Alexandru Dumitrache1,∗

1 Institute of Mathematical Statistics and Applied Mathematics, P.O. Box 1-24, 010145 Bucharest, Romania.

This paper investigates the motivation for the n-dimensional absorption-diffusion equation. First, the one-dimensional, non-linear absorption-diffusion equation was mathematically developed. A numerical method to computationally solve it anddiffusion-free equations were also developed. For the cases tested, the numerical diffusion solutions are always less than thenumerical diffusion-free solutions.

1 The Mathematical problem

Let us consider a non-linear absorption-diffusion equation:

ut = ∇2Φ(u)− F (u), x ∈ Ω, 0 < t < ∞ (1)

where Ω is a bounded domain in Rn, Φ is diffusion terms, F is absorption terms and u(x, t) is the temperature or concentrationprofile. The goal of paper is to analyse the types of diffusion and absorption terms, and in which conditions the equation (1)produce or not extinction in finite time. Φ and F are assumed nondecreasing, nonnegative C1((0,∞)) ∩ C[0,∞)) functions:Φ(0) = 0, F (0) = 0, Φ(x) > 0, F (x) > 0, if x > 0.

The considered boundary conditions are homogeneous Dirichlet condition: u = 0, on ∂Ω× [0,∞) and initial conditions:u(x, 0) = u0(x) ≥ 0 on Ω, where u0 is the initial temperature of the material.

By considering a related problem given by the diffusion-free equation,

w′(t) = −[Φ(w(t)) + F (w(t))], 0 < t < ∞, w(0) = T > 0, (2)

can be shown that the solution, w(t), to equation (2), has finite extinction time if and only if∫ ε

0

ds

Φ(s) + F (s)< ∞ with ε > 0. (3)

If we have a Φ and F that satisfy condition (3), then w will have a finite extinction time. If we find that u(x, t) ≤ w(t) for allx and t, then u will also have a finite extinction time.

We know that all nonnegative solutions of (1) have extinction time, but we wish to determine whether extinction timeoccurs in finite or infinite time for any Φ and F . This can be done directly by numerically calculating the solution of theone-dimensional version of equation (1). Indirectly this can be done by attempting to show numerically that any solution u of(1) must to satisfy u ≤ w where w is the solution to (2).

2 Numerical analysis

A numerical method for the solution of equation (1) in one-dimension is derived, after rewriting it in terms of partial deriva-tives: ut = Φ′(u)uxx + Φ′′(u)(ux)2 − F (u), that allow us to create a finite difference scheme. A predictor-corrector methodis used in order to ensure the accuracy of solution.

Corrector for absorption-diffusion problem. First, we will discretize in space the above equation, with an error on theapproximations O(�x2). Our space is the closed interval [0, 1]. Then, we need to discretize in time, with the time interval[0,∞), tn = n�t, n = 0, 1, .... Notice that

∫ t+�t

tut(xi, t)dt = ui(t +�t) − ui(t). We will let uin = ui(tn) = u(xi, tn.

After some calculation we have arranged the equation in a matrix form, to allow us to solve entire x interval at once for eachtime step. Note that we can incorporate our boundary conditions of u0n = 0 and uNn = 0. We have

An+1−→u n+1 = Bn

−→u n − �t

2

−→F n. (4)

∗ Corresponding author E-mail: alex dumitrache@yahoo.com, Phone: +00 40 21 3182433, Fax: +00 40 21 3182439

PAMM · Proc. Appl. Math. Mech. 7, 4070041–4070042 (2007) / DOI 10.1002/pamm.200701050

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

where An+1 and Bn+1 are representing the matrix of terms from equation in finite difference, −→u n is representing the uin

vector, and−→F n the F (uin+1) + F (uin). Finally, it can be solved for −→u n+1 to get

−→u n+1 = A−1n+1Bn

−→u n − �t

2A−1

n+1

−→F n (5)

When solving equation (5), we will choose a forward difference method as our predictor because it a good, standard explicitmethod with a local error of O(�x2).

Corrector for absorption-diffusion problem. We then follow the same way as in previous subsection. It is used theforward rectangle rule and incorporating the boundary conditions we can obtain the matrix form of equation:

−→u (p)n+1 = Cn

−→u (p)n −�t

−→F (6)

where the superscript (p) denotes predictor.Predictor-corrector for absorption-diffusion problem. Now we can put it all together. By choosing a �x and �t that

satisfy the stability condition, Φ′(u)�t/(�x2) ≤ (1/2) it is assured that the predictor method is computationally stable.Next, because the initial condition is u(x, 0) = u0(x), can be create initial vector,−→u (p)

0 . We then solve equation (8) for−→u (p)1 .

Now, we substitute−→u (p)1 and−→u (p)

0 into An+1, Bn, −→u n and−→F n in equation (4) for−→u 1 as shown in equation (5). Then, we let

−→u 1 = −→u (p)1 and we use this our new predictor in equation (6). Finally, we repeat the process for each time step until we have

reached the final time step. In cases with finite extinction time, the final time step is at the time of extinction. Alternatively,we allow the computer to run until uin ≤ 10−324.

In a similar way can be produced a numerical method which can create a numerical method for solving diffusion-freeequation (2): w

(p)n+1 = w

(p)n −�t[Φ(w

(p)n ) + F (w

(p)n )]. We use the predictor-corrector method to increase the accuracy.

3 Analysis of numerical results

It is proved that this numerical method for the solution of the one-dimensional version of equation (1) is an accurate solutionwhen the problem is linear. It is also seen that this method for certain cases of Φ and F , the solution has finite extinctiontime, consistent with the analytical results of Lair [1]. The solutions of equation (1) and (2) are compared to check inequalityu(x, t) ≤ w(t). For this applications it was considered the equation ut = [up]xx − uq , for some p > 0 and q ≥ 0. We knowthis equation has finite extinction time if and only if min(p, q) < 1. It was assumed the zero Dirichlet boundary conditionsand the initial conditions u(x, 0) = sin(πx), for 0 ≤ x ≤ 1. One can see that Φ seems to have a much greater influence onthe solution than does F .

If increasing the power of Phi and F , increases extinction time, then making both p and q less than 1 should causeextinction to occur at an earlier time. We get an extinction time of t = 0.192 with Φ =

√u and F =

√u.

The code can handle any Φ = up, where p ≥ 1/2 and also any F = uq where q ≥ 0. The only exception to this is if thedifference between p and q is large [4].

It was made a comparison between one-dimensional version of equation (1) and the numerical solution of equation (2) withT = 1. It was compared the maxi[u(xi, t)] to w(t),at each t. We have found u ≤ w for all Φ and F with the same boundaryand initial conditions.

4 Conclusions

We developed a numerical method in order to calculate solutions to the one-dimensional, non-linear absorption-diffusionequation. We tested the accuracy of this method and we saw that the numerical solution generated only a moderate amountof error. With such a small amount of error in our method, we were confident that our numerical scheme was accurate andproceeded to test various types of diffusion and absorption terms in order to see which ones produced extinction in finitetime [2], [3]. Our extinction results were consistent with the analytical results of Lair and Oxley [1]. Next, we developeda numerical method to computationally solve the diffusion-free equation, in a similar manner to the method for the one-dimensional, non-linear absorption-diffusion equation. This ensured that we had a high degree of accuracy and that we couldcompare the numerical solutions of the two.

References[1] V. A. Lair,M. and E. Oxley, Extinction in finite time of solutions to nonlinear absorption-diffusion equations, J. of Math. Anal. and

Appl., 182, 857-866 (1994).[2] A. S. Kalashnikov, The propagation of disturbances in problems of non-linear heat conduction with absorption, USSR Comp. Math.

and Math. Phys., 14, 70-85 (1974).[3] B. Catherine and I. Stakgold, The formation of the dead core in a parabolic reaction-diffusion equation, Trans. of the Amer. Math.

Soc., 286, 275-293 (1984).[4] D. M. Walker, The expected time till absorption when absorption is not certain, J. Appl. Prob., 10, 447–450 (1998).

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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