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26Математические заметки СВФУ Октябрь—декабрь, 2015. Том 22, № 4
УДК 519.633
FINITE DIFFERENCE METHOD FOR THE INVERSE PROBLEM OF THE SIMULTANEOUS DETERMINATION OF THE RIGHT-HAND SIDE AND THE LOWEST COEFFICIENTS IN PARABOLIC EQUATIONS L.-D. Su
Су Л.-Д. Конечно-разностный метод для одновременного определения в обратной задаче правой части и младших коэффициентов в параболическом уравнении.
Аннотация. Предлагается разностная схема для решения обратной задачи определения двух младших коэффициентов, зависящих от времени только в параболическом уравнении. Зависимость от времени в правой части параболического уравнения с использованием значений дополнительного решения в точках области вычислений. Для решения нелинейной обратной задачи строятся линейные аппроксимации по времени с использованием стандартной конечно-разностной процедуры в пространстве. Представлены результаты численных экспериментов, подтверждающих возможность применения предлагаемых алгоритмов для решения обратной задачи нахождения коэффициентов.
Ключевые слова: обратная задача, конечно-разностный метод, параболическое уравнение, нахождение коэффициентов.
Abstract: We propose a numerical scheme to solve the inverse problem of determining two lower coefficients that depends on time only in the parabolic equation. The time dependence of the right-hand side of a parabolic equation is determined using additional solution values at points of the computational domain. For solving the nonlinear inverse problem, linearized approximations in time are constructed using standard finite difference procedures in space. The results of numerical experiments are presented, confirming the capabilities of the proposed computational algorithms for solving the coefficients inverse problem.
Keywords: inverse problem, finite difference method, parabolic partial differential equation, identification of the coefficients.
In this paper, we consider an inverse problem of finding the two coefficients y(t) and f (t) in the following equation
Ut - uxx + Y(t) • u = f (t) • g(x,t), (1)
where g(x,t) is known analytic function, the solution u(x,t) and the two coefficients Y (t), f (t) are unknown.
Many inverse problems arise throughout engineering and mathematical sciences [1—4]. In a direct problem, it is required to find a solution that satisfies some given
© 2015 Su L.-D.
partial differential equation and some initial and boundary conditions. Different from direct problem, in inverse problems, the master equation, initial conditions and boundary conditions are not fully specified, instead, some additional information is available.
There are many different kinds of inverse problems in physics: coefficient inverse problems (in which the equation is not specified completely as some equation coefficients are unknown), boundary inverse problems (in which boundary conditions are unknown), and evolutionary inverse problems (in which initial conditions are unknown) [5, 6]. In this paper, we concentrate on the inverse problem of the parabolic equations is a coefficient inverse problem.
Many inverse problems are formulated as non-classical problems for PDEs. In other words, most standard numerical methods cannot achieve good accuracy in solving this problems. To solve approximately these problems, emphasis is on the development of stable computational algorithms that take into account peculiarities of inverse problems [7,8]. Several regularization methods have been developed for solving ill-conditioning problems [8-10].
Much attention is paid to the problem of determining the lower coefficient of a parabolic equation of second order. The existence and uniqueness of the solution for such an inverse problem and well-posed of this problem in various functional classes were examined [11,12]. Numerical methods for solving the problem of the identification of the lower coefficient of parabolic equations are also considered in many works [13-15].
In this paper, we consider the problem of determining the two lower coefficient that depends on time only. Approximation in space is performed using standard finite difference. The main features of the nonlinear inverse problem are taken into account via a proper choice of the linearized approximation in time. Linear problems at a particular time level are solved on the basis of a special decomposition into new standard elliptic problems.
The layout of the article is as follows: In section 1, we briefly introduce the formulation of the problem. In section 2, we introduce the method and apply this method on the time-dependent problems. The results of numerical experiments are presented in section 3. Section 4 is dedicated to a brief conclusion. Finally, some references are introduced at the end.
1. The problem formulation
Let Q is a bounded domain with a smooth boundary dQ. The direct problem is formulated as follows
ut - uxx + y(t) • u = f (t) • g(x, t), x € Q, 0 < t < T, (2)
with initial condition
u(x, 0) = uo(x), x£il, (3)
and the Dirichlet boundary condition
u(0, t) = $1 (t), u(1,t)= $2(t), 0 <t < T, (4)
where 7(t), f (t), g(x, t) and $i(t) (i = 1, 2) are known functions, u(x,t) is the solution of the parabolic equation of second order (2), the initial condition uo(x) is known.
In this paper, we consider the coefficients inverse problem, in (2), where the coefficients 7(t) and f (t) are unknown. The additional conditions often formulated as:
J u(x, t)w(x) dx = ^(t), J u(x,t)x(x) dx = ^(t), 0 <t ^ T, (5)
o o
where w(x) and x(x) are weight functions. Specifically, choosing w(x) = ¿(x — x*) and x(x) = ¿(x — x**), (x* ,x** £ O), where ¿(x) is the Dirac ¿-function, from (5), we get
u(x*,t) = <^(t), u(x**,t)= ^(t), 0 <t < T. (6)
The inverse problem is to find u(x, t), 7(t) and f (t) from problems (2)—(4) and additional conditions (5) or (6) is well-posed. The corresponding conditions for existence and uniqueness of the solution are available in the above-mentioned works.
In this paper, we consider only the numerical solution of these inverse problems with one dimension omitting theoretical issues of the convergence of an approximate solution to the exact one.
2. The computational algorithm
The inverse problem (2)-(5) (or (2)-(4), (6)) is nonlinear. The standard approach is based on the simplest approximations in time and involves the iterative solution of the corresponding nonlinear problem for the evaluation of the approximate solution at a new level. In our work, we apply such approximations in time that lead to linear problems for evaluating the solution at the new time level.
2.1. The inverse problem. We consider the inverse problem (2)-(4), (6) where O = [0,1]. To solve the parabolic problem numerically, we introduce the grid in space
iJh = {x\Xl= ih, 2 = 0,1,2,..., M, Mh = I}, for the time we also have
UT = {tn | tn = nr, n = 0,1, 2,..., N, Nt = T}. For all grid nodes, except boundary ones, using the operator D written as
„ u(x + h, t) — 2u(x, t) + u(x — h, t)
Du =--^-^--^-
h2
Finite difference approximations in space are employed. Using the fully implicit scheme for approximation in time and the operator D, the notation un = u(x, tn), tn+1 = tn + t (n = 0,1, 2,..., N — 1), we obtain the following variational problem
--^ + Dun+1 + 7n+1 ■ un = fn+1 ■ gn+1, xGu;h, (7)
the initial condition
u(x, 0) = uq(x), xGUh, (8)
and the boundary conditions
u(0,t) =$i(i), u{l,t) =$2{t), tenJT, (9)
the additional conditions (6) take the form:
un(x*) = un(x**)= n = 0,1, 2,..., N. (10)
We use the following decomposition for the solution un+1 at the new time level un+1(x) = yn+1(x)+ 7n+V+1 (x) + /n+1wn+1(x). (11)
To find yn+1(x), substituted (11) in (7), we employ the equation
yn+1 _
--+ = 0, n = 0,1,..., N - 1, x G c^, (12)
T
with the boundary conditions
y(0,t) = $1(i), y(1,t)= $2(i). The functions vn+1(x) and wn+1(x) are determined from
„,n+1
+ Dvn+1 = -un, n = 0,1, 2,..., N - 1, x G coh, (13)
T
w
n+1
+ Dwn+1 = gn+\ n = 0,1, 2,..., N - 1, x G (14)
/
with the boundary condition v(0, t) = v(1, t) = 0 and w(0, t) = w(1, t) = 0.
To evaluate and /n+1, the addition conditions (10) are used, substituted (11) into (10), we get
Yn+V+1(x*) + /n+1wn+1(x*) = <^+1 - yn+1(x*), 7n+1vn+1(x**) + /n+1 wn+1 (x**) = ^n+1 - yn+1 (x**), (15)
where x*,x** G [0,1] and x* = x**. To solve Yn+1 and /"+1 from (15), we assume vn+1(x*)wn+1(x**) - vn+1(x**)wn+1(x*) = 0, where vn+1 and wn+1 are determined from (13) and (14).
Thus, the computational algorithm for solving the inverse problem (2)-(4), (6) based on the linearized scheme (7)-(10) involves the solution of three standard grid elliptic equations for the auxiliary functions yn+1(x) from equation (12), vn+1(x) form equation (13) and wn+1(x) from equation (14), the further evaluation of Yn+1 and /"+1 from (15), and the final calculation un+1(x) from the relation (11).
2.2. The solutions of yn+1(x), vn+1(x) and wn+1(x). To calculate un+1(x) = yn+1(x) + yn+1vn+1(x) + /n+1wn+1(x), n =0,1,..., N - 1, we should find yn+1(x), vn+1(x) and wn+1(x) from (12)-(14).
To find yn+1(x), vn+1 (x) and wn+1 (x), using the notation yn = y(xi;in), tn+1 = tn+T, n = 0,1, 2,..., N-1, x, = xi_1+h, i = 1, 2,..., M, converting (12)—(14) to the following forms with the boundary conditions:
/i2 T
y^i1 " ( 2 + ^ ) + V?i1 + -< = 0,
T
1, 2.....M - 1,
yn+1 = $1(in+1), yM+1 = $2(i"+1),
-r+Y - ( 2 + - V+1 + + = 0,
T
vn+1 = vn+1 = 0 vo = vM = 0,
w.
n+1 i+1
h2
-(2 + -)<+1+Wri1+^
n+1 'i-1
l2 n+1
W0-+1 = WM+1 =0.
1, 2.....M - 1,
1, 2, . . . , M 1,
Using the following decompositions for the solutions y0+1, v0+1 and w.
(16)
(17)
(18)
,,n+1.
y
n+1
n+1 n+1
■ yn+L1 + A
n+1
n+1
n+1,vn+1 xi vi+1
+ Y
n+1
w
n+1 = an+1 ■ wn+1 + ¿n+i i = M - 1, M - 2, . . . , 0,
w
(19)
i+1
and
n+1
y^ = X), Vm
By combining (16)-(19), we can get:
n+1
w
n+1 M
0.
an+1 =
1
2 + k - an_+11 ! ,,n+1
,n+l = I1
Pl 2 + K - a^1 :
n+1 + ^V+' + ei1
2 + k - «n+11 !
(20)
2 + k - «n+11
i = 1, 2, 3, ...,M.
with the conditions ao = 70 = So = 0, /?o = where n = From (20) we can get all the a, A, Y and so we can get all the solutions of y, v, w.
3. Numerical examples
In this section we present numerical results to test the efficiency of the new scheme for solving the coefficients inverse problems, in the example, we put x G [0,1] with the conditions
u0 = 5exp(-100(x - 0.5)2), g(x) = 4(1 - x)x, u(0, t) = u(1, t) = 0, 0 <t < T.
The coefficients Y(t) and f (t) are taken in the forms
Y(t) = -
1 + exp (Z1 (t - 0.7T))'
f(t)
0.1(T -1)
1 + exp (Z2((T - t) - 0.8T))■
(21)
(22)
0
3
t
The solution at the observation points.
The solution u at T=0.5.
Рис. 1. The solution at the observation point and the solution at T = 0.5
We consider the inverse problem with h = jj, t = jj, M = 200, N = 400. The observation points are x* = 0.5 and x** =0.1. The solution of the direct problem at the observation point are depicted in fig. 1(a), the solution at the final time moment (T = 0.5) is presented in fig. 1(b), with Z1 = 1000, C2 = 250.
We also give the solution of 7(t) (fig. 2(a)) and /(t) (fig. 2(b)) of the inverse problem with Zi = 1000, Z2 = 250. The graphs show the very good accuracy and efficiency of the new approximate scheme.
Рис. 2. The exact and numerical solutions of the inverse problem
The solution 7(t) of the inverse problem with different Zi are present in fig. 3(a). For large Zi (see fig. 3(a)), Y(t) approaches discontinuous functions with a discontinuity point at t = 0.7T.
The solution f (t) of the inverse problem with different Z2 are present in fig. 3(b). For large Z2 (see fig. 3(b)), f(t) also approaches discontinuous functions with a discontinuity point at t = 0.2T.
4. Conclusions
In this paper, we proposed a numerical scheme to solve the inverse problems
Рис. 3. The solutions of the inverse problem with different variables
using the finite difference method. The numerical implementation using the linearized approximations in time and standard finite difference procedures in space, based on a decomposition of the approximate solution, where the transition to a new time level involves the solutions of three standard elliptic problems. Numerical solutions of the model problem demonstrate the convergence of the approximate solution of the inverse problem.
5. Acknowledgments
Professors V. I. Vasil'ev and P. N. Vabishchevich have carefully reviewed this paper. As a result of their careful analysis, this paper has been improved. The author would like to express his thankfulness to them for their helpful constructive comments.
ЛИТЕРАТУРА
1. Colton D., Ewing R., Rundell W. Inverse problems in partial differential equations. Philadelphia, PA: SIAM, 1990.
2. Engl H. W., Rundell W. Inverse problems in diffusion processes. Philadelphia, PA: SIAM, 1995.
3. Conca C., Manasevich R., Uhlmann G., Vogelius M. S. Partial differential equations and inverse problems // Contemp. Math. 2004. V. 362. P. 410.
4. Muller T. G., Timmer J. Parameter identification in partial differential equations // Int. J. Bifurcation Chaos Appl. Sci. Eng. 2004. V. 14, N 6. P. 2053-2060.
5. Samarskii A. A., Vabishchevich P. N. Numerical methods for solving inverse problems of mathematical physics. Utrecht: VSP, 2007.
6. Jiang T. S., Li M., Chen C. S. The Method of particular solutions for solving inverse problems of a nonhomogeneous convection-diffusion equation with variable coefficients // Numerical Heat Transfer. Part A: Applications. , 2012. V. 61. P. 338-352.
7. Hansen P. C. The L-curve and its use in the numerical treatment of inverse problems // Computational inverse problems in electrocardiology (ed. P. Johnston). (Adv. Comput. Bioeng. Ser.) Southampton: WITPress, 2000. P. 119-142.
8. Hansen P. C. Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems // Numer. Algorithms. 1994. V. 6, N 1. P. 1-35.
9. Hansen P. C., O'Leary D. P. The use of the L-curve in the regularization of discrete ill-posed problems // SIAM J. Sci. Comput. 1993. V. 14, N 6. P. 1487-1503.
10. Hao D. N. A mollification method for ill-posed problem // Numer. Math. 1994. V. 68, N 4. P. 469-506.
11. Isakov V. Inverse problems for partial differential equations. Berlin: Springer-Verl., 2006. (Appl. Math. Sci.; V. 127).
12. Prilepko A. I., Orlovsky D. G., Vasin I. A. Methods for solving inverse problems in mathematical physics. Berlin: Marcel Dekker, 2000.
13. Dehghan M. An inverse problem of finding a source parameter in a semilinear parabolic equation // Appl. Math. Modelling. 2001. V. 25, N 9. P. 743-754.
14. Chan T. F., Tai X. C. Level Set And Total Variation Regularization For Elliptic inverse problems with discontinuous coefficients // J. Comput. Phys. 2004. V. 193, N 1. P. 40-66.
15. Vabishchevich P. N., Vasil'ev V. I., Vasil'eva M. V. Computational identification of the right-hand side of a parabolic equation // Comput. Math. Math. Phys. 2015. V. 55, N 9. P. 1015-1021.
Статья поступила 18 сентября 201.5 г. Ling-De Su (Су Линг-Де)
Северо-Восточный федеральный университет имени М. К. Аммосова, Научно-исследовательский институт математики, ул. Кулаковского, 48, Якутск 677891, Республика Саха (Якутия) sulingde@gmail•com
