Thai Journal of Mathematics
Volume 6 (2008) Number 1 : 117–137 www.math.science.cmu.ac.th/thaijournal
Finite Difference Methods for Finding a Control Parameter in Two-Dimensional Parabolic Equation with Neumann Boundary Conditions T. Mookum and M. Khebchareon
Abstract : We study the finite difference methods for finding u = u(x, y, t) and p = p(t) which satisfy ∂u ∂2u ∂2u = + 2 + pu + φ(x, y, t) ∂t ∂x2 ∂y in R × [0, T ], R = [0, 1]2 with Neumann boundary conditions. The finite difference methods are discussed based on the explicit finite difference method and the implicit finite difference method. Keywords : Finite difference methods, Control parameter, Parabolic equation, Neumann boundary conditions, Inverse problem Keywords:
1
Inroduction
In this paper, we study two finite difference methods for finding solution u(x, y, t) and control parameter p(t) of two-dimensional parabolic equation in a unit square, R = [0, 1]2 : ∂2u ∂2u ∂u = + 2 + p(t)u + φ(x, y, t), ∂t ∂x2 ∂y
(x, y, t) ∈ R × (0, T ],
(1.1)
with the initial condition u(x, y, 0) = f (x, y),
(x, y) ∈ R,
(1.2)
and Neumann boundary conditions ∂u(0, y, t) ∂x ∂u(1, y, t) ∂x ∂u(x, 0, t) ∂y ∂u(x, 1, t) ∂y
= g0 (y, t),
0 ≤ y ≤ 1,
0 < t ≤ T,
(1.3)
= g1 (y, t),
0 ≤ y ≤ 1,
0 < t ≤ T,
(1.4)
= q0 (x, t),
0 ≤ x ≤ 1,
0 < t ≤ T,
(1.5)
= q1 (x, t),
0 ≤ x ≤ 1,
0 0, then the range of stability is 2 + lpn lpn ≤s≤ . 8 8 If
(3.3)
8 2 − ≤ pn ≤ 0, then the range of stability is h2 l 0 0, lpn β1 h β2 h 2 + lpn ≤ s[sin2 ( ) + sin2 ( )] ≤ . 4 2 2 4 Because 0 ≤ sin2 (
βi h ) ≤ 1 for i = 1, 2, we obtain the range of stability as follow 2 lpn 2 + lpn ≤s≤ . 8 8
Case 2:
2 8 − ≤ pn ≤ 0, we get the range of stability be 2 h l 00 Thus the implicit method is unconditionally stable.
4
Numerical Results
In this section, two parabolic problems for which exact solution and control parameter are known are now used to test both explicit and implicit methods.
4.1
Problem 1
Consider (1.1)-(1.7) with
φ(x, y, t)
=
g0 (y, t)
=
g1 (y, t)
=
q0 (x, t)
=
q1 (x, t)
=
E(t)
=
f (x, y)
=
5π 2 π − 5t) exp(t) sin (x + 2y), 16 4 π πy exp(t) cos , 4 2 π π exp(t) cos (1 + 2y), 4 4 π πx exp(t) cos , 2 4 π π exp(t) cos (x + 2), 2 4 exp(t) sin(0.2π), π sin (x + 2y), 4 (
(4.1) (4.2) (4.3) (4.4) (4.5) (4.6) (4.7)
for which the exact solution are u(x, y, t) = exp(t) sin
π (x + 2y), 4
(4.8)
and p(t) = 1 + 5t.
127
(4.9)
−4 The maximum error obtained for uN and (x0 , y0 ) = i,j at T = 1.0, with l = 10 (0.4, 0.2), for various values of h using the explicit method and the implicit method are shown in Table 1, when the error is calculated by
εni,j = |u(xi , yj , tn ) − uni,j |.
(4.10)
The numerical results obtained for uN i,j at T = 1.0, computed for s = 1/4, h = 1/50, using the explicit method and the implicit method are illustrated in Figure 4 and Figure 6, respectively. And their error are illustrated in Figure 5 and Figure 7, respectively. The maximum error obtained for p(t) with l = 10−4 , for various values of h using the explicit method and the implicit method are shown in Table 2, when the error is calculated by εn = |p(tn ) − pn |.
(4.11)
The numerical results obtained for p(t) with s = 1/4, h = 1/50, using the explicit method and the implicit method are illustrated in Figure 8. And their error are illustrated in Figure 9.
3
u(x,y,1)
2.5 2 1.5 1 0.5 0 1 0.8 0.6 0.4 0.2 y
0 0
0.2
0.4
0.6
0.8
1
x
π Figure 3: The exact solution of u(x, y, 1) = exp(1) sin (x + 2y). 4
128
3 2.5
u(x,y,1)
2 1.5 1 0.5 0 −0.5 1 0.8
1 0.6
0.8 0.6
0.4
y
0.4
0.2
x
0.2 0
0
1 Figure 4: The approximated solution of u(x, y, 1) for problem 1 with s = 4 1 and h = by explicit method. 50 Table 1: Results for maximum error of u for problem 1 with l = 10−4 and T = 1.0. h
s
1/10 1/20 1/30 1/40 1/50
1/100 1/25 9/100 4/25 1/4
Max Explicit 0.2068844859 0.1057844859 0.0707844859 0.0527844859 0.0417844859
Error Implicit 0.0136730000 0.0099684000 0.0092870000 0.0090488000 0.0089387000
error 0f u(x,y,1)
0.05 0.04 0.03 0.02 0.01 0 1 0.8
1 0.6
0.8 0.6
0.4
0.4
0.2
y
0.2 0
0
x
1 1 by Figure 5: The error of u(x, y, 1) for problem 1 with s = and h = 4 50 explicit method. 129
3 2.5
u(x,y,1)
2 1.5 1 0.5 0 −0.5 1 0.8
1 0.6
0.8 0.6
0.4
0.4
0.2
0.2 0
y
0
x
1 Figure 6: The approximate solution of u(x, y, 1) for problem 1 with s = 4 1 and h = by implicit method. 50 −3
x 10
error of u(x,y,1)
9.2 9 8.8 8.6 8.4 8.2 1 0.8
1 0.6
0.8 0.6
0.4
0.4
0.2
y
0.2 0
0
x
1 1 Figure 7: The error of u(x, y, 1) for problem 1 with s = and h = by 4 50 implicit method. Table 2: Results for maximum error of p for problem 1 with l = 10−4 . h
s
1/10 1/20 1/30 1/40 1/50
1/100 1/25 9/100 4/25 1/4
Max Explicit 0.1754677992 0.0895181394 0.0600383795 0.0451300287 0.0100353196
130
Error Implicit 0.0095762113 0.0027834118 0.0015414624 0.0011079126 0.0009074380
7
6
p(t)
5
4
3
2
1 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
Figure 8: The exact and approximate solution of p(t) for problem 1 with 1 1 s = and h = .(-*-) exact, (-¤-) explicit method, (-◦-) implicit method. 4 50
0.012
0.01
error of p(t)
0.008
0.006
0.004
0.002
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
Figure 9: The error of p(t) for problem 1 with s = explicit method, (-◦-) implicit method.
131
1 1 and h = . (-¤-) 4 50
4.2
Problem 2 If φ(x, y, t) = (
5π 2 π + 5t + 2) exp(t) sin (x + 2y), 16 4
(4.12)
we get the exact solution be u(x, y, t) = exp(t) sin
π (x + 2y), 4
(4.13)
and p(t) = −1 − 5t.
(4.14)
−4 The maximum error obtained for uN and (x0 , y0 ) = i,j at T = 1.0, with l = 10 (0.4, 0.2), for various values of h using the explicit method and the implicit method are shown in Table 3. The numerical results obtained for uN i,j at T = 1.0, computed for s = 1/4, h = 1/50, using the explicit method and the implicit method are illustrated in Figure 10 and Figure 12, respectively. Their error are illustrated in Figure 11 and Figure 13, respectively. The maximum error obtained for p(t) with l = 10−4 , for various values of h using the explicit method and the implicit method are shown in Table 4. The numerical results obtained for p(t) with s = 1/4, h = 1/50, using the explicit method and the implicit method are illustrated in Figure 14. Their error are illustrated in Figure 15.
3 2.5
u(x,y,1)
2 1.5 1 0.5 0 −0.5 1 0.8
1 0.6
0.8 0.6
0.4
0.4
0.2
y
0.2 0
0
x
1 Figure 10: The approximate solution of u(x, y, 1) for Problem 2 with s = 4 1 and h = by explicit method. 50
132
error of u(x,y,1)
0.05 0.04 0.03 0.02 0.01 0 1 0.8
1 0.6
0.8 0.6
0.4
0.4
0.2
0.2 0
y
0
x
1 1 by Figure 11: The error of u(x, y, 1) for Problem 2 with s = and h = 4 50 explicit method.
3
u(x,y,1)
2.5 2 1.5 1 0.5 0 1 0.8
1 0.6
0.8 0.6
0.4
0.4
0.2
y
0.2 0
0
x
1 Figure 12: The approximate solution of u(x, y, 1) for Problem 2 with s = 4 1 by implicit method. and h = 50
133
−3
x 10
error of u(x,y,1)
1 0.8 0.6 0.4 0.2 0 1 0.8
1 0.6
0.8 0.6
0.4
0.4
0.2
0.2 0
y
0
x
1 1 Figure 13: The error of u(x, y, 1) for problem 2 with s = and h = by 4 50 implicit method.
−1
−2
p(t)
−3
−4
−5
−6
−7 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
t
Figure 14: The exact and approximate solution of p(t) for problem 2 with 1 1 s = and h = . (-*-) exact, (-¤-) explicit method, (-◦-) implicit method. 4 50
134
0.8
0.7
error of p(t)
0.6
0.5
0.4
0.3
0.2
0.1
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
Figure 15: The error of p(t) for problem 2 with s = explicit method, (-◦-) implicit method.
1 1 and h = . (-¤-) 4 50
Table 3: Results for maximum error of u for problem 2 with l = 10−4 and T = 1.0. h
s
1/10 1/20 1/30 1/40 1/50
1/100 1/25 9/100 4/25 1/4
Max Explicit 0.1882844859 0.1028844859 0.0711844859 0.0546844859 0.0444844859
Error Implicit 0.0136730000 0.0015181715 0.0010847316 0.0009181715 0.0008245148
Table 4: Results for maximum error of p for problem 2 with l = 10−4 . h
s
1/10 1/20 1/30 1/40 1/50
1/100 1/25 9/100 4/25 1/4
Max Explicit 0.1386540353 0.0735859116 0.0503461184 0.0382842836 0.7704000029
135
Error Implicit 0.0101379878 0.0026283249 0.0012460209 0.0007627972 0.0005392346
5
Conclusion
In this paper, two finite difference methods were applied to a two-dimensional parabolic equation and a control parameter with Neumann boundary conditions. For approximation at the boundary we use second-order central difference. The numerical experiments shown that the approximate solution is reliable if the value of s lies in stability region. In addition, the numerical results obtained from both numerical scheme are closed to the exact solution. In the stability analysis of the explicit method, the stability region depends on the value of p(t). However, numerical results of problem 2, show that the small value of p(t) yields small effect on stability of the numerical scheme. Indeed, the value of p(t) in our test problem is a monotonic function in [0, 1], thus this provide that the numerical result is predictable. For the case of p(t) is non–monotonic function require more investigation.
References [1] Burden, R.L., Faires, J.D., (1993). Numerical Analysis, 5th Ed., PWS Publishing Company. [2] Dehghan, M., (2000). A finite difference method for non-local boundary value problem for two-dimensional heat equation. Appl. Math. Comp. 112, 133-142. [3] Dehghan, M., (2000). Fourth-order techniques for identifying a control parameter in the parabolic equations. Internat. J. Engrg. Sci., 40, 433447. [4] Fatullayev, A., Can, E., (2000). Numerical procedures for determining unknown source parameter in parabolic equations. Mathematics and Computers in Simulation. 54, 159-167. [5] Hoffman, J.D., (1992). Numerical Methods for Engineers and Scientists. McGraw-Hill. [6] Kharab, A., Guenther, R.B., (2002). An Introduction to Numerical Method : a MAT-LAB Approach. Chapman & Hall/CRC. [7] Kincaid, D., Chency, W., (2002). Numerical Analysis: Mathematics of Scientific Computing. 3rd Ed., Wadswarth Group. [8] Lapidus, L., Pinder, G.F., (1982). Numerical Solution of Partial Differential Equations in Science and Engineering. J. Wiley&Sons. [9] Patel, V.A., (1994). Numerical Analysis, Harcourt Brace College Publishers. [10] Warming, R.F., Hyett, B.J., (1974). The modified equation approach to the stability and accuracy analysis of finite difference methods. J. Comput. Phys., 14, 159-179.
136
(Received 30 May 2007) Theeradech Mookum Department of Mathematics, Faculty of Science, Mahidol University THAILAND email
[email protected]
Morrakot Khebchareon Department of Mathematics, Faculty of Science, Chiang Mai University THAILAND email
[email protected]
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