Research Journal of Applied Sciences, Engineering and Technology 4(22): 4630-4635, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: March 16, 2012
Accepted: April 13, 2012
Published: November 15, 2012
A New Algorithm that Developed Finite Difference Method for Solving Laplace
Equation for a Plate with Four Different Constant Temperature
Boundary Conditions
H. Ahmadi and M. Manteghian
Department of Chemical Engineering, Tarbiat Modares University, Iran
Abstract: Solving Laplace equation L2T = 0 using analytical methods is difficult, so numerical methods are
used. One of the numerical methods for solving Laplace equation is finite difference method. We know that
knotting and writing finite difference method for a specific body, eventually will give rise to linear algebraic
equations. In this study, a new algorithm use for develop finite difference method for solving Laplace equation.
In this algorithm, the temperature of the nodes of a specific figure quickly will be evaluated using finite
difference method and the number of equations would be reducing significantly. By this method, a new formula
for solving Laplace equation for a plate with four different constant temperature boundary conditions (Dirichlet
condition) derived.
Keywords: Finite difference, laplace equation, numerical methods
boundary conditions the temperature of nodes could be
drive of this equation:
INTRODUCTION
In 1928 Friedrichs, and Lewy uses numerical analysis
for solving partial deferential equations. During the 1950
and 1960s, finite difference developed for initial value
problems, (Vidar, 1999). Lord Rayleigh (1894, 1896) and
Ritz (1908) using a variation formulation of boundary
value problem. For finite difference methods there are
references for example textbooks of Collatz, (1955)
Forsythe and Wasow (1960) Vidar, (1999) LeVeque,
(1955) and David and David (1977) and etc. Many studies
have been in field finite difference method. In finite
difference in a region R, we choose h and introduce R in
a grid consisting of equidistant horizontal and vertical
straight lines of distance h, their intersections are called
mesh point. Using finite difference since for obtaining
high accuracy one needs many mesh points and a
1000*1000 matrix or larger may cause a storage problem.
Erwin, (2002) For solving this problem, many methods
gave. In this study, we developed the finite difference
method for solving Laplace equation for a plate with four
different constant temperature boundary conditions
(Dirichlet condition). This new algorithm that extract of
finite difference method could be reducing the volume of
calculations:
DEVELOPMENT OF FINITE DIFFERENCE
METHOD FOR SOLVING LAPLACE EQUATION
We know that knotting and writing finite difference
method for a specific body, eventually will give rise to
linear algebraic equations. Thus for the Laplace equation
in steady state with four different constant temperature
T = G8i Ti
where, 8i are the weighting coefficients of the nodes, and
Ti is the boundary condition.
Consider Laplace equation in steady state:
 2T  2T

0
x 2 y 2
for a plate with four constant temperature boundary
conditions. With use of finite difference method and the
formula of T = G8i Ti at the same time for any node, the
following relations for any number of nodes would obtain:
4in 

n
j
j
1

0
(1)
where, 8i is the weighting coefficient of the node i,8j is
the weighting coefficient of neighborhood nodes around
node i. n has the values of 1 to 4. 1 for weighting
coefficient of nodes in the left side. 2 for weighting
coefficient of nodes in the right side, 3 for weighting
coefficient of nodes in the down side and 4 for weighting
coefficient of nodes in the up side, for example 824 is the
weighting coefficient of node 2 in the up side. 1 In the
right side of Eq. (1) is for nodes near the walls in which
8s are directed toward the walls and 0 for all other 8s.
And for a node:
Corresponding Author: M. Manteghian, Department of Chemical Engineering, Tarbiat Modares University, Iran
4630
Res. J. Appl. Sci. Eng. Technol., 4(22): 4630-4635, 2012
1
0 1 1 0
0
0
0  1 
 1   4
   0


4
0
0
1
1
0
0   0
  
 2 
 3    1 0
3
0
0
0  1 0  1 
  
  
0
4  1 0  1 0   0
 4     1 0
 5   0  1 0  1 4
0
0  1  0
  
  


0
1
0
0
0
3
0  1  0
 6 
   0
0 1 1 0
0
3  1  0
  
 7 
0
0
0  1  1  1 3   0
 8   0
Fig. 1: The square plate meshes point and the weighting
coefficients of the nodes
 in  1
(2)
Important advantage of these equations is that the
results contain only 1 and 0 digits. Thus for solving these
equations you need only drive inverse matrix of the
entries column 1 and 3 of the coefficients matrix. Thus,
the volumes of calculation very reduce. Moreover, for
solving these linear algebraic equations, you may use the
equations below that derived by formula (2):
81+82 = 0.5
83+84+85+86 = 1
87+88= 0.5
Moreover, with finding 8s we can using following
relation and calculate temperature of the nodes:
T =  in Ti
(3)
Consider Laplace equation in steady state
 2T  2T

 0 with the following constant temperature
x 2 y 2
boundary conditions.
As Fig. 1 shows, we only need to write the relation
for the colored nodes. We write the following equations
easily using figure or Eq. (1):
Node a:
(a, b, e, T1 or T2)Y481- 83- 84 = 1
(a, b, e, T3 or T4)Y482-85- 86 = 0
Thus with this method the 16 of equations and
unknowns of finite difference, convert to a problem of 5
equations and unknowns in this method, thus volume of
calculations reduce and with finding 8s we can by
following relation calculate temperature of the nodes:
T = G8i Ti
In Table 1a, b, c and d, we give the value of different
8 for a plate with 360 nodes. In Fig. 2 and 3 temperature
calculated by finite difference and this new algorithm
solution for a plate with dimensions 1.3*3.1 with 360
nodes and two different constant temperature boundary
conditions was compared.
Node b:
CONCLUSION
(a, b, c, f, T1)Y483- 81- 83- 87 = 1
(a, b, c, f, T2)Y484- 87- 85- 81 = 0
(a, b, c, f, T4)Y485- 82- 84- 88 = 0
(a, b, c, f, T3)Y486- 86- 88- 82 = 0
In this study, a new algorithm (new formula)
extracted of finite differences method for solution Laplace
equation for a plate with constant temperature boundary
conditions. The important advantages of new algorithm
compared with finite difference are:
Node f:
  b, e, j , g , f , T1 or T2   47  8  7  3  4  0

  b, e, j , g , f , T3 or T4   48  6  7  5  8  0
C
C
AX=B YX= A-1BY
4631
This new algorithm would have a good agreement
with finite difference and the volume of calculations
by this method reduced significantly.
New algorithm considered more expensive and timeconsuming than finite difference. In this method, a
new formula for solving Laplace equation for a plate
Res. J. Appl. Sci. Eng. Technol., 4(22): 4630-4635, 2012
Table 1: The value of different 8 for a plate with 360 nodes
(a) The value of 8 in upward direction
--------------------------------------------------------------------------------------------------------------------------------------------------------------------Node
1
2
3
4
5
6
7
8
9
10
11
12
1
0.4938
0.6849
0.7708
0.8137
0.8357
0.8451
0.8451
0.8357
0.8137
0.7708
0.6849
0.4938
2
0.2901
0.4751
0.5846
0.6484
0.6838
0.6997
0.6997
0.6838
0.6484
0.5846
0.4751
0.2901
3
0.1917
0.3406
0.4442
0.5115
0.5515
0.5702
0.5702
0.5515
0.5115
0.4442
0.3406
0.1917
4
0.1360
0.2516
0.3400
0.4019
0.4407
0.4592
0.4592
0.4407
0.4019
0.3400
0.2516
0.1360
5
0.1005
0.1900
0.2622
0.3154
0.3500
0.3669
0.3669
0.3500
0.3154
0.2622
0.1900
0.1005
6
0.0761
0.1455
0.2034
0.2475
0.2769
0.2915
0.2915
0.2769
0.2475
0.2034
0.1455
0.0761
7
0.0585
0.1125
0.1586
0.1943
0.2186
0.2308
0.2308
0.2186
0.1943
0.1586
0.1125
0.0585
8
0.0454
0.0876
0.1240
0.1526
0.1723
0.1822
0.1822
0.1723
0.1526
0.1240
0.0876
0.0454
9
0.0353
0.0684
0.0971
0.1199
0.1357
0.1437
0.1437
0.1357
0.1199
0.0971
0.0684
0.0353
10
0.0276
0.0535
0.0762
0.0942
0.1068
0.1132
0.1132
0.1068
0.0942
0.0762
0.0535
0.0276
11
0.0216
0.0420
0.0598
0.0740
0.0840
0.0891
0.0891
0.0840
0.0740
0.0598
0.0420
0.0216
12
0.0170
0.0330
0.0470
0.0582
0.0661
0.0701
0.0701
0.0661
0.0582
0.0470
0.0330
0.0170
13
0.0133
0.0259
0.0369
0.0458
0.0519
0.0551
0.0551
0.0519
0.0458
0.0369
0.0259
0.0133
14
0.0105
0.0203
0.0290
0.0360
0.0408
0.0434
0.0434
0.0408
0.0360
0.0290
0.0203
0.0105
15
0.0082
0.0160
0.0228
0.0283
0.0321
0.0341
0.0341
0.0321
0.0283
0.0228
0.0160
0.0082
16
0.0065
0.0126
0.0179
0.0222
0.0252
0.0268
0.0268
0.0252
0.0222
0.0179
0.0126
0.0065
17
0.0051
0.0099
0.0141
0.0175
0.0198
0.0211
0.0211
0.0198
0.0175
0.0141
0.0099
0.0051
18
0.0040
0.0077
0.0111
0.0137
0.0156
0.0165
0.0165
0.0156
0.0137
0.0111
0.0077
0.0040
19
0.0031
0.0061
0.0087
0.0108
0.0122
0.0130
0.0130
0.0122
0.0108
0.0087
0.0061
0.0031
20
0.0025
0.0048
0.0068
0.0085
0.0096
0.0102
0.0102
0.0096
0.0085
0.0068
0.0048
0.0025
21
0.0019
0.0037
0.0053
0.0066
0.0075
0.0080
0.0080
0.0075
0.0066
0.0053
0.0037
0.0019
22
0.0015
0.0029
0.0042
0.0052
0.0059
0.0063
0.0063
0.0059
0.0052
0.0042
0.0029
0.0015
23
0.0012
0.0023
0.0033
0.0040
0.0046
0.0049
0.0049
0.0046
0.0040
0.0033
0.0023
0.0012
24
0.0009
0.0018
0.0025
0.0031
0.0036
0.0038
0.0038
0.0036
0.0031
0.0025
0.0018
0.0009
25
0.0007
0.0014
0.0019
0.0024
0.0027
0.0029
0.0029
0.0027
0.0024
0.0019
0.0014
0.0007
26
0.0005
0.0010
0.0015
0.0018
0.0021
0.0022
0.0022
0.0021
0.0018
0.0015
0.0010
0.0005
27
0.0004
0.0008
0.0011
0.0013
0.0015
0.0016
0.0016
0.0015
0.0013
0.0011
0.0008
0.0004
28
0.0003
0.0005
0.0008
0.0009
0.0011
0.0011
0.0011
0.0011
0.0009
0.0008
0.0005
0.0003
29
0.0002
0.0003
0.0005
0.0006
0.0007
0.0007
0.0007
0.0007
0.0006
0.0005
0.0003
0.0002
30
0.0001
0.0002
0.0002
0.0003
0.0003
0.0004
0.0004
0.0003
0.0003
0.0002
0.0002
0.0001
(b) The value of 8 in right side direction
1
0.0093
0.0189
0.0291
0.0402
0.0529
0.0679
0.0866
0.1111
0.1458
0.1999
0.2960
0.4969
2
0.0183
0.0372
0.0571
0.0789
0.1035
0.1323
0.1673
0.2120
0.2721
0.3578
0.4874
0.6914
3
0.0268
0.0543
0.0833
0.1148
0.1499
0.1903
0.2384
0.2975
0.3727
0.4717
0.6045
0.7812
4
0.0346
0.0700
0.1071
0.1470
0.1910
0.2408
0.2984
0.3668
0.4497
0.5518
0.6776
0.8291
5
0.0415
0.0839
0.1281
0.1752
0.2264
0.2833
0.3476
0.4215
0.5076
0.6082
0.7251
0.8575
6
0.0475
0.0960
0.1462
0.1992
0.2562
0.3184
0.3872
0.4642
0.5508
0.6484
0.7572
0.8756
7
0.0527
0.1062
0.1615
0.2193
0.2808
0.3468
0.4186
0.4971
0.5832
0.6774
0.7794
0.8879
8
0.0570
0.1148
0.1742
0.2359
0.3007
0.3696
0.4433
0.5224
0.6075
0.6986
0.7953
0.8964
9
0.0606
0.1218
0.1845
0.2492
0.3167
0.3876
0.4625
0.5418
0.6257
0.7142
0.8068
0.9026
10
0.0634
0.1275
0.1927
0.2598
0.3293
0.4016
0.4772
0.5564
0.6393
0.7258
0.8152
0.9070
11
0.0657
0.1319
0.1992
0.2680
0.3390
0.4123
0.4884
0.5675
0.6495
0.7342
0.8214
0.9102
12
0.0674
0.1352
0.2040
0.2742
0.3462
0.4203
0.4967
0.5755
0.6568
0.7403
0.8257
0.9125
13
0.0686
0.1376
0.2075
0.2786
0.3513
0.4259
0.5025
0.5811
0.6619
0.7445
0.8287
0.9141
14
0.0694
0.1392
0.2097
0.2815
0.3546
0.4295
0.5061
0.5847
0.6651
0.7472
0.8306
0.9151
15
0.0698
0.1399
0.2108
0.2828
0.3562
0.4312
0.5079
0.5864
0.6667
0.7485
0.8316
0.9155
16
0.0698
0.1399
0.2108
0.2828
0.3562
0.4312
0.5079
0.5864
0.6667
0.7485
0.8316
0.9155
17
0.0694
0.1392
0.2097
0.2815
0.3546
0.4295
0.5061
0.5847
0.6651
0.7472
0.8306
0.9151
18
0.0686
0.1376
0.2075
0.2786
0.3513
0.4259
0.5025
0.5811
0.6619
0.7445
0.8287
0.9141
19
0.0674
0.1352
0.2040
0.2742
0.3462
0.4203
0.4967
0.5755
0.6568
0.7403
0.8257
0.9125
20
0.0657
0.1319
0.1992
0.2680
0.3390
0.4123
0.4884
0.5675
0.6495
0.7342
0.8214
0.9102
21
0.0634
0.1275
0.1927
0.2598
0.3293
0.4016
0.4772
0.5564
0.6393
0.7258
0.8152
0.9070
22
0.0606
0.1218
0.1845
0.2492
0.3167
0.3876
0.4625
0.5418
0.6257
0.7142
0.8068
0.9026
23
0.0570
0.1148
0.1742
0.2359
0.3007
0.3696
0.4433
0.5224
0.6075
0.6986
0.7953
0.8964
24
0.0527
0.1062
0.1615
0.2193
0.2808
0.3468
0.4186
0.4971
0.5832
0.6774
0.7794
0.8879
25
0.0475
0.0960
0.1462
0.1992
0.2562
0.3184
0.3872
0.4642
0.5508
0.6484
0.7572
0.8756
26
0.0415
0.0839
0.1281
0.1752
0.2264
0.2833
0.3476
0.4215
0.5076
0.6082
0.7251
0.8575
27
0.0346
0.0700
0.1071
0.1470
0.1910
0.2408
0.2984
0.3668
0.4497
0.5518
0.6776
0.8291
28
0.0268
0.0543
0.0833
0.1148
0.1499
0.1903
0.2384
0.2975
0.3727
0.4717
0.6045
0.7812
29
0.0183
0.0372
0.0571
0.0789
0.1035
0.1323
0.1673
0.2120
0.2721
0.3578
0.4874
0.6914
30
0.0093
0.0189
0.0291
0.0402
0.0529
0.0679
0.0866
0.1111
0.1458
0.1999
0.2960
0.4969
4632
Res. J. Appl. Sci. Eng. Technol., 4(22): 4630-4635, 2012
Table 1: (Continue)
(c) The value of 8 in downward direction
--------------------------------------------------------------------------------------------------------------------------------------------------------------------Node
1
2
3
4
5
6
7
8
9
10
11
12
1
0.0001
0.0002
0.0002
0.0003
0.0003
0.0004
0.0004
0.0003
0.0003
0.0002
0.0002
0.0001
2
0.0002
0.0003
0.0005
0.0006
0.0007
0.0007
0.0007
0.0007
0.0006
0.0005
0.0003
0.0002
3
0.0003
0.0005
0.0008
0.0009
0.0011
0.0011
0.0011
0.0011
0.0009
0.0008
0.0005
0.0003
4
0.0004
0.0008
0.0011
0.0013
0.0015
0.0016
0.0016
0.0015
0.0013
0.0011
0.0008
0.0004
5
0.0005
0.0010
0.0015
0.0018
0.0021
0.0022
0.0022
0.0021
0.0018
0.0015
0.0010
0.0005
6
0.0007
0.0014
0.0019
0.0024
0.0027
0.0029
0.0029
0.0027
0.0024
0.0019
0.0014
0.0007
7
0.0009
0.0018
0.0025
0.0031
0.0036
0.0038
0.0038
0.0036
0.0031
0.0025
0.0018
0.0009
8
0.0012
0.0023
0.0033
0.0040
0.0046
0.0049
0.0049
0.0046
0.0040
0.0033
0.0023
0.0012
9
0.0015
0.0029
0.0042
0.0052
0.0059
0.0063
0.0063
0.0059
0.0052
0.0042
0.0029
0.0015
10
0.0019
0.0037
0.0053
0.0066
0.0075
0.0080
0.0080
0.0075
0.0066
0.0053
0.0037
0.0019
11
0.0025
0.0048
0.0068
0.0085
0.0096
0.0102
0.0102
0.0096
0.0085
0.0068
0.0048
0.0025
12
0.0031
0.0061
0.0087
0.0108
0.0122
0.0130
0.0130
0.0122
0.0108
0.0087
0.0061
0.0031
13
0.0040
0.0077
0.0111
0.0137
0.0156
0.0165
0.0165
0.0156
0.0137
0.0111
0.0077
0.0040
14
0.0051
0.0099
0.0141
0.0175
0.0198
0.0211
0.0211
0.0198
0.0175
0.0141
0.0099
0.0051
15
0.0065
0.0126
0.0179
0.0222
0.0252
0.0268
0.0268
0.0252
0.0222
0.0179
0.0126
0.0065
16
0.0082
0.0160
0.0228
0.0283
0.0321
0.0341
0.0341
0.0321
0.0283
0.0228
0.0160
00082
17
0.0105
0.0203
0.0290
0.0360
0.0408
0.0434
0.0434
0.0408
0.0360
00290
0.0203
0.0105
18
0.0133
0.0259
0.0369
0.0458
0.0519
0.0551
0.0551
0.0519
0.0458
0.0369
0.0259
0.0133
19
0.0170
0.0330
0.0470
0.0582
0.0661
0.0701
0.0701
0.0661
0.0582
0.0470
0.0330
0.0170
20
0.0216
0.0420
0.0598
0.0740
0.0840
0.0891
0.0891
0.0840
0.0740
0.0598
0.0420
0.0216
21
0.0276
0.0535
0.0762
0.0942
0.1068
0.1132
0.1132
0.1068
0.0942
0.0762
0.0535
0.0276
22
0.0353
0.0684
0.0971
0.1199
0.1357
0.1437
0.1437
0.1357
0.1199
0.0971
0.0684
0.0353
23
0.0454
0.0876
0.1240
0.1526
0.1723
0.1822
0.1822
0.1723
0.1526
0.1240
0.0876
0.0454
24
0.0585
0.1125
0.1586
0.1943
0.2186
0.2308
0.2308
0.2186
0.1943
0.1586
0.1125
0.0585
25
0.0761
0.1455
0.2034
0.2475
0.2769
0.2915
0.2915
0.2769
0.2475
0.2034
0.1455
0.0761
26
0.1005
0.1900
0.2622
0.3154
0.3500
0.3669
0.3669
0.3500
0.3154
0.2622
0.1900
0.1005
27
0.1360
0.2516
0.3400
0.4019
0.4407
0.4592
0.4592
0.4407
0.4019
0.3400
0.2516
0.1360
28
0.1917
0.3406
0.4442
0.5115
0.5515
0.5702
0.5702
0.5515
0.5115
0.4442
0.3406
0.1917
29
0.2901
0.4751
0.5846
0.6484
0.6838
0.6997
0.6997
0.6838
0.6484
0.5846
0.4751
0.2901
30
0.4938
0.6849
0.7708
0.8137
0.8357
0.8451
0.8451
0.8357
0.8137
0.7708
0.6849
0.4938
(d) The value of 8 in left side direction
1
0.4969
0.2960
0.1999
0.1458
0.1111
0.0866
0.0679
0.0529
0.0402
0.0291
0.0189
0.0093
2
0.6914
0.4874
0.3578
0.2721
0.2120
0.1673
0.1323
0.1035
0.0789
0.0571
0.0372
0.0183
3
0.7812
0.6045
0.4717
0.3727
0.2975
0.2384
0.1903
0.1499
0.1148
0.0833
0.0543
0.0268
4
0.8291
0.6776
0.5518
0.4497
0.3668
0.2984
0.2408
0.1910
0.1470
0.1071
0.0700
0.0346
5
0.8575
0.7251
0.6082
0.5076
0.4215
0.3476
0.2833
0.2264
0.1752
0.1281
0.0839
0.0415
6
0.8756
0.7572
0.6484
0.5508
0.4642
0.3872
0.3184
0.2562
0.1992
0.1462
0.0960
0.0475
7
0.8879
0.7794
0.6774
0.5832
0.4971
0.4186
0.3468
0.2808
0.2193
0.1615
0.1062
0.0527
8
0.8964
0.7953
0.6986
0.6075
0.5224
0.4433
0.3696
0.3007
0.2359
0.1742
0.1148
0.0570
9
0.9026
0.8068
0.7142
0.6257
0.5418
0.4625
0.3876
0.3167
0.2492
0.1845
0.1218
0.0606
10
0.9070
0.8152
0.7258
0.6393
0.5564
0.4772
0.4016
0.3293
0.2598
0.1927
0.1275
0.0634
11
0.9102
0.8214
0.7342
0.6495
0.5675
0.4884
0.4123
0.3390
0.2680
0.1992
0.1319
0.0657
12
0.9125
0.8257
0.7403
0.6568
0.5755
0.4967
0.4203
0.3462
0.2742
0.2040
0.1352
0.0674
13
0.9141
0.8287
0.7445
0.6619
0.5811
0.5025
0.4259
0.3513
0.2786
0.2075
0.1376
0.0686
14
0.9151
0.8306
0.7472
0.6651
0.5847
0.5061
0.4295
0.3546
0.2815
0.2097
0.1392
0.0694
15
0.9155
0.8316
0.7485
0.6667
0.5864
0.5079
0.4312
0.3562
0.2828
0.2108
0.1399
0.0698
16
0.9155
0.8316
0.7485
0.6667
0.5864
0.5079
0.4312
0.3562
0.2828
0.2108
0.1399
0.0698
17
0.9151
0.8306
0.7472
0.6651
0.5847
0.5061
0.4295
0.3546
0.2815
0.2097
0.1392
0.0694
18
0.9141
0.8287
0.7445
0.6619
0.5811
0.5025
0.4259
0.3513
0.2786
0.2075
0.1376
0.0686
19
0.9125
0.8257
0.7403
0.6568
0.5755
0.4967
0.4203
0.3462
0.2742
0.2040
0.1352
0.0674
20
0.9102
0.8214
0.7342
0.6495
0.5675
0.4884
0.4123
0.3390
0.2680
0.1992
0.1319
0.0657
21
0.9070
0.8152
0.7258
0.6393
0.5564
0.4772
0.4016
0.3293
0.2598
0.1927
0.1275
0.0634
22
0.9026
0.8068
0.7142
0.6257
0.5418
0.4625
0.3876
0.3167
0.2492
0.1845
0.1218
0.0606
23
0.8964
0.7953
0.6986
0.6075
0.5224
0.4433
0.3696
0.3007
0.2359
0.1742
0.1148
0.0570
24
0.8879
0.7794
0.6774
0.5832
0.4971
0.4186
0.3468
0.2808
0.2193
0.1615
0.1062
0.0527
25
0.8756
0.7572
0.6484
0.5508
0.4642
0.3872
0.3184
0.2562
0.1992
0.1462
0.0960
0.0475
26
0.8575
0.7251
0.6082
0.5076
0.4215
0.3476
0.2833
0.2264
0.1752
0.1281
0.0839
0.0415
27
0.8291
0.6776
0.5518
0.4497
0.3668
0.2984
0.2408
0.1910
0.1470
0.1071
0.0700
0.0346
28
0.7812
0.6045
0.4717
0.3727
0.2975
0.2384
0.1903
0.1499
0.1148
0.0833
0.0543
0.0268
29
0.6914
0.4874
0.3578
0.2721
0.2120
0.1673
0.1323
0.1035
0.0789
0.0571
0.0372
0.0183
30
0.4969
0.2960
0.1999
0.1458
0.1111
0.0866
0.0679
0.0529
0.0402
0.0291
0.0189
0.0093
4633
Res. J. Appl. Sci. Eng. Technol., 4(22): 4630-4635, 2012
(a)
(b)
Fig. 2: Comparison temperature calculated by finite difference with new algorithm solution for a plate (1.3*3.1) with 360 nodes and
T1 = 48.25(ºC), T2 = 0(ºC), T3 = 69.4(ºC), T4 = 142.5(ºC). (a) Finite difference, (b) new algorithm
(a)
(b)
Fig. 3: Comparison temperature calculated by finite difference with new algorithm solution for a plate (1.3*3.1) with 360 nodes and
T1 = 15(ºC), T2 = 36(ºC), T3 = 100(ºC), T4 = 4(ºC). (a) Finite difference, (b) new algorithm
C
C
with four different constant temperature boundary
conditions (Dirichlet condition) derived. Using of
this formula rather than finite difference, the volume
of calculations reduce.
The relation of this method for finding the value of
8s are independent of boundary condition and
temperature of the walls, but in finite difference for
finding temperature of nodes boundary condition
used directly in the relation.
With deriving different 8s for a plate with n nodes
scheme and four different constant temperature
boundary condition, they could be using for
calculated temperature of nodes for any plate with n
nodes and four different constant temperature
boundary condition, thus with one step we could
derive temperature of that nodes by relation T =
G8iTi. However, in finite difference for any four
different temperature of boundary condition the stage
must repeat.
REFERENCES
Collatz, L.N., 1955. Treatment of Differential Equations.
Springer, Berlin, pp: 15+526.
David, R.C. and G.L. David, 1977. Heat Transfer
Calculations using Finite Difference Equations.
Applied Science Publishers, London, pp: 283.
Erwin, K., 2002. Advanced Engineering Mathematics,
Ohio State University of Columbus Ohio. 9th Edn.,
John Wiley and Sons, United States of America,
pp: 1093.
4634
Res. J. Appl. Sci. Eng. Technol., 4(22): 4630-4635, 2012
Forsythe, G.E. and W.R. Wasow, 1960. Finite Difference
Methods for Partial Differential Equations. Applied
Mathematics Series, John Wiley and Sons Inc., New
York, London, pp: 1960 x+444.
LeVeque, R.J., 1955. Finite Difference Method for
Ordinary and Partial Differential Equations. Society
for Industrial and Applied Mathematics, 6 Sep.,
University of Washington Seattle, Washington,
pp: 341.
Vidar, T., 1999. From finite differences to finite elements
a short history of numerical analysis of partial
differential equations. J. Comput. Appl. Math.,
128(1-2): 1-54.
4635