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Again the solution obtained by proposed method i: -ompared with the exact solution. Results are given ir ables 5 and 6, and plots for this example are shown ir igure 3.

Figure 5 Again the solution obtained by proposed method i: -ompared with the exact solution. Results are given ir ables 5 and 6, and plots for this example are shown ir igure 3.

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The residual R(t; r,C1, C2,---, Cn) is then orthogonal- ized with the functions ¢; for j = 1, 2,..., 2. This gives
The residual R(t; r,C1, C2,---, Cn) is then orthogonal- ized with the functions ¢; for j = 1, 2,..., 2. This gives
Eqs. (28) and (29) are now written as
Eqs. (28) and (29) are now written as
Table 2. Fuzzy solution of Example 1 at tf = 0.01.
Table 2. Fuzzy solution of Example 1 at tf = 0.01.
Table 1. Fuzzy solution of Example | at tf = 0.1.
Table 1. Fuzzy solution of Example | at tf = 0.1.
Figure 2. Fuzzy solution of Example 2 using the proposed method.
Figure 2. Fuzzy solution of Example 2 using the proposed method.
Table 4. Fuzzy solution of Example 2 at t = 0.01.
Table 4. Fuzzy solution of Example 2 at t = 0.01.
Table 3. Fuzzy solution of Example 2 at tf = 0.1.
Table 3. Fuzzy solution of Example 2 at tf = 0.1.
Table 5. Fuzzy solution of Example 2 at tf = 0.1.
Table 5. Fuzzy solution of Example 2 at tf = 0.1.
Figure 3. Fuzzy solution of Example 3 for Case 2 using the proposed method. Hence the solutions are
Figure 3. Fuzzy solution of Example 3 for Case 2 using the proposed method. Hence the solutions are
Table 6. Fuzzy solution of Example 2 at t = 0.01. The exact fuzzy solution for this problem may be obtained as follows:
Table 6. Fuzzy solution of Example 2 at t = 0.01. The exact fuzzy solution for this problem may be obtained as follows:
Here also above results are compared with the exact solution and are incorporated in tables 7 and 8. Corre- sponding plots for this example are shown in figure 4. Example 5 Let us now consider the following FDE (Case 2):
Here also above results are compared with the exact solution and are incorporated in tables 7 and 8. Corre- sponding plots for this example are shown in figure 4. Example 5 Let us now consider the following FDE (Case 2):
Figure 4. Fuzzy solution of Example 4 for Case 3 using the proposed method. Table 8. Fuzzy solution of Example 4 at f = 0.01.
Figure 4. Fuzzy solution of Example 4 for Case 3 using the proposed method. Table 8. Fuzzy solution of Example 4 at f = 0.01.
applying the procedure as discussed in section 3, one may have Here also above results are compared with the special case r= 1 [26] and are incorporated in table 9. Corre- sponding plots for this example are shown in figure 5.
applying the procedure as discussed in section 3, one may have Here also above results are compared with the special case r= 1 [26] and are incorporated in table 9. Corre- sponding plots for this example are shown in figure 5.
Table 7. Fuzzy solution of Example 4 at t = 0.1.
Table 7. Fuzzy solution of Example 4 at t = 0.1.
Figure 5. Fuzzy solution of Example 3 for Case 2 using the proposed method. Table 9. Fuzzy solution of Example 4.
Figure 5. Fuzzy solution of Example 3 for Case 2 using the proposed method. Table 9. Fuzzy solution of Example 4.
Tere the considered parameters are defined as below.
Tere the considered parameters are defined as below.
figure 6 used to enhance the heat transfer from a surface wall maintained at 300°C. The governing differential equation is
figure 6 used to enhance the heat transfer from a surface wall maintained at 300°C. The governing differential equation is
Table 10. Comparison of various lower bound solutions for uncertain temperature in a fin for r = 0.6. Table 11. Comparison of various upper bound solutions for uncertain temperature in a fin for r = 0.6.
Table 10. Comparison of various lower bound solutions for uncertain temperature in a fin for r = 0.6. Table 11. Comparison of various upper bound solutions for uncertain temperature in a fin for r = 0.6.
Table 12. Comparison of the solutions for uncertain temperature in a fin for r = 1.
Table 12. Comparison of the solutions for uncertain temperature in a fin for r = 1.
incorporates the solution bounds for x = 0.2 with different values of r. the main contribution is the new Galerkin type method to handle nth order FDEs. As such some known FDEs are solved as test problems to have the confidence on the proposed method. The solutions by the proposed method in
incorporates the solution bounds for x = 0.2 with different values of r. the main contribution is the new Galerkin type method to handle nth order FDEs. As such some known FDEs are solved as test problems to have the confidence on the proposed method. The solutions by the proposed method in
Table 13. Comparison of various bound solutions for uncertain temperature in a fin for x = 0.2. all the test problems are found to be very close to the exact solutions. Finally the proposed method has been applied to an application problem too. The proposed method may be found to be a straightforward and alternate way to handle nth order FBVPs.
Table 13. Comparison of various bound solutions for uncertain temperature in a fin for x = 0.2. all the test problems are found to be very close to the exact solutions. Finally the proposed method has been applied to an application problem too. The proposed method may be found to be a straightforward and alternate way to handle nth order FBVPs.
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