Direct Integration of Boundary Value Problems Using the Block Method via the Shooting Technique Combined with Steffensen’s Strategy
Abstract
:1. Introduction
- ,
- for some constant, ,
- For the boundary conditions of Equation (2), assume:
2. Methodology
3. Analysis of the Block Method
3.1. Order and Error Constant
3.2. Stability
- all roots, , of satisfy ,
- for those roots with , the multiplicity must not exceed two.
3.3. Consistency and Convergence of the Method
3.4. Stability Analysis
4. Implementation
- Set , and compute the numerical solution using the 2PDD6 formulae.
- At , verify if nearly satisfies or not within the specified set of tolerance, TOL.for Type 1 and Type 2 are represented as and , respectively.
- If the prescribed stopping condition, TOL, is satisfied, then the required numerical solution is achieved. Otherwise, set the new guessing value, , for using Steffensen’s method. The entire process is repeated.
Algorithm 1. The 2PDD6 method. | |
Step 1: | Set TOL, step size, h, and initial estimate, |
Step 2: | Calculate for . |
Step 3: | For , calculate the starting values using the one step method. |
Step 4: | For , and to 2, do |
, and compute , and using the predictor and corrector formulas | |
with PE(CE) modes where until convergence. | |
The calculation of the corrector formula as in Equations (13) to (14) involved in the convergence test. | |
Step 5: | For , set |
, , , . | |
Step 6: | If , then repeat Step 4. Else, go to Step 7. |
Step 7: | At , verify the stopping condition, . If satisfied, then go to Step 9. |
Else, continue Step 8. | |
Step 8: | Correct a new set of guessing values, , for using the formula in Equation (25). |
Repeat Step 2. | |
Step 9: | Compute the results. Complete. |
5. Results and Discussion
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
MAXER: | Maximum error |
AVERR: | Average error |
h: | Step size |
TStep: | Total steps taken at the last iteration |
TFC: | Total function call at the last iteration |
TG: | Total number of guesses |
Time: | Execution time in seconds |
2PDD6: | Direct two point diagonal block method of order six proposed in this study |
2PDAM6: | Direct two step Adams–Moulton block method of order six as in Phang et al. [6] |
DAM6: | Direct Adams–Moulton method of order six as in Majid et al. [13] |
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Method | h | MAXER | AVERR | TStep | TFC | TG | Time |
---|---|---|---|---|---|---|---|
DAM6 | 0.10 | 1.0564(−4) | 5.5715(−5) | 20 | 65 | 2 | 0.459 |
0.05 | 4.5582(−6) | 2.1032(−6) | 40 | 73 | 3 | 0.485 | |
0.01 | 3.9852(−9) | 1.4829(−9) | 200 | 228 | 2 | 0.544 | |
0.001 | 2.5269(−13) | 1.8016(−13) | 2000 | 2028 | 2 | 0.647 | |
2PDAM6 | 0.10 | 9.1588(−5) | 2.7699(−5) | 12 | 96 | 3 | 0.138 |
0.05 | 1.5746(−5) | 1.0187(−5) | 22 | 116 | 3 | 0.147 | |
0.01 | 6.6570(−9) | 3.2354(−9) | 102 | 424 | 2 | 0.182 | |
0.001 | 4.3965(−14) | 2.0196(−14) | 1002 | 4024 | 2 | 0.478 | |
2PDD6 | 0.10 | 2.3596(−4) | 1.4009(−4) | 12 | 68 | 2 | 0.126 |
0.05 | 6.1990(−6) | 3.9173(−6) | 22 | 74 | 2 | 0.136 | |
0.01 | 3.7837(−9) | 1.2019(−9) | 102 | 228 | 2 | 0.172 | |
0.001 | 2.1760(−13) | 1.4017(−13) | 1002 | 2028 | 2 | 0.219 |
Method | h | MAXER | AVERR | TStep | TFC | TG | Time |
---|---|---|---|---|---|---|---|
DAM6 | 0.10 | 5.5585(−6) | 3.8778(−6) | 10 | 44 | 2 | 0.396 |
0.05 | 6.8134(−8) | 4.0499(−8) | 20 | 48 | 2 | 0.410 | |
0.01 | 4.7002(−12) | 3.0072(−12) | 100 | 128 | 2 | 0.442 | |
0.001 | 1.9984(−15) | 6.3048(−16) | 1000 | 1028 | 2 | 0.541 | |
2PDAM6 | 0.10 | 2.3984(−6) | 1.4746(−6) | 7 | 56 | 2 | 0.146 |
0.05 | 2.1534(−7) | 1.0869(−7) | 12 | 64 | 2 | 0.182 | |
0.01 | 1.1817(−11) | 6.3774(−12) | 52 | 224 | 2 | 0.203 | |
0.001 | 1.8874(−15) | 6.7210(−16) | 502 | 2014 | 2 | 0.497 | |
2PDD6 | 0.10 | 1.7657(−6) | 1.2702(−6) | 7 | 44 | 2 | 0.120 |
0.05 | 1.0605(−8) | 4.7569(−9) | 12 | 48 | 2 | 0.141 | |
0.01 | 3.4963(−13) | 8.8978(−14) | 52 | 128 | 2 | 0.156 | |
0.001 | 2.3315(−15) | 9.3690(−16) | 502 | 1028 | 2 | 0.367 |
Method | h | MAXER | AVERR | TStep | TFC | TG | Time |
---|---|---|---|---|---|---|---|
DAM6 | 0.10 | 4.2518(−6) | 2.2424(−6) | 10 | 39 | 1 | 0.387 |
0.05 | 2.0221(−7) | 1.0901(−7) | 20 | 48 | 1 | 0.394 | |
0.01 | 8.3222(−11) | 4.8364(−11) | 100 | 128 | 1 | 0.403 | |
0.001 | 8.8818(−16) | 4.3506(−16) | 1000 | 1028 | 1 | 0.449 | |
2PDAM6 | 0.10 | 3.3869(−6) | 1.9785(−6) | 7 | 48 | 1 | 0.125 |
0.05 | 1.7258(−7) | 9.8088(−8) | 12 | 64 | 1 | 0.139 | |
0.01 | 7.8293(−11) | 4.6356(−11) | 52 | 224 | 1 | 0.143 | |
0.001 | 1.9984(−15) | 1.0042(−15) | 502 | 2024 | 1 | 0.272 | |
2PDD6 | 0.10 | 3.0436(−6) | 1.9095(−6) | 7 | 40 | 1 | 0.111 |
0.05 | 1.4687(−7) | 8.9600(−8) | 12 | 48 | 1 | 0.125 | |
0.01 | 7.5328(−11) | 4.5165(−11) | 52 | 128 | 1 | 0.135 | |
0.001 | 1.9984(−15) | 1.0142(−15) | 502 | 1028 | 1 | 0.235 |
Method | h | MAXER | AVERR | TStep | TFC | TG | Time |
---|---|---|---|---|---|---|---|
DAM6 | 0.10 | 2.4032(−3) | 1.2027(−3) | 10 | 46 | 3 | 0.421 |
0.05 | 2.0730(−5) | 1.2485(−5) | 20 | 64 | 2 | 0.439 | |
0.01 | 6.5416(−9) | 1.8124(−9) | 100 | 128 | 1 | 0.419 | |
0.001 | 7.1180(−14) | 2.2245(−14) | 1000 | 1028 | 1 | 0.442 | |
2PDAM6 | 0.10 | 3.5119(−3) | 1.7495(−3) | 7 | 56 | 2 | 0.143 |
0.05 | 6.4821(−5) | 3.2109(−5) | 12 | 92 | 2 | 0.185 | |
0.01 | 6.5757(−9) | 1.9641(−9) | 52 | 224 | 1 | 0.215 | |
0.001 | 7.3618(−14) | 2.3120(−14) | 502 | 2024 | 1 | 0.395 | |
2PDD6 | 0.10 | 5.1071(−4) | 2.7006(−4) | 7 | 46 | 2 | 0.137 |
0.05 | 2.7670(−5) | 1.4512(−5) | 12 | 64 | 2 | 0.162 | |
0.01 | 6.4668(−9) | 2.0067(−9) | 52 | 128 | 1 | 0.172 | |
0.001 | 7.2182(−14) | 2.2759(−14) | 502 | 1028 | 1 | 0.364 |
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Mohd Nasir, N.; Abdul Majid, Z.; Ismail, F.; Bachok, N. Direct Integration of Boundary Value Problems Using the Block Method via the Shooting Technique Combined with Steffensen’s Strategy. Mathematics 2019, 7, 1075. https://doi.org/10.3390/math7111075
Mohd Nasir N, Abdul Majid Z, Ismail F, Bachok N. Direct Integration of Boundary Value Problems Using the Block Method via the Shooting Technique Combined with Steffensen’s Strategy. Mathematics. 2019; 7(11):1075. https://doi.org/10.3390/math7111075
Chicago/Turabian StyleMohd Nasir, Nadirah, Zanariah Abdul Majid, Fudziah Ismail, and Norfifah Bachok. 2019. "Direct Integration of Boundary Value Problems Using the Block Method via the Shooting Technique Combined with Steffensen’s Strategy" Mathematics 7, no. 11: 1075. https://doi.org/10.3390/math7111075
APA StyleMohd Nasir, N., Abdul Majid, Z., Ismail, F., & Bachok, N. (2019). Direct Integration of Boundary Value Problems Using the Block Method via the Shooting Technique Combined with Steffensen’s Strategy. Mathematics, 7(11), 1075. https://doi.org/10.3390/math7111075