# Fixed Coeficient A(α) Stable Block Backward Differentiation Formulas for Stiff Ordinary Differential Equations

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## Abstract

**:**

## 1. Introduction

- $Real\left({\lambda}_{i}\left(x\right)\right)<0,$$i=1,2,\dots ,n$
- $\underset{t}{\mathrm{max}}\left|Real\left({\lambda}_{i}\left(x\right)\right)\right|\ll \underset{t}{\mathrm{min}}\left|Real\left({\lambda}_{i}\left(x\right)\right)\right|$

## 2. Derivation of the Method

#### 2.1. Order of the Method

## 3. Stability Analysis

#### 3.1. Zero Stability

**Definition**

**1.**

#### 3.2. Stability Region

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

#### 3.3. Convergence of the Method

- ${\sum}_{\mathrm{j}=0}^{\mathrm{k}}{\mathsf{\alpha}}_{\mathrm{j}}=0$
- ${\sum}_{\mathrm{j}=0}^{\mathrm{k}}{\mathrm{j}\mathsf{\alpha}}_{\mathrm{j}}={\sum}_{\mathrm{j}=0}^{\mathrm{k}}{\mathsf{\beta}}_{\mathrm{j}}$

- ${\sum}_{j=0}^{5}{A}_{j}=\left[\begin{array}{c}{\alpha}_{0,1}\\ {\alpha}_{0,2}\\ {\alpha}_{0,3}\end{array}\right]+\left[\begin{array}{c}{\alpha}_{1,1}\\ {\alpha}_{1,2}\\ {\alpha}_{1,3}\end{array}\right]+\left[\begin{array}{c}{\alpha}_{2,1}\\ {\alpha}_{2,2}\\ {\alpha}_{2,3}\end{array}\right]+\left[\begin{array}{c}{\alpha}_{3,1}\\ {\alpha}_{3,2}\\ {\alpha}_{3,3}\end{array}\right]+\left[\begin{array}{c}{\alpha}_{4,1}\\ {\alpha}_{4,2}\\ {\alpha}_{4,3}\end{array}\right]+\left[\begin{array}{c}{\alpha}_{5,1}\\ {\alpha}_{5,2}\\ {\alpha}_{5,3}\end{array}\right]\phantom{\rule{0ex}{0ex}}=\left[\begin{array}{c}\frac{1}{116}\\ \frac{1}{73}\\ -\frac{15}{236}\end{array}\right]+\left[\begin{array}{c}-\frac{9}{58}\\ -\frac{11}{146}\\ \frac{23}{59}\end{array}\right]+\left[\begin{array}{c}-\frac{31}{29}\\ \frac{6}{73}\\ -1\end{array}\right]+\left[\begin{array}{c}1\\ -\frac{82}{73}\\ \frac{78}{59}\end{array}\right]+\left[\begin{array}{c}\frac{27}{116}\\ 1\\ -\frac{389}{236}\end{array}\right]+\left[\begin{array}{c}-\frac{1}{58}\\ \frac{15}{146}\\ 1\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right].$
- ${\sum}_{j=0}^{5}{jA}_{j}=\left(0\right)\left[\begin{array}{c}{\alpha}_{0,1}\\ {\alpha}_{0,2}\\ {\alpha}_{0,3}\end{array}\right]+\left(1\right)\left[\begin{array}{c}{\alpha}_{1,1}\\ {\alpha}_{1,2}\\ {\alpha}_{1,3}\end{array}\right]+\left(2\right)\left[\begin{array}{c}{\alpha}_{2,1}\\ {\alpha}_{2,2}\\ {\alpha}_{2,3}\end{array}\right]+\left(3\right)\left[\begin{array}{c}{\alpha}_{3,1}\\ {\alpha}_{3,2}\\ {\alpha}_{3,3}\end{array}\right]+\left(4\right)\left[\begin{array}{c}{\alpha}_{4,1}\\ {\alpha}_{4,2}\\ {\alpha}_{4,3}\end{array}\right]+\left(5\right)\left[\begin{array}{c}{\alpha}_{5,1}\\ {\alpha}_{5,2}\\ {\alpha}_{5,3}\end{array}\right]\phantom{\rule{0ex}{0ex}}=\left(0\right)\left[\begin{array}{c}\frac{1}{116}\\ \frac{1}{73}\\ -\frac{15}{236}\end{array}\right]+\left(1\right)\left[\begin{array}{c}-\frac{9}{58}\\ -\frac{11}{146}\\ \frac{23}{59}\end{array}\right]+\left(2\right)\left[\begin{array}{c}-\frac{31}{29}\\ \frac{6}{73}\\ -1\end{array}\right]+\left(3\right)\left[\begin{array}{c}1\\ -\frac{82}{73}\\ \frac{78}{59}\end{array}\right]+\left(4\right)\left[\begin{array}{c}\frac{27}{116}\\ 1\\ -\frac{389}{236}\end{array}\right]+\left(5\right)\left[\begin{array}{c}-\frac{1}{58}\\ \frac{15}{146}\\ 1\end{array}\right]=\left[\begin{array}{c}\frac{45}{29}\\ \frac{90}{73}\\ \frac{45}{59}\end{array}\right]={\displaystyle {\displaystyle \sum}_{j=0}^{5}}{B}_{j}.$

## 4. Implementation

- Step 1.
- Predict: ${y}_{n+1}^{p},{y}_{n+2}^{p},{y}_{n+3}^{p}$ are developed explicitly.
- Step 2.
- Evaluate: ${f}_{n+1}^{}=f\left({x}_{n+1},{y}_{n+1}^{p}\right)$$${f}_{n+2}^{}=f\left({x}_{n+2},{y}_{n+2}^{p}\right)$$$${f}_{n+3}^{}=f\left({x}_{n+3},{y}_{n+3}^{p}\right)$$
- Step 3.
- Correct: ${y}_{n+1},{y}_{n+2},{y}_{n+3}$ by
- Step 4.
- Evaluate: ${f}_{n+1}^{}=f\left({x}_{n+1},{y}_{n+1}^{}\right)$$${f}_{n+2}=f\left({x}_{n+2},{y}_{n+2}\right)$$$${f}_{n+3}=f\left({x}_{n+3},{y}_{n+3}\right)$$

## 5. Numerical Results

- The fifth order method given by [12].
- The fifth order method derived by [13].
- The new $A\left(\alpha \right)$-BBDF.
**Problem****1:**${y}^{\prime}=-20\left(y-{x}^{2}\right)+2x$ with initial value $y\left(0\right)=1/3$ and $x\in \left[0,1\right]$Exact solution: $y\left(x\right)={x}^{2}+\frac{1}{3}{e}^{-20x}$Source: Burden and Faires [21].**Problem****2:**${y}^{\prime}=\frac{y\left(1-y\right)}{2y-1}$ with initial value $y\left(0\right)=5/9$ and $x\in \left[0,5\right]$Exact solution: $y\left(x\right)=\frac{1}{2}+\sqrt{\frac{1}{4}-\frac{5}{36}{e}^{-x}}$Source: Alvarez and Rojo [22].**Problem****3:**$${{y}^{\prime}}_{1}=-21{y}_{1}+19{y}_{2}-20{y}_{3}$$$${{y}^{\prime}}_{2}=19{y}_{1}-21{y}_{2}-20{y}_{3}$$$${{y}^{\prime}}_{3}=40{y}_{1}-40{y}_{2}-40{y}_{3}$$$${y}_{1}\left(0\right)=1,{y}_{2}\left(0\right)=0,{y}_{3}\left(0\right)=-1,x\in \left[0,1\right]$$Exact solution:$${y}_{1}\left(x\right)=0.5\left[{e}^{-2x}+{e}^{\left(-40+40i\right)x}\right]$$$${y}_{2}\left(x\right)=0.5\left[{e}^{-2x}-{e}^{\left(-40+40i\right)x}\right]$$$${y}_{3}\left(x\right)=-{e}^{\left(-40+40i\right)x}$$Source: Lambert [18].

BBDF(5) | : | Fifth order Block Backward Differentiation Formula in [12] |

3SBBDF | : | 3-point Superclass of Block Backward Differentiation Formula in [14] |

$A\left(\alpha \right)$-BBDF | : | New 3-point BBDF |

NS | : | Number of steps |

$h$ | : | Step size |

T | : | Computing time in seconds |

MAXE | : | Maximum error |

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Bui, T.D. Solving stiff differential equations in the simulation of physical systems. Simulation
**1981**, 37, 37–46. [Google Scholar] [CrossRef] - Aiken, J.C. Stiff Review. In Proceedings of the International Conference on Stiff Computation, Park City, UT, USA, 12–14 April 1982. [Google Scholar]
- Chu, M.T.; Hamilton, H. Parallel solution of ODE’s by multi-block methods. SIAM J. Sci. Stat. Comput.
**1987**, 8, 342–353. [Google Scholar] [CrossRef] - Aceto, L.; Trigiante, D. On the A-stable methods in the GBDF class. Nonlinear Anal. Real World Appl.
**2002**, 3, 9–23. [Google Scholar] [CrossRef] - Dahlquist, G. A Special Stability Problem for Linear Multistep Methods. BIT
**1963**, 3, 27–43. [Google Scholar] [CrossRef] - Majid, Z.A.; Suleiman, M.B.; Ismail, F.; Othman, M. 2-Point Implicit Block One-Step Method Half Gauss-Seidel For Solving First Order Ordinary Differential Equations. Matematika
**2003**, 19, 91–100. [Google Scholar] - Ibrahim, Z.B.; Othman, K.I.; Suleiman, M. Implicit r-point block backward differentiation formula for solving first-order stiff ODEs. Appl. Math. Comput.
**2007**, 186, 558–565. [Google Scholar] [CrossRef] - Majid, Z.A.; Suleiman, M.; Omar, Z. 3-Point Implicit Block Method for solving ordinary differential equations. Bull. Malays. Math. Sci. Soc. Second Ser.
**2006**, 29, 29–31. [Google Scholar] - Ismail, F.; Ken, Y.L.; Othman, M. Explicit and implicit 3-point block methods for solving special second order ordinary differential equations directly. Int. J. Math. Anal.
**2009**, 3, 239–254. [Google Scholar] - Ibrahim, Z.B.; Othman, K.I.; Suleiman, M. 2-point block predictor-corrector of backward differentiation formulas for solving second order ordinary differential equations directly. Chiang Mai J. Sci.
**2012**, 39, 502–510. [Google Scholar] - Ibrahim, Z.B.; Othman, K.I.; Suleiman, M. Derivation of diagonally implicit block backward differentiation formulas for solving stiff initial value problems. Math. Probl. Eng.
**2015**, 2015, 179231. [Google Scholar] [CrossRef] - Shampine, L.F.; Watts, H.A. Block implicit one-step methods. Math. Comp.
**1969**, 23, 731–740. [Google Scholar] [CrossRef] - Nasir, N.A.A.; Ibrahim, Z.B.; Othman, K.I.; Suleiman, M.B. Numerical Solution of First Order Ordinary Differential Equations Using Fifth Order Block Backward Differentiation Formulas. Sains Malays.
**2012**, 41, 489–492. [Google Scholar] - Asnor, A.I.; Yatim, S.A.M.; Ibrahim, Z.B. Formulation of Modified Variable Step Block Backward Differentiation Formulae for Solving Stiff Ordinary Differential Equations. Indian J. Sci. Technol.
**2017**, 10, 1–7. [Google Scholar] [CrossRef] - Iskandar Shah, M.Z.; Ibrahim, Z.B. Convergence Properties of pth Order Diagonally Implicit Block Backward Differentiation Formulas. Chiang Mai J. Sci.
**2018**, 45, 601–606. [Google Scholar] - Musa, H.; Suleiman, M.B.; Senu, N. Fully Implicit 3-point Block Extended Backward Differentiation Formulas for Stiff Initial value Problems. Appl. Math. Sci.
**2012**, 6, 4211–4228. [Google Scholar] - Babaginda, B.; Musa, H.; Ibrahim, L.K. A New Numerical Method For Solving Stiff Initial Value Problems. Fluid Mech. Open Access
**2016**, 3, 1–5. [Google Scholar] - Lambert, J.D. Computational Methods in Ordinary Differential Equations, 3rd ed.; John Wiley and Sons: New York, NY, USA, 1973; pp. 154–196. [Google Scholar]
- Fatunla, S.O. Block methods for second order ODEs. Int. J. Comput. Math.
**1991**, 41, 55–63. [Google Scholar] [CrossRef] - Butcher, J.C. Forty-Five Years of A-stability. J. Numer. Anal. Ind. Appl. Math.
**2009**, 4, 1–9. [Google Scholar] - Burden, R.L.; Faires, J.D. Numerical Analysis, 7th ed.; Brooks/Cole: Belmont, CA, USA, 2001. [Google Scholar]
- Alvarez, J.; Rojo, J. An improved class of generalized Runge- Kutta methods for stiff problems. I. The scalar case. Appl. Math. Comput.
**2002**, 130, 537–560. [Google Scholar] [CrossRef]

${\mathit{\alpha}}_{0,\mathit{i}}$ | ${\mathit{\alpha}}_{1,\mathit{i}}$ | ${\mathit{\alpha}}_{2,\mathit{i}}$ | ${\mathit{\alpha}}_{3,\mathit{i}}$ | ${\mathit{\alpha}}_{4,\mathit{i}}$ | ${\mathit{\alpha}}_{5,\mathit{i}}$ | ${\mathit{\beta}}_{\mathit{i}}$ | ||
---|---|---|---|---|---|---|---|---|

$i=1$ | ${y}_{n+1}$ | $1/116$ | $-9/58$ | $-31/29$ | 1 | $27/116$ | $-1/58$ | $24/29$ |

$i=2$ | ${y}_{n+2}$ | $1/73$ | $-11/146$ | $6/73$ | $-82/73$ | 1 | $15/146$ | $48/73$ |

$i=3$ | ${y}_{n+3}$ | $-15/236$ | $23/59$ | $-1$ | $78/59$ | $-389/236$ | 1 | $24/59$ |

$\mathit{h}.$ | Method | NS | MAXE | TIME |
---|---|---|---|---|

${10}^{-2}$ | BBDF(5) | 50 | 1.22077 ($-$02) | 3.38123 × 10^{−5} |

3SBBDF | 34 | 1.25970 ($-$02) | 7.97504 × 10^{−6} | |

$A\left(\alpha \right)$-BBDF | 34 | 9.80872 ($-$03) | 5.74054 × 10^{−6} | |

${10}^{-4}$ | BBDF(5) | 5,000 | 3.61596 ($-$06) | 1.95107 × 10^{−4} |

3SBBDF | 3334 | 2.78963 ($-$06) | 2.43208 × 10^{−5} | |

$A\left(\alpha \right)$-BBDF | 3334 | 2.10240 ($-$06) | 1.81288 × 10^{−5} | |

${10}^{-6}$ | BBDF(5) | 500,000 | 3.65378 ($-$10) | 1.19412 × 10^{−2} |

3SBBDF | 333,334 | 2.84503 ($-$10) | 2.40965 × 10^{−3} | |

$A\left(\alpha \right)$-BBDF | 333,334 | 2.15115 ($-$10) | 1.24343 × 10^{−3} |

$\mathit{h}.$ | Methods | NS | MAXE | TIME |
---|---|---|---|---|

${10}^{-2}$ | BBDF(5) | 250 | 8.22989 ($-$05) | 3.13736 × 10^{−5} |

3SBBDF | 167 | 6.27205 ($-$05) | 1.42650 × 10^{−5} | |

$A\left(\alpha \right)$-BBDF | 167 | 4.80218 ($-$05) | 5.55058 × 10^{−6} | |

${10}^{-4}$ | BBDF(5) | 25,000 | 9.13120 ($-$09) | 6.23244 × 10^{−4} |

3SBBDF | 16,667 | 7.10257 ($-$09) | 2.84914 × 10^{−4} | |

$A\left(\alpha \right)$-BBDF | 16,667 | 5.36673 ($-$09) | 3.00451 × 10^{−5} | |

${10}^{-6}$ | BBDF(5) | 2,500,000 | 4.75320 ($-$11) | 8.99541 × 10^{−2} |

3SBBDF | 1,666,667 | 3.20597 ($-$11) | 1.71905 × 10^{−2} | |

$A\left(\alpha \right)$-BBDF | 1,666,667 | 2.04591 ($-$11) | 3.01268 × 10^{−3} |

$\mathit{h}.$ | Methods | NS | MAXE | TIME |
---|---|---|---|---|

${10}^{-2}$ | BBDF(5) | 50 | 8.31685 ($-$02) | 4.61774 × 10^{−5} |

3SBBDF | 34 | 1.60854 ($-$01) | 2.57089 × 10^{−5} | |

$A\left(\alpha \right)$-BBDF | 34 | 1.46790 ($-$01) | 1.13065 × 10^{−5} | |

${10}^{-4}$ | BBDF(5) | 5000 | 8.63685 ($-$05) | 7.02594 × 10^{−4} |

3SBBDF | 3334 | 6.71328 ($-$05) | 2.17249 × 10^{−4} | |

$A\left(\alpha \right)$-BBDF | 3334 | 5.06905 ($-$05) | 1.56282 × 10^{−4} | |

${10}^{-6}$ | BBDF(5) | 500,000 | 8.64038 ($-$09) | 1.33371 × 10^{−2} |

3SBBDF | 333,334 | 6.72941 ($-$09) | 1.62222 × 10^{−2} | |

$A\left(\alpha \right)$-BBDF | 333,334 | 5.08898 ($-$09) | 1.07846 × 10^{−2} |

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**MDPI and ACS Style**

Ibrahim, Z.B.; Mohd Noor, N.; Othman, K.I.
Fixed Coeficient *A*(*α*) Stable Block Backward
Differentiation Formulas for Stiff Ordinary
Differential Equations. *Symmetry* **2019**, *11*, 846.
https://doi.org/10.3390/sym11070846

**AMA Style**

Ibrahim ZB, Mohd Noor N, Othman KI.
Fixed Coeficient *A*(*α*) Stable Block Backward
Differentiation Formulas for Stiff Ordinary
Differential Equations. *Symmetry*. 2019; 11(7):846.
https://doi.org/10.3390/sym11070846

**Chicago/Turabian Style**

Ibrahim, Zarina Bibi, Nursyazwani Mohd Noor, and Khairil Iskandar Othman.
2019. "Fixed Coeficient *A*(*α*) Stable Block Backward
Differentiation Formulas for Stiff Ordinary
Differential Equations" *Symmetry* 11, no. 7: 846.
https://doi.org/10.3390/sym11070846