# Variable Step Exponentially Fitted Explicit Sixth-Order Hybrid Method with Four Stages for Spring-Mass and Other Oscillatory Problems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**A**= [a

_{ij}],

**b**

^{T}= [b

_{1}b

_{2}… b

_{s}] and

**c**

^{T}= [c

_{1}c

_{2}… c

_{s}]. Order conditions for this class of methods are as listed in [13] while the leading term of the local truncation error for a pth-order hybrid method is defined to be

_{i}∈ T

_{2}and ρ(t

_{i}) = p + 2. The error constant is the quantity

## 2. Stability Analysis

^{T}. Applying the hybrid methods defined in Equation (1) with constant coefficients to the differential Equation (2) yields the following equation

**Definition**

**1.**

_{p}) is called the interval of periodicity if$P({H}^{2})=1$and$\left|S({H}^{2})\right|<2$for all H ∈ (0, H

_{p}).

**Definition**

**2.**

_{a}) is called the interval of absolute stability if$\left|P({H}^{2})\right|<1$and$\left|S({H}^{2})\right|<1+P({H}^{2})$for all H ∈ (0, H

_{a}).

**Definition**

**3.**

## 3. Derivation of the New Method

#### 3.1. Exponentially Fitted Sixth-Order Method

^{−3}[14]. Using these coefficients, the characteristic polynomial (Equation (4)) is a Schur polynomial if H < 4.42. The Schur polynomial is symmetric and a basis of all symmetric polynomials.

_{i}[y(t)] as follows:

#### 3.2. Exponentially Fitted Fourth-Order Method

**c**and

**A**values as the sixth-order method with constant coefficients described in Section 3.1. Using the order conditions for a fourth-order explicit hybrid method as listed in [13], we obtain

## 4. Results

**Exact solution:**${y}_{1}(t)=\mathrm{sin}(t)-\mathrm{sin}(5t)+\mathrm{cos}(2t),{y}_{2}(t)=\mathrm{sin}(t)+\mathrm{sin}(5t)+\mathrm{sin}(2t)$

_{1}= 0.200179477536, A

_{3}= 2.46946143 ⋅ 10

^{−4}, A

_{5}= 3.04014 ⋅ 10

^{−7}, A

_{7}= 3.74 ⋅ 10

^{−10}, A

_{9}= 0.000000000000.

**Exact solution:**${y}_{1}(t)=\mathrm{cos}({t}^{2}),{y}_{2}(t)=\mathrm{sin}({t}^{2})$

**,**m = 80 and ρ = 0.001 for 0 ≤ t ≤ 100. For all methods, the selected w was $\sqrt{9.633357907}$.

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Franco, J.M.; Randez, L. Eight-order explicit two-step hybrid methods with symmetric nodes and weights for solving orbital and oscillatory IVPs. Int. J. Mod. Phys. C
**2018**, 29, 18. [Google Scholar] [CrossRef] - Senu, N.; Suleiman, M.; Ismail, F.; Othman, M. A zero dissipative Runge-Kutta-Nystrom method with minimal phase-lag. Math. Probl. Eng.
**2010**, 591341. [Google Scholar] [CrossRef] - Chawla, M.M. A new class of explicit two-step fourth order methods for y″ = f(t, y) with extended intervals of periodicity. J. Comput. Appl. Math.
**1986**, 14, 467–470. [Google Scholar] [CrossRef] [Green Version] - Stavroyiannis, S.; Simos, T.E. Optimization as a function of the phase-lag order of nonlinear explicit two-step P-stable method for linear periodic IVPs. Appl. Numer. Math.
**2009**, 59, 2467–2474. [Google Scholar] [CrossRef] - Franco, J.M.; Palacios, M. High-order P-stable multistep methods. J. Comput. Appl. Math.
**1990**, 30, 1–10. [Google Scholar] [CrossRef] [Green Version] - Ahmad, N.A.; Senu, N.; Ismail, F. Phase-fitted and amplification-fitted higher order two-derivative Runge-Kutta method for the numerical solution of orbital and related periodical IVPs. Math. Probl. Eng.
**2017**, 1871278. [Google Scholar] [CrossRef] - Yang, H.; Wu, X. Trigonometrically-fitted ARKN methods for perturbed oscillators. Appl. Numer. Math.
**2008**, 58, 1375–1395. [Google Scholar] [CrossRef] - Franco, J.M. Exponentially fitted explicit Runge-Kutta-Nystrom methods. J. Comput. Appl. Math.
**2004**, 167, 1–19. [Google Scholar] [CrossRef] [Green Version] - Van de Vyver, H. A Runge-Kutta-Nystrom pair for the numerical integration of perturbed oscillators. Comput. Phys. Commun.
**2005**, 167, 129–142. [Google Scholar] [CrossRef] - Raptis, A.D.; Cash, J.R. A variable step method for the numerical integration of the one-dimensional Schrodinger equation. Comput. Phys. Commun.
**1985**, 36, 113–119. [Google Scholar] [CrossRef] - Simos, T.E.; Williams, P.S. New insights in the development of Numerov-type methods with minimal phase-lag for the numerical solution of the Schrodinger equation. Comput. Chem.
**2001**, 25, 77–82. [Google Scholar] [CrossRef] - Franco, J.M. A class of explicit two-step hybrid methods for second-order IVPs. J. Comput. Appl. Math.
**2006**, 187, 41–57. [Google Scholar] [CrossRef] [Green Version] - Coleman, J.P. Order conditions for a class of two-step methods for y” = f(x,y). IMA J. Numer. Anal.
**2003**, 23, 197–220. [Google Scholar] [CrossRef] - Samat, F.; Razak, N. Derivation of explicit 6(4) pair of hybrid methods for special second order ordinary differential equations. In Proceedings of the 25th National Symposium on Mathematical Sciences, AIP Conference Proceedings, Pahang, Malaysia, 27–29 August 2017; American Institute of Physics: College Park, MD, USA, 2018; Volume 1974, p. 020077. [Google Scholar]
- Tsitouras, C.; Simos, T.E. Trigonometric-fitted explicit Numerov-type method with vanishing phase-lag and its first and second derivatives. Mediterr. J. Math.
**2018**, 15, 168. [Google Scholar] [CrossRef] - Samat, F.; Ismail, F.; Suleiman, M.B. A variable step-size exponentially fitted explicit hybrid method for solving oscillatory problems. Int. J. Math. Math. Sci.
**2011**, 2011, 328197. [Google Scholar] [CrossRef] [Green Version] - Simos, T.E.; Dimas, E.; Sideridis, A.B. A Runge-Kutta-Nystrom method for the numerical integration of special second-order periodic initial-value problem. J. Comput. Appl. Math.
**1994**, 51, 317–326. [Google Scholar] [CrossRef] [Green Version] - Geyer, H.; Seyfarth, A.; Blickhan, R. Spring-mass running: Simple approximate solution and application to gait stability. J. Theor. Biol
**2005**, 232, 315–328. [Google Scholar] [CrossRef] [PubMed] [Green Version]

−1 | 0 | 0 | 0 | 0 | 0 |

0 | 0 | 0 | 0 | 0 | 0 |

${c}_{3}$ | ${a}_{31}$ | ${a}_{32}$ | 0 | 0 | 0 |

${c}_{4}$ | ${a}_{41}$ | ${a}_{42}$ | ${a}_{43}$ | 0 | 0 |

${c}_{5}$ | ${a}_{51}$ | ${a}_{52}$ | ${a}_{53}$ | ${a}_{54}$ | 0 |

${b}_{1}$ | ${b}_{2}$ | ${b}_{3}$ | ${b}_{4}$ | ${b}_{5}$ |

−1 | 0 | 0 | 0 | 0 |

0 | 0 | 0 | 0 | 0 |

${c}_{3}$ | ${a}_{31}$ | ${a}_{32}$ | 0 | 0 |

${c}_{4}$ | ${a}_{41}$ | ${a}_{42}$ | ${a}_{43}$ | 0 |

${\overline{b}}_{1}$ | ${\overline{b}}_{2}$ | ${\overline{b}}_{3}$ | ${\overline{b}}_{4}$ |

TOL | METHOD | SSTEP | FSTEP | NFE | MAXGE |
---|---|---|---|---|---|

${10}^{-2}$ | EXH6 | 62 | 0 | 248 | 6.91104 × 10^{−2} |

TSI6 | 766 | 3 | 3076 | 5.36708 × 10^{−1} | |

EEHM6(4) | 50 | 1 | 200 | 1.60731 × 10^{−2} | |

${10}^{-4}$ | EXH6 | 132 | 0 | 528 | 5.60303 × 10^{−8} |

TSI6 | 7444 | 0 | 29,776 | 3.81115 × 10^{−13} | |

EEHM6(4) | 67 | 0 | 268 | 4.99527 × 10^{−5} | |

${10}^{-6}$ | EXH6 | 282 | 0 | 1128 | 3.81414 × 10^{−11} |

TSI6 | 112248 | 0 | 448,992 | 2.23796 × 10^{−11} | |

EEHM6(4) | 142 | 0 | 568 | 2.21358 × 10^{−9} | |

${10}^{-8}$ | EXH6 | 606 | 0 | 2424 | 3.80414 × 10^{−13} |

TSI6 | 1209137 | 0 | 4,836,548 | 1.37566 × 10^{−9} | |

EEHM6(4) | 304 | 0 | 1216 | 2.06565 × 10^{−11} | |

${10}^{-10}$ | EXH6 | 1304 | 0 | 5216 | 3.42059 × 10^{−14} |

EEHM6(4) | 653 | 0 | 2612 | 2.12689 × 10^{−13} | |

${10}^{-12}$ | EXH6 | 2808 | 0 | 11,232 | 8.79681 × 10^{−14} |

EEHM6(4) | 1405 | 0 | 5620 | 3.29937 × 10^{−14} |

TOL | METHOD | SSTEP | FSTEP | NFE | MAXGE |
---|---|---|---|---|---|

${10}^{-2}$ | EXH6 | 42 | 0 | 168 | 2.74183 × 10^{−3} |

TSI6 | 485 | 0 | 1940 | 1.23705 × 10^{−8} | |

EEHM6(4) | 50 | 0 | 200 | 5.52299 × 10^{−4} | |

${10}^{-4}$ | EXH6 | 88 | 0 | 352 | 1.99249 × 10^{−5} |

TSI6 | 5212 | 0 | 20,848 | 2.51907 × 10^{−12} | |

EEHM6(4) | 106 | 0 | 424 | 5.30432 × 10^{−6} | |

${10}^{-6}$ | EXH6 | 189 | 0 | 756 | 1.92665 × 10^{−7} |

TSI6 | 56125 | 0 | 224,500 | 8.95829 × 10^{−11} | |

EEHM6(4) | 226 | 0 | 904 | 5.32751 × 10^{−8} | |

${10}^{-8}$ | EXH6 | 405 | 0 | 1620 | 1.92570 × 10^{−9} |

TSI6 | 1209137 | 0 | 4,836,548 | 5.85237 × 10^{−9} | |

EEHM6(4) | 485 | 0 | 1940 | 5.37504 × 10^{−10} | |

${10}^{-10}$ | EXH6 | 870 | 0 | 3480 | 1.92941 × 10^{−11} |

EEHM6(4) | 1044 | 0 | 4176 | 5.59090 × 10^{−12} | |

${10}^{-12}$ | EXH6 | 1872 | 0 | 7488 | 3.10657 × 10^{−13} |

EEHM6(4) | 2246 | 0 | 8984 | 1.29793 × 10^{−12} |

TOL | METHOD | SSTEP | FSTEP | NFE | MAXGE |
---|---|---|---|---|---|

${10}^{-2}$ | EXH6 | 14 | 1 | 60 | 9.25756 × 10^{−2} |

TSI6 | 21 | 0 | 84 | 9.29852 × 10^{−3} | |

EEHM6(4) | 11 | 0 | 44 | 8.40394 × 10^{−3} | |

${10}^{-4}$ | EXH6 | 22 | 0 | 88 | 3.45117 × 10^{−5} |

TSI6 | 476 | 3 | 1916 | 9.82270 × 10^{−3} | |

EEHM6(4) | 22 | 0 | 88 | 3.09200 × 10^{−5} | |

${10}^{-6}$ | EXH6 | 58 | 0 | 232 | 4.72255 × 10^{−8} |

TSI6 | 4491 | 0 | 17,964 | 3.71608 × 10^{−12} | |

EEHM6(4) | 46 | 0 | 184 | 1.55125 × 10^{−7} | |

${10}^{-8}$ | EXH6 | 122 | 0 | 488 | 3.73456 × 10^{−10} |

TSI6 | 48367 | 0 | 193,468 | 6.42794 × 10^{−12} | |

EEHM6(4) | 98 | 0 | 392 | 9.31549 × 10^{−10} | |

${10}^{-10}$ | EXH6 | 262 | 0 | 1048 | 6.78776 × 10^{−12} |

EEHM6(4) | 210 | 0 | 840 | 6.60339 × 10^{−12} | |

${10}^{-12}$ | EXH6 | 563 | 0 | 2252 | 4.27902 × 10^{−12} |

EEHM6(4) | 450 | 0 | 1800 | 4.39979 × 10^{−12} |

TOL | METHOD | SSTEP | FSTEP | NFE | MAXGE |
---|---|---|---|---|---|

${10}^{-2}$ | EXH6 | 42 | 0 | 168 | 1.40533 × 10^{−3} |

TSI6 | 304 | 0 | 1216 | 5.85004 × 10^{−9} | |

EEHM6(4) | 42 | 0 | 168 | 1.61920 × 10^{−3} | |

${10}^{-4}$ | EXH6 | 88 | 0 | 352 | 1.31231 × 10^{−5} |

TSI6 | 2607 | 0 | 10,428 | 2.05918 × 10^{−13} | |

EEHM6(4) | 88 | 0 | 352 | 1.22888 × 10^{−5} | |

${10}^{-6}$ | EXH6 | 189 | 0 | 756 | 1.30796 × 10^{−7} |

TSI6 | 28063 | 0 | 112,252 | 6.81904 × 10^{−12} | |

EEHM6(4) | 189 | 0 | 756 | 1.19089 × 10^{−7} | |

${10}^{-8}$ | EXH6 | 405 | 0 | 1620 | 1.27003 × 10^{−9} |

TSI6 | 604569 | 0 | 2,418,276 | 2.61264 × 10^{−10} | |

EEHM6(4) | 405 | 0 | 1620 | 1.14692 × 10^{−9} | |

${10}^{-10}$ | EXH6 | 870 | 0 | 3480 | 1.24588 × 10^{−11} |

EEHM6(4) | 870 | 0 | 3480 | 1.12312 × 10^{−11} | |

${10}^{-12}$ | EXH6 | 1872 | 0 | 7488 | 1.90808 × 10^{−13} |

EEHM6(4) | 1872 | 0 | 7488 | 1.47056 × 10^{−13} |

TOL | METHOD | SSTEP | FSTEP | NFE | MAXGE |
---|---|---|---|---|---|

${10}^{-2}$ | EXH6 | 672 | 1 | 2692 | 1.50399 × 10^{−2} |

TSI6 | 302 | 2 | 1216 | 1.95744 × 10^{−2} | |

EEHM6(4) | 122 | 0 | 488 | 1.36083 × 10^{−1} | |

${10}^{-4}$ | EXH6 | 175 | 0 | 700 | 3.80609 × 10^{−3} |

TSI6 | 1423 | 5 | 5712 | 3.18389 × 10^{−3} | |

EEHM6(4) | 132 | 0 | 528 | 8.61690 × 10^{−3} | |

${10}^{-6}$ | EXH6 | 376 | 0 | 1504 | 2.67053 × 10^{−9} |

TSI6 | 13027 | 1 | 52,112 | 1.67782 × 10^{−5} | |

EEHM6(4) | 142 | 0 | 568 | 6.45235 × 10^{−3} | |

${10}^{-8}$ | EXH6 | 808 | 0 | 3232 | 7.32747 × 10^{−15} |

TSI6 | 241829 | 0 | 967,316 | 2.33693 × 10^{−11} | |

EEHM6(4) | 153 | 0 | 612 | 8.84274 × 10^{−3} | |

${10}^{-10}$ | EXH6 | 1738 | 0 | 6952 | 1.86517 × 10^{−14} |

EEHM6(4) | 327 | 0 | 1308 | 7.12467 × 10^{−3} | |

${10}^{-12}$ | EXH6 | 3743 | 0 | 14,972 | 8.48210 × 10^{−14} |

EEHM6(4) | 352 | 0 | 1408 | 6.32892 × 10^{−3} |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Samat, F.; Ismail, E.S.
Variable Step Exponentially Fitted Explicit Sixth-Order Hybrid Method with Four Stages for Spring-Mass and Other Oscillatory Problems. *Symmetry* **2020**, *12*, 387.
https://doi.org/10.3390/sym12030387

**AMA Style**

Samat F, Ismail ES.
Variable Step Exponentially Fitted Explicit Sixth-Order Hybrid Method with Four Stages for Spring-Mass and Other Oscillatory Problems. *Symmetry*. 2020; 12(3):387.
https://doi.org/10.3390/sym12030387

**Chicago/Turabian Style**

Samat, Faieza, and Eddie Shahril Ismail.
2020. "Variable Step Exponentially Fitted Explicit Sixth-Order Hybrid Method with Four Stages for Spring-Mass and Other Oscillatory Problems" *Symmetry* 12, no. 3: 387.
https://doi.org/10.3390/sym12030387