# On Two-Derivative Runge–Kutta Type Methods for Solving u‴ = f(x,u(x)) with Application to Thin Film Flow Problem

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## Abstract

**:**

## 1. Introduction

## 2. The Formulation of TDRKT Methods

## 3. Construction of TDRKT Methods

**Definition**

**1.**

- (i)
- The graph expressed as ${\tau}_{1}$, with one meagre vertex (root of the rooted tree); the graph expessed as ${\tau}_{2}$; the graph denoted as ${\tau}_{3}$; and lastly, denoted as ${\tau}_{4}$;
- (ii)
- If ${t}_{1},\dots ,{t}_{r},{t}_{r+1},\dots ,{t}_{m},{t}_{m+1},\dots ,{t}_{n},{t}_{n+1},\dots ,{t}_{s}\in RT,{t}_{n+1},\dots ,{t}_{s}$ are different from ${\tau}_{1}$, then the graph can be obtained as the roots of ${t}_{1},\dots ,{t}_{r}$ connecting downward to white circle vertex, combining the roots of ${t}_{r+1},\dots ,{t}_{m}$ into this black triangle vertex, followed by joining the roots ${t}_{m+1},\dots ,{t}_{n}$ downward to white rectangle vertex and subsequently to the roots ${t}_{n},{t}_{n+1},\dots ,{t}_{s}$ into a new black circle vertex (root of $RT$). It is expressed as$$\begin{array}{c}\hfill \begin{array}{c}\hfill t={\left[{t}_{1},\dots ,{t}_{r},<{t}_{r+1},\dots ,{t}_{m}>,<{t}_{m+1},\dots ,{t}_{n}>,<{t}_{n+1},\dots ,{t}_{s}>\right]}_{4},\end{array}\end{array}$$

- (i)
- The meagre vertex is the root of every rooted tree.
- (ii)
- The offspring of meagre vertex must consist of only one white circle vertex.
- (ii)
- The offspring of white circle vertex must consist of only one black triangle vertex.
- (iv)
- The offspring of black triangle vertex must consist of only one white rectangle vertex.

**Definition**

**2.**

- (i)
- $\rho \left({\tau}_{1}\right)=1,\phantom{\rule{4pt}{0ex}}\rho \left({\tau}_{2}\right)=2,\phantom{\rule{4pt}{0ex}}\rho \left({\tau}_{3}\right)=3,\phantom{\rule{4pt}{0ex}}\rho \left({\tau}_{4}\right)=4,$
- (ii)
- for $t={\left[{t}_{1},\dots ,{t}_{r},<{t}_{r+1},\dots ,{t}_{m}>,<{t}_{m+1},\dots ,{t}_{n}>\right]}_{3}\in RT$,$$\begin{array}{c}\hfill \rho \left(t\right)=4+\sum _{i=1}^{r}\rho \left({t}_{i}\right)+\sum _{i=r+1}^{m}(\rho \left({t}_{i}\right)-1)+\sum _{i=m+1}^{n}(\rho \left({t}_{i}\right)-2).\end{array}$$For every $t\in RT$, the order ρ represents the amount of vertices t. The set comprised of all rooted trees with order k is expressed as $R{T}_{k}$.

**Definition**

**3.**

- (i)
- $\begin{array}{c}F\left({\tau}_{1}\right)(u,{u}^{\prime},{u}^{\u2033})={u}^{\prime},F\left({\tau}_{2}\right)(u,{u}^{\prime},{u}^{\u2033})={u}^{\u2033},F\left({\tau}_{3}\right)(u,{u}^{\prime},{u}^{\u2033})=f\left(u\right),\hfill \\ F\left({\tau}_{4}\right)(u,{u}^{\prime},{u}^{\u2033})={f}^{\prime}\left(u\right)=g(u,{u}^{\prime})\hfill \end{array}$;
- (ii)
- for $t={\left[{t}_{1},\dots ,{t}_{r},<{t}_{r+1},\dots ,{t}_{m}>,<{t}_{m+1},\dots ,{t}_{n}>,<{t}_{n+1},\dots ,{t}_{s}>\right]}_{4}\in RT$,$$\begin{array}{c}\hfill \begin{array}{c}\hfill F\left(t\right)(u,{u}^{\prime},{u}^{\u2033})=\frac{{\partial}^{n}g}{\partial {u}^{r}\partial {u}^{\prime m-r}\partial {u}^{\u2033n-m}}\left(u,{u}^{\prime},{u}^{\u2033}\right)\left[F\left({t}_{1}\right)(u,{u}^{\prime},{u}^{\u2033}),...,F\left({t}_{n}\right)(u,{u}^{\prime},{u}^{\u2033})\right].\end{array}\end{array}$$

**Definition**

**4.**

- (i)
- $\sigma \left({\tau}_{1}\right)=\sigma \left({\tau}_{2}\right)=\sigma \left({\tau}_{3}\right)=\sigma \left({\tau}_{4}\right)=1$;
- (ii)
- for $t=\left[{t}_{1}^{{\mu}_{1}},\dots ,{t}_{r}^{{\mu}_{r}},<{t}_{r+1}^{{\mu}_{r+1}},\dots ,{t}_{m}^{{\mu}_{m}}>,<{t}_{m+1}^{{\mu}_{m+1}},\dots ,{t}_{n}^{{\mu}_{n}}>\right]\in RT,$ with ${t}_{1},\dots ,{t}_{r}$, ${t}_{r+1},\dots ,{t}_{m}$ and ${t}_{m+1},\dots ,{t}_{n}$ distinct,$$\begin{array}{c}\hfill \sigma \left(t\right)=\prod _{i=1}^{n}{\mu}_{i}!\left(\sigma {\left({t}_{i}\right)}^{{\mu}_{i}}\right),\end{array}$$

**Theorem**

**1.**

**Proof.**

**Proposition**

**1.**

- (i)
- $\gamma \left({\tau}_{1}\right)=1,\phantom{\rule{4pt}{0ex}}\gamma \left({\tau}_{2}\right)=2,\phantom{\rule{4pt}{0ex}}\gamma \left({\tau}_{3}\right)=6$;
- (ii)
- for $t={\left[{t}_{1},\dots ,{t}_{r},<{t}_{r+1},\dots ,{t}_{m}>,<{t}_{m+1},\dots ,{t}_{n}>\right]}_{3}\in RT,$$\gamma \left(t\right)=\rho \left(t\right)(\rho \left(t\right)-1)(\rho \left(t\right)-2){\prod}_{i=1}^{r}\gamma \left({t}_{i}\right){\prod}_{i=r+1}^{m}\frac{\gamma \left({t}_{i}\right)}{\rho \left({t}_{i}\right)}{\prod}_{i=m+1}^{n}\frac{\gamma \left({t}_{i}\right)}{\rho \left({t}_{i}\right)(\rho \left({t}_{i}\right)-1)}.$

**Proposition**

**2.**

- (i)
- $\alpha \left({\tau}_{1}\right)=1,\phantom{\rule{4pt}{0ex}}\alpha \left({\tau}_{2}\right)=1,\alpha \left({\tau}_{3}\right)=1$;
- (ii)
- for $t={\left[{t}_{1}^{{\mu}_{1}},\dots ,{t}_{r}^{{\mu}_{r}},<{t}_{r+1}^{{\mu}_{r+1}},\dots ,{t}_{m}^{{\mu}_{m}}>,<{t}_{m+1}^{{\mu}_{m+1}},\dots ,{t}_{n}^{{\mu}_{n}}>\right]}_{3}\in RT,$ whereby ${t}_{1},\dots ,{t}_{r}$ distinct, ${t}_{r+1},\dots ,{t}_{m}$ distinct and ${t}_{m+1},\dots ,{t}_{n}$ distinct,$$\begin{array}{c}\hfill \alpha \left(t\right)=(\rho \left(t\right)-3)!\prod _{i=1}^{r}\frac{1}{{\mu}_{i}!}{\left(\frac{\alpha \left({t}_{i}\right)}{\rho \left({t}_{i}\right)!}\right)}^{{\mu}_{i}}\prod _{i=r+1}^{m}\frac{1}{{\mu}_{i}!}{\left(\frac{\alpha \left({t}_{i}\right)}{(\rho \left({t}_{i}\right)-1)!}\right)}^{{\mu}_{i}}\xb7\prod _{i=m+1}^{n}\frac{1}{{\mu}_{i}!}{\left(\frac{\alpha \left({t}_{i}\right)}{(\rho \left({t}_{i}\right)-2)!}\right)}^{{\mu}_{i}},\end{array}$$

#### 3.1. Analytical Solution and Exact Derivative on B-Series

**Theorem**

**2.**

**Proof.**

#### 3.2. Numerical Solution and Numerical Derivative on B-Series

- The order conditions for u:
- Fourth order:$${b}^{T}e=\frac{1}{24}.$$
- Fifth order:$${b}^{T}c=\frac{1}{120}.$$
- Sixth order:$${b}^{T}{c}^{2}=\frac{1}{360}.$$
- The order conditions for ${u}^{\prime}$:
- Third order:$${b}^{\prime T}e=\frac{1}{6}.$$
- Fourth order:$${b}^{\prime T}c=\frac{1}{24}.$$
- Fifth order:$${b}^{\prime T}{c}^{2}=\frac{1}{60}.$$
- Sixth order:$$\begin{array}{c}\hfill {b}^{\prime T}{c}^{3}=\frac{1}{120}\phantom{\rule{4pt}{0ex}},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{b}^{\prime T}\widehat{a}e=\frac{1}{720}.\end{array}$$
- The order conditions for ${u}^{\u2033}$:
- Second order:$${b}^{\u2033T}e=\frac{1}{2}.$$
- Third order:$${b}^{\u2033T}c=\frac{1}{6}.$$
- Fourth order:$${b}^{\u2033T}{c}^{2}=\frac{1}{12}.$$
- Fifth order:$$\begin{array}{c}\hfill {b}^{\u2033T}{c}^{3}=\frac{1}{20}\phantom{\rule{4pt}{0ex}},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{b}^{\u2033T}\widehat{a}e=\frac{1}{120}.\end{array}$$
- Sixth order:$$\begin{array}{c}{b}^{\u2033T}{c}^{4}=\frac{1}{30}\phantom{\rule{4pt}{0ex}},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{b}_{i}^{\u2033T}\widehat{a}c=\frac{1}{720},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{b}^{\u2033T}Ae=\frac{1}{720},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{b}^{\u2033T}c\widehat{a}e=\frac{1}{180}.\hfill \end{array}$$

#### 3.3. Two-Stage TDRKT Method of Order Four

#### 3.4. Three-Stage TDRKT Method of Order Five

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

## 4. Problem Testing and Numerical Result

**TDRKT4**—Explicit two-derivative Runge–Kutta type method with two stage fourth-order.**TDRKT5**—Explicit two-derivative Runge–Kutta type method with three stage fifth-order.**RK4**—Runge–Kutta fourth-order method as given in Hossain et al. [13].**RK5**—Runge–Kutta fifth-order method as given in Goeken and Johnson [14].**Mechee4**—Explicit two stage fourth-order direct method proposed by Mechee et al. [15].**Mechee5**—Explicit three stage fifth-order direct method proposed by Mechee et al. [16].**Hussain4**—Fourth-order improved Runge–Kutta direct method proposed by Hussain et al. [17].**Hussain5**—Fifth-order improved Runge–Kutta direct method proposed by Hussain et al. [18].

**Problem**

**1**

**Problem**

**2**

**Problem**

**3**

**Problem**

**4**

**Problem**

**5**

**Problem**

**6**

## 5. Numerical Results

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

- Omar, A.A.; Zaer, A.; Ramzi, A.; Shaher, M. A reliable analytical method for solving higher-order initial value problems. Discret. Dyn. Nat. Soc.
**2013**, 2013, 673829. [Google Scholar] - Agboola, O.O.; Opanuga, A.A.; Gbadeyan, J.A. Solution of third order ordinary differential equations using differential transform method. Glob. J. Pure Appl. Math.
**2015**, 11, 2511–2516. [Google Scholar] - Khataybeh, S.N.; Hashim, I.; Alshbool, M. Solving directly third-order ODEs using operational matrices of Bernstein polynomials method with applications to fluid flow equations. J. King Saud-Univ.-Sci.
**2019**, 31, 822–826. [Google Scholar] [CrossRef] - Tang, W.S.; Zhang, J.J. Symmetric integrator based on continuous-stage Runge–Kutta–Nyström methods for reversible systems. Appl. Math. Comput.
**2019**, 361, 1–12. [Google Scholar] [CrossRef] [Green Version] - Fang, Y.L.; You, X.; Ming, Q.H. Trigonometrically fitted two-derivative Runge-Kutta methods for solving oscillatory differential equations. Numer. Algorithms
**2014**, 63, 651–667. [Google Scholar] [CrossRef] - Chen, Z.; Qiu, Z.; Li, J.; You, X. Two-derivative Runge-Kutta-Nyström methods for second-order ordinary differential equations. Numer. Algorithms
**2015**, 70, 897–927. [Google Scholar] [CrossRef] - Ehigie, J.O.; Zou, M.M.; Hou, X.L.; You, X. On modified TDRKN methods for second-order systems of differential equations. Int. J. Comput. Math.
**2017**, 95, 1–15. [Google Scholar] [CrossRef] - Mohamed, T.S.; Senu, N.; Ibrahim, Z.B.; Nik Long, N.M.A. Efficient two-derivative Runge-Kutta-Nyström for solving general second-order ordinary differential equations. Discret. Dyn. Nat. Soc.
**2018**, 2018, 2393015. [Google Scholar] [CrossRef] [Green Version] - Henrici, P. Discrete Variable Methods in Ordinary Differential Equations; John Wiley & Sons: New York, NY, USA, 1962. [Google Scholar]
- Suli, E.; Mayers, D.F. Discrete Variable Methods in Ordinary Differential Equations; Cambridge University Press: Cambridge, UK, 2003; pp. 337–340. [Google Scholar]
- Lambert, J.D. Numerical Methods for Ordinary Differential Systems: The Initial Value Problem; John Wiley & Sons, Inc.: New York, NY, USA, 1991. [Google Scholar]
- Atkinson, K.; Han, W.; Stewart, D. Numerical Solution of Ordinary Differential Equations: Convergence, Stability and Asymptotic Error; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2009. [Google Scholar]
- Hossain, B.; Hossain, J.; Miah, M.; Alam, S. A comparative study on fourth order and butcher’s fifth order runge-kutta methods with third order initial value problem (IVP). Appl. Comput. Math.
**2017**, 6, 243–253. [Google Scholar] [CrossRef] [Green Version] - Goeken, D.; Johnson, O. Fifth-order Runge-Kutta with higher order derivation approximations. Electron. J. Differ. Equ.
**1999**, 95, 1–9. [Google Scholar] - Mechee, M.; Ismail, F.; Siri, Z.; Senu, N. A third-order direct integrators of Runge-Kutta type for special third-order ordinary and delay differential equations. Asian J. Appl. Sci.
**2014**, 7, 102–116. [Google Scholar] [CrossRef] [Green Version] - Mechee, M.; Senu, N.; Ismail, F.; Nikouravan, B.; Siri, Z. Three-stage fifth-order Runge-Kutta method for directly solving special third-order differential equation with application to thin film flow problem. Math. Probl. Eng.
**2013**, 2013, 795397. [Google Scholar] [CrossRef] - Hussain, K.A.; Ismail, F.; Senu, N.; Rabiei, F. Fourth-order improved Runge–Kutta method for directly solving special third-order ordinary differential equations. Iran. J. Sci. Technol. Trans. Sci.
**2017**, 41, 429–437. [Google Scholar] [CrossRef] - Hussain, K.A.; Ismail, F.; Senu, N.; Rabiei, F.; Ibrahim, R. Integration for special third-order ordinary differential equations using improved Runge-Kutta direct method. Malays. J. Sci.
**2015**, 34, 172–179. [Google Scholar] [CrossRef] - Yap, L.K.; Ismail, F.; Senu, N. An accurate block hybrid collocation method for third order ordinary differential equations. J. Appl. Math.
**2014**, 2014, 673829. [Google Scholar] [CrossRef] [Green Version] - Momoniat, E.; Mahomed, F.M. Symmetry reduction and numerical solution of third-order ode from thin film flow. Math. Comput. Appl.
**2015**, 15, 709–719. [Google Scholar] [CrossRef] [Green Version] - Butcher, J.C. Numerical methods for ordinary differential methods in the 20th century. J. Comput. Appl. Math.
**2000**, 125, 1–29. [Google Scholar] [CrossRef] [Green Version] - Butcher, J.C. Numerical Methods for Ordinary Differential Equations; John Wiley & Sons: Chichester, UK, 2008. [Google Scholar]
- Butcher, J.C. Trees, stumps, and applications. Axioms
**2018**, 7, 52. [Google Scholar] [CrossRef] [Green Version]

C | a | $\widehat{\mathit{a}}$ | |

${b}^{T}$ | ${b}^{{}^{\prime}T}$ | ${b}^{{}^{\u2033}T}$ |

0 | 0 | 0 | ||||

$\frac{1}{2}$ | $\frac{1}{384}$ | 0 | $\frac{1}{48}$ | 0 | ||

$\frac{1}{40}$ | $\frac{1}{60}$ | $\frac{1}{12}$ | $\frac{1}{12}$ | $\frac{1}{6}$ | $\frac{1}{3}$ |

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

$\frac{3}{10}$ | $\frac{27}{80,000}$ | 0 | $\frac{9}{2000}$ | 0 | |||||

$\frac{3}{4}$ | $\frac{27}{2048}$ | 0 | 0 | $\frac{3}{128}$ | $\frac{3}{64}$ | 0 | |||

$\frac{1}{72}$ | $\frac{1}{36}$ | 0 | $\frac{5}{108}$ | $\frac{35}{324}$ | $\frac{1}{81}$ | $\frac{5}{54}$ | $\frac{25}{81}$ | $\frac{8}{81}$ |

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## Share and Cite

**MDPI and ACS Style**

Lee, K.C.; Senu, N.; Ahmadian, A.; Ibrahim, S.N.I.
On Two-Derivative Runge–Kutta Type Methods for Solving *u*‴ = *f*(*x*,*u*(*x*)) with Application to Thin Film Flow Problem. *Symmetry* **2020**, *12*, 924.
https://doi.org/10.3390/sym12060924

**AMA Style**

Lee KC, Senu N, Ahmadian A, Ibrahim SNI.
On Two-Derivative Runge–Kutta Type Methods for Solving *u*‴ = *f*(*x*,*u*(*x*)) with Application to Thin Film Flow Problem. *Symmetry*. 2020; 12(6):924.
https://doi.org/10.3390/sym12060924

**Chicago/Turabian Style**

Lee, Khai Chien, Norazak Senu, Ali Ahmadian, and Siti Nur Iqmal Ibrahim.
2020. "On Two-Derivative Runge–Kutta Type Methods for Solving *u*‴ = *f*(*x*,*u*(*x*)) with Application to Thin Film Flow Problem" *Symmetry* 12, no. 6: 924.
https://doi.org/10.3390/sym12060924