On Two-Derivative Runge–Kutta Type Methods for Solving u‴ = f(x,u(x)) with Application to Thin Film Flow Problem
Abstract
:1. Introduction
2. The Formulation of TDRKT Methods
3. Construction of TDRKT Methods
- (i)
- The graph
expressed as , with one meagre vertex (root of the rooted tree); the graph
expessed as ; the graph
denoted as ; and lastly,
denoted as ;
- (ii)
- If are different from , then the graph can be obtained as the roots of connecting downward to white circle vertex, combining the roots of into this black triangle vertex, followed by joining the roots downward to white rectangle vertex and subsequently to the roots into a new black circle vertex (root of ). It is expressed as
is root of the rooted tree t.
- (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.
- (i)
- (ii)
- for ,For every , the order ρ represents the amount of vertices t. The set comprised of all rooted trees with order k is expressed as .
- (i)
- ;
- (ii)
- for ,
- (i)
- ;
- (ii)
- for with , and distinct,
- (i)
- ;
- (ii)
- for
- (i)
- ;
- (ii)
- for whereby distinct, distinct and distinct,
3.1. Analytical Solution and Exact Derivative on B-Series
3.2. Numerical Solution and Numerical Derivative on B-Series
- The order conditions for u:
- Fourth order:
- Fifth order:
- Sixth order:
- The order conditions for :
- Third order:
- Fourth order:
- Fifth order:
- Sixth order:
- The order conditions for :
- Second order:
- Third order:
- Fourth order:
- Fifth order:
- Sixth order:
3.3. Two-Stage TDRKT Method of Order Four
3.4. Three-Stage TDRKT Method of Order Five
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].
5. Numerical Results
6. Discussion and Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A
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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
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 StyleLee, 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
APA StyleLee, K. C., Senu, N., Ahmadian, A., & Ibrahim, S. N. I. (2020). On Two-Derivative Runge–Kutta Type Methods for Solving u‴ = f(x,u(x)) with Application to Thin Film Flow Problem. Symmetry, 12(6), 924. https://doi.org/10.3390/sym12060924