Ababneh, O. Y. & Ahmad, R. (2009). Construction of third-order diagonal implicit
Runge-Kutta methods for stiff problems. Chinese Physics Letters, 26(8), 080503.
Abia, L. & Sanz-Serna, J. M. (1993). Partitioned Runge-Kutta methods for separable Hamiltonian
problems. Mathematics of Computation, 60(202), 617-634.
Agam, S. A. & Yahaya, Y. A. (2014). A highly efficient implicit Runge-Kutta method for first order
ordinary differential equations. African Journal of Mathematics and Computer Science Research,
7(5), 55-60.
Aiken, R. C. (1985). Stiff Computation. Oxford: Oxford University Press.
Antoñana, M., Makazaga, J. & Murua, A. (2018). Efficient implementation of symplectic
implicit Runge-Kutta schemes with simplified Newton iterations. Numerical Algorithms, 78(1), 63-86.
Araz, S. ?. (2020). Numerical analysis of a new Volterra integro-differential equation involving
fractal-fractional operators. Chaos, Solitons & Fractals, 130, 109396.
Atangana, A. & Araz, S. ?. (2020). New numerical method for ordinary differential equations:
Newton polynomial. Journal of Computational and Applied Mathematics, 372, 112622.
Atkinson, K., Han, W. & Stewart, D. E. (2009). Numerical Solution of Ordinary
Differential Equations. New Jersey: John Wiley & Sons, Inc.
Baboulin, M., Buttari, A., Dongarra, J., Kurzak, J., Langou, J., Langou, J., et al. (2009).
Accelerating scientific computations with mixed precision
algorithms. Computer Physics Communications, 180(12), 2526-2533.
Bader, G. & Deuflhard, P. (1983). A semi-implicit mid-point rule for stiff systems of ordinary
differential equations. Numerische Mathematik, 41(3), 373-398.
Berghe, G. V. & Daele, M. V. (2011). Symplectic exponentially-fitted four-stage
Runge-Kutta methods of the Gauss type. Numerical Algorithms, 56(4), 591- 608.
Bjurel, G., Dahlquist, G., Lindberg, B., Linde, S. & Oden, L. (1970). Survey of stiff ordinary
differential equations. Department of Information Processing, Royal
Institute of Technology, Stockholm.
Blanes, S., Casas, F. & Thalhammer, M. (2019). Splitting and composition methods with embedded
error estimators. Applied Numerical Mathematics. 146 (2019), 400-415.
Boom, P. D. & Zingg, D. W. (2015). Investigation of efficient high-order implicit
Runge-Kutta methods based on generalized summation-by-parts operators. 22nd AIAA
Computational Fluid Dynamics Conference, 2757, 1-15.
Branch, M. & Mahshahr, I. R. I. (2016). Computing simulation of the generalized duffing
oscillator basedon EBM and MHPM. Mechanics and Mechanical Engineering, 20(4), 595-604.
Burrage, K. & Butcher, J. C. (1979). Stability criteria for implicit Runge-Kutta
methods. SIAM Journal on Numerical Analysis, 16(1), 46-57.
Butcher, J. C. (1964). Implicit Runge-Kutta processes. Mathematics of
Computation, 18(85), 50-64.
Butcher, J. C. (1996). A history of Runge-Kutta methods. Applied Numerical Mathematics,
20(3), 247-260.
Butcher, J. C. (1997). An introduction to “Almost Runge-Kutta” methods. Applied Numerical
Mathematics, 24(2-3), 331-342.
Butcher, J. C. (2016). Numerical Methods for Ordinary Differential Equations. United Kingdom: John
Wiley & Sons.
Calvo, M., Franco, J. M., Montijano, J. I. & Rández, L. (2009). Sixth-order symmetric and
symplectic exponentially fitted Runge-Kutta methods of the Gauss type. Journal of Computational and
Applied Mathematics, 223(1), 387–398.
Cash, J. R. (1975). A class of implicit Runge-Kutta methods for the numerical
integration of stiff ordinary differential equations. Journal of the ACM (JACM),
22(4), 504-511.
Cerrolaza, M., Shefelbine, S. & Garzón-Alvarado, D. (Eds.). (2018). Numerical Methods and
Advanced Simulation in Biomechanics and Biological Processes. London, UK: Academic Press.
Chan, R. P. K. (1990). On symmetric Runge-Kutta methods of high order. Computing, 45(4), 301–309.
Chan, R. P. K. & Gorgey, A. (2013). Active and passive symmetrization of Runge- Kutta Gauss
methods. Applied Numerical Mathematics, 67, 64–77.
Chan, R. P. K. & Razali, N. (2014). Smoothing effects on the IMR and ITR. Numerical
Algorithms, 65(3), 401-420.
Cong, N. H. (1994). Parallel iteration of symmetric Runge-Kutta methods for nonstiff initial-value
problems. Journal of Computational and Applied Mathematics, 51(1), 117–125.
Cooper, G. J. & Butcher, J. C. (1983). An iteration scheme for implicit Runge-Kutta methods. IMA
Journal of Numerical Analysis, 3(2), 127-140.
Dahlquist, G. G. (1963). A special stability problem for linear multistep methods. BIT Numerical
Mathematics, 3(1), 27-43.
Dekker, K. & Verwer J.G. (1984). Stability of Runge-Kutta methods for stiff nonlinear differential
equations. North-Holland: CWI Monographs.
DeVries, P. & Hasbun, J. (2011). A First Course in Computational Physics. Sudbury, MA: Jones and
Bartlett Publishers.
Dormand, J. R. (2018). Numerical Methods for Differential Equations: A
Computational Approach. Boca Raton, FL: CRC Press.
Ehle, B. L. (1973). A-stable methods and Padé approximations to the
exponential. SIAM Journal on Mathematical Analysis, 4(4), 671-680.
Enright, W. H., Hull, T. E. & Lindberg, B. (1975). Comparing numerical methods for stiff systems of
ODEs. BIT Numerical Mathematics, 15(1), 10-48.
Fan, Z., Song, M. & Liu, M. (2009). The αth moment stability for the stochastic
pantograph equation. Journal of Computational and Applied
Mathematics, 233(2), 109-120.
Faragó, I., Havasi, Á. & Zlatev, Z. (2013). The convergence of diagonally implicit
Runge-Kutta methods combined with Richardson extrapolation. Computers and Mathematics with
Applications, 65(3), 395–401.
Fatunla, S. O. (2014). Numerical Methods for Initial Value Problems in Ordinary
Differential Equations. San Diego, CA: Academic Press, Inc.
Gear, C. W. (1980). Runge-Kutta starters for multistep methods. ACM Transactions on Mathematical
Software (TOMS), 6(3), 263-279.
González-Pinto, S., González-Concepción, C. & Montijano, J. I. (1994). Iterative schemes
for Gauss methods. Computers and Mathematics with Applications, 27(7),
67-81.
González-Pinto, S., Hernández-Abreu, D. & Montijano, J. I. (2019). Variable step-size control
based on two-steps for Radau IIA methods. Preprint:
https://www.researchgate.net/publication/331687918_Variable_step-
l_based_on_two-steps_for_Radau_IIA_methods
González-Pinto, S., Montijano, J. I. & Rández, L. (1995). Iterative schemes for three-
stage implicit Runge-Kutta methods. Applied Numerical Mathematics, 17(4), 363–382.
Gorgey, A. (2012). Extrapolation of symmetrized Runge-Kutta methods. PhD thesis, ResearchSpace @
University of Auckland.
Gorgey, A. (2015). Extrapolation of symmetrized Runge-Kutta methods in the variable stepsize
setting. International Journal of Applied Mathematics and Statistics, 55(2),
14-22.
Gorgey, A. & Chan, R. P. K. (2015). Choice of strategies for extrapolation with
symmetrization in the constant stepsize setting. Applied Numerical
Mathematics, 87, 31-37.
Gorgey, A. & Mat, N. A. A. (2018). Efficiency of Runge-Kutta methods in solving simple harmonic
oscillators. Matematika, 34(1), 1-12.
Gorgey, A. & Muhammad, H. (2017). Efficiency of Runge-Kutta methods in solving Kepler problem. AIP
Conference Proceedings. 1847(1), 020016.
Guo, P. & Li, C. J. (2019). Razumikhin-type technique on stability of exact and
numerical solutions for the nonlinear stochastic pantograph differential equations.
BIT Numerical Mathematics, 59(1), 77-96.
Gustafsson, K. (1994). Control-theoretic techniques for stepsize selection in implicit Runge-Kutta
methods. ACM Transactions on Mathematical Software (TOMS), 20(4), 496-517.
Hairer, E. & Wanner, G. (1981). Algebraically stable and implementable Runge-Kutta methods of high
order. SIAM Journal on Numerical Analysis, 18(6), 1098-1108.
Hairer, E. & Wanner, G. (1996). Solving Ordinary Differential Equations II. London: Springer-Verlag
Berlin Heidelberg.
Hairer, E. & Wanner, G. (1999). Stiff differential equations solved by Radau methods.
Journal of Computational and Applied Mathematics, 111(1-2), 93-111.
Higham, N. J. (1993). The accuracy of floating point summation. SIAM Journal on Scientific
Computing, 14(4), 783-799.
Hindmarsh, A. C. (1980). LSODE and LSODI, two new initial value ordinary differential
equation solvers. ACM Signum Newsletter, 15(4), 10-11.
Hitchens, F. (2015). Propeller Aerodynamics: The History, Aerodynamics & Operation of Aircraft
Propellers. Wellington, NZ: Andrews UK Limited.
Holder, A. & Eichholz, J. (2019). Modeling with delay differential equations. In An
Introduction to Computational Science. Switzerland: Springer, Cham. 377-387.
Hussain, E. A. & Abdul-Abbass, Y. M. (2019). On Fuzzy differential equation. Journal
of Al-Qadisiyah for Computer Science and Mathematics, 11(2), 1-9.
Iserles, A. (2009). A first course in the numerical analysis of differential equations.
New York, US: Cambridge University Press.
Ismail, A. & Gorgey, A. (2015). Behaviour of the extrapolated implicit midpoint and implicit
trapezoidal rules with and without compensated summation. Matematika, 31(1), 47–57.
Kennedy, C. A. & Carpenter, M. H. (2019). Diagonally implicit Runge-Kutta methods for stiff ODEs.
Applied Numerical Mathematics, 146(6), 221-244.
Kim, I. P. & Kräuter, A. R. (2018). Decompositions of a matrix by means of its dual matrices with
applications. Linear Algebra and its Applications, 537, 100-117.
Kulikov, G. Y. (2015). Embedded symmetric nested implicit Runge-Kutta methods of Gauss and Lobatto
types for solving stiff ordinary differential equations and Hamiltonian systems. Computational
Mathematics and Mathematical Physics, 55(6), 983–1003.
Kuntzmann, J. (1961). Neuere entwicklungen der methode von runge und kutta.
ZAMM?Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und
Mechanik, 41(1), 28-31.
Lambert, J. D. (1991). Numerical Methods for Ordinary Differential Systems: The Initial
Value Problem. New York, US: John Wiley & Sons, Inc.
Lapidus, L. & Schiesser, W. E. (1976). Numerical Methods for Differential Systems: Recent
Developments in Algorithms, Software, and Applications. United Kingdom: Academic Press
Inc.
Liu, Y. (1995). Stability analysis of θ-methods for neutral functional-differential
equations. Numerische Mathematik, 70(4), 473-485.
Liu, M. Y., Zhang, L. & Zhang, C. F. (2019). Study on banded implicit Runge-Kutta methods for
solving stiff differential equations. Mathematical Problems in Engineering, 2019, 4850872.
Muhammad, M. H. (2018). An Efficient Implementation Technique for Implicit Runge- Kutta Methods in
Solving the Stiff Problems. Universiti Pendidikan Sultan Idris.
Muhammad, M. H. & Gorgey, A. (2018). Investigation on the most efficient ways to solve the implicit
equations for Gauss methods in the constant stepsize setting. Applied Mathematical Sciences, 12(2),
93-103.
Najafi, R. & Nemati, S. B. (2017). Numerical solution of the forced Duffing equations using
Legendre multiwavelets. Computational Methods for Differential
tions, 5(1), 43-55.
Nazari, F., Mohammadian, A., Charron, M. & Zadra, A. (2014). Optimal high-order
diagonally-implicit Runge-Kutta schemes for nonlinear diffusive systems on atmospheric
boundary layer. Journal of Computational Physics, 271, 118-130.
Owolabi, K. M. (2019). Mathematical modelling and analysis of love dynamics: A fractional
approach. Physica A: Statistical Mechanics and its Applications, 525, 849-865.
Peat, K. D. & Thomas, R. M. (1989). Implementation of iteration schemes for implicit Runge-Kutta
methods. University of Manchester. Department of Mathematics.
Prothero, A., & Robinson, A. (1974). On the stability and accuracy of one-step methods for solving
stiff systems of ordinary differential equations. Mathematics of Computation, 28(125),
145-162.
Ramos, H. (2019). Development of a new Runge?Kutta method and its economical implementation.
Computational and Mathematical Methods, 1(2), e1016.
Rang, J. (2016). The Prothero and Robinson example: Convergence studies for Runge- Kutta and
Rosenbrock-Wanner methods. Applied Numerical Mathematics, 108, 37-56.
Rasedee, A. F. N., Ishak, N., Hamzah, S. R., Ijam, H. M., Suleiman, M., Ibrahim, Z. B., et al.
(2017). Variable order variable stepsize algorithm for solving nonlinear Duffng oscillator. Journal
of Physics: Conference Series, 890, 012045.
Razali, N., Nopiah, Z. M. & Othman, H. (2018). Comparison of one-step and two-step symmetrization
in the variable stepsize setting. Sains Malaysiana, 47(11), 2927-2932.
Rechenberg, H. (2001). The Historical Development of Quantum Theory, Volume 1.
New York, US: Springer Science & Business Media.
Robertson, H. H. (1966). The solution of a set of reaction rate equations. Cambridge,
Massachusetts: Academic Press. 178-182.
Roussel, M. R. (2019). Nonlinear Dynamics: A hands-on introductory survey. Bristol, UK: Morgan &
Claypool Publishers.
Rushanan, J. J. (1989). On the Vandermonde matrix. The American Mathematical Monthly,
96(10), 921-924.
Sanderse, B. & Koren, B. (2012). Accuracy analysis of explicit Runge-Kutta methods applied to
the incompressible Navier-Stokes equations. Journal of Computational Physics,
231(8), 3041– 3063.
Sanz-Serna, J. (1988). Runge-Kutta schemes for Hamiltonian systems. BIT Numerical
Mathematics, 28(4), 877-883.
Sanz-Serna, J. M. (2016). Symplectic Runge-Kutta schemes for adjoint equations,
automatic differentiation, optimal control, and more. SIAM Review, 58(1), 3-33.
Schäfer, E. (1975). A new approach to explain the “high irradiance responses” of
photomorphogenesis on the basis of phytochrome. Journal of Mathematical Biology, 2(1),
41-56.
Shampine, L. F. (1984). Stability of explicit Runge-Kutta methods. Computers &
Mathematics with Applications, 10(6), 419-432.
Shampine, L. F. (1985). Local error estimation by doubling. Computing, 34(2), 179- 190.
Shampine, L. F. (2018). Numerical Solution of Ordinary Differential Equations. Boca Raton, FL:
Routledge.
Skvortsov, L. M. & Kozlov, O. S. (2014). Efficient implementation of diagonally implicit
Runge-Kutta methods. Mathematical Models and Computer Simulations, 6(4), 415-424.
Sun, G. A. (2000). Simple way constructing symplectic Runge-Kutta methods. Journal of Computational
Mathematics, 18(1), 61–68.
Swart, D., Jacques, J. B. & Lioen, W. M. (1998). Collecting real-life problems to test solvers for
implicit differential equations. CWI Quarterly, 11(1), 83-100.
Toufik, M. & Atangana, A. (2017). New numerical approximation of fractional derivative
with non-local and non-singular kernel: application to chaotic models. The European Physical
Journal Plus, 132(10), 444.
Varah, J. M. (1979). On the efficient implementation of implicit Runge-Kutta methods.
Mathematics of Computation, 33(146), 557.
Wang, P., Zhou, J., Wang, R. & Chen, J. (2017). New generalized variable stepsizes of the CQ
algorithm for solving the split feasibility problem. Journal of Inequalities and
Applications, 2017(1), 135.
Wang, Y. & Chen, Y. (2020). Shifted Legendre Polynomials algorithm used for the dynamic analysis of
viscoelastic pipes conveying fluid with variable fractional order model. Applied Mathematical
Modelling, 81, 159-176.
Wilkie, J. & Çetinba?, M. (2005). Variable-stepsize Runge-Kutta methods for
stochastic Schrödinger equations. Physics Letters A, 337(3), 166-182.
Williams, G. (2017). Linear Algebra with Applications. United State of America: Jones & Barlett
Learning.
Willoughby, R. A. (1974). Stiff Differential Systems. International Symposium on Stiff
Differential Systems. Boston, MA: Springer. 1-19.
Xu, P., Yuan, Z., Jian, W. & Zhao, W. (2015). Variable step-size method based on a reference
separation system for source separation. Journal of Sensors, 2015. 964098.
Yaici, M. & Hariche, K. (2019). A particular block Vandermonde matrix. ITM Web of Conferences, 24.
01008.
Yang, H., Yang, Z., Wang, P. & Han, D. (2019). Mean-square stability analysis for nonlinear
stochastic pantograph equations by transformation approach. Journal of Mathematical Analysis and
Applications, 479(1), 977-986.
Yang, X., Yang, Z. & Xiao, Y. (2020). Asymptotical mean-square stability of linear θ- methods for
stochastic pantograph differential equations: variable stepsize and transformation
approach. Unpublished manuscript. DOI: 10.22541/au.159023888.86381071
Ycart, B. (2012). A Case of Mathematical Eponymy: The Vandermonde Determinant.
arXiv preprint arXiv:1204.4716.
Ye, K. (2017). New classes of matrix decompositions. Linear Algebra and its
Applications, 514, 47-81.
Yu, W. & Jafari, R. (2019). Fuzzy Differential Equations. In Modeling and Control of Uncertain
Nonlinear Systems with Fuzzy Equations and Z-Number, Piscataway, NJ: Wiley-IEEE Press. 21-37.
Zhang, D. K. (2019). Discovering New Runge-Kutta Methods using Unstructured Numerical
Search. arXiv preprint arXiv:1911.00318.
Zhang, H., Sandu, A. & Tranquilli, P. (2015). Application of approximate matrix
factorization to high order linearly implicit Runge-Kutta methods. Journal of Computational and
Applied Mathematics, 286, 196–210.
Zhu, B., Hu, Z., Tang, Y. & Zhang, R. (2016). Symmetric and symplectic methods for gyrocenter
dynamics in time-independent magnetic fields. International
eling, Simulation, and Scientific Computing, 7(02), 1650008.