**
Vijeyata Chauhan **
Department of Mathematics, Jaypee Institute of Information Technology, Noida, India.

**
Pankaj Kumar Srivastava **
Department of Mathematics, Jaypee Institute of Information Technology, Noida, India.

DOI https://dx.doi.org/10.33889/IJMEMS.2019.4.2-030

**Abstract**

The Runge-Kutta method is a one step method with multiple stages, the number of stages determine order of method. The method can be applied to work out on differential equation of the type’s explicit, implicit, partial and delay differential equation etc. The present paper describes a review on recent computational techniques for solving differential equations using Runge-Kutta algorithm of various order. This survey includes the summary of the articles of last decade till recent years based on third; fourth; fifth and sixth order Runge-Kutta methods. Along with this a combination of these methods and various other type of Runge-Kutta algorithm based articles are included. Comparisons of methods with own critical comments as remarks have been included.

**Keywords-** Runge-Kutta algorithm, Convergence of method, Implicit-Explicit method, Ordinary and partial differential equations.

**Citation**

Chauhan, V., & Srivastava, P. K. (2019). Computational Techniques Based on Runge-Kutta Method of Various Order and Type for Solving Differential Equations. *International Journal of Mathematical, Engineering and Management Sciences*, *4*(2), 375-386. https://dx.doi.org/10.33889/IJMEMS.2019.4.2-030.

**Conflict of Interest**

Both authors have contributed equally in this work. The authors declare that there is no conflict of interest for this publication.

**Acknowledgements**

The authors extend their appreciation to the anonymous reviewers for their valuable suggestions.

**References**

Ababneh, O. Y., Ahmad, R., & Ismail, E. S. (2009). New multi-step Runge-Kutta method. Applied Mathematical Sciences, 3(45), 2255-2262.

Abbasbandy, S., & Viranloo, T. A. (2002). Numerical solutions of fuzzy differential equations by Taylor method. Computational Methods in Applied Mathematics, 2(2), 113-124.

Ademiluyi, R. A., & Babatola, P. O. (2001). Semi implicit rational Runge-Kutta formulas of approximation of stiff initial value problems in ODEs. Journal of Mathematical Science and Education, 3, 1-25.

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.

Ahuja, J., & Gupta, U. (2019). Rayleigh-Bénard convection for nanofluids for more realistic boundary conditions (rigid-free and rigid-rigid) using darcy model. International Journal of Mathematical, Engineering and Management Sciences, 4(1), 139-156.

Akanbi, M. A. (2011). On 3-stage geometric explicit Runge–Kutta method for singular autonomous initial value problems in ordinary differential equations. Computing, 92(3), 243-263.

Akanbi, M. A., & Okunuga, S. A. (2005). On region of absolute stability and convergence of 3-stage multiderivative explicit Runge-Kutta methods. Journal of the Sciencea Research and Development Institute, 10, 2005-2006.

Arora, G., & Pratiksha (2019). A cumulative study on differential transform method. International Journal of Mathematical, Engineering and Management Sciences, 4(1), 170-181.

Butcher, J. C. (2000). Numerical methods for ordinary differential equations in the 20th century. Journal of Computational and Applied Mathematics, 125(1-2), 1-29.

Butcher, J. C. (2009). On fifth and sixth order explicit Runge-Kutta methods: order conditions and order barriers. Canadian Applied Mathematics Quarterly, 17(3), 433-445.

Butcher, J. C. (2016). Numerical methods for ordinary differential equations. John Wiley & Sons.

Byrne, G. D., & Lambert, R. J. (1966). Pseudo-Runge-Kutta methods involving two points. Journal of the ACM, 13(1), 114-123.

Chauhan, V., & Srivastava, P. K. (2018). Trio-Geometric mean-based three-stage Runge–Kutta algorithm to solve initial value problem arising in autonomous systems. International Journal of Modeling, Simulation, and Scientific Computing, 9(04), 1850026.

de la Cruz, H., Biscay, R. J., Jiménez, J. C., & Carbonell, F. (2013). Local linearization—Runge–Kutta methods: A class of A-stable explicit integrators for dynamical systems. Mathematical and Computer Modelling, 57(3-4), 720-740.

Dormand, J. R., & Prince, P. J. (1980). A family of embedded Runge-Kutta formulae. Journal of Computational and Applied Mathematics, 6(1), 19-26.

Fatunla, S. O. (1988). Numerical methods for initial value problems in ordinary differential equations. Academic Press. ISBN: 9781483269269.

Ghanaie, Z. A., & Moghadam, M. M. (2011). Solving fuzzy differential equations by Runge-Kutta method. The Journal of Mathematics and Computer Science, 2(2), 208-221.

Gottlieb, S. (2005). On high order strong stability preserving Runge-Kutta and multi step time discretizations. Journal of Scientific Computing, 25(1-2), 105-128.

Gunwant, D. (2019). Stress concentration studies in flat plates with rectangular cut-outs using finite element method. International Journal of Mathematical, Engineering and Management Sciences, 4(1), 66-76.

Heun, K. (1900), Neue methoden zur approximativen integration der differentialgleichungen eincr unable ver'anderliehen. Zeitschrift für angewandte Mathematik und Physik, 45, 23-38.

Huta, A. (1956). Une amélioration de la méthode de Runge-Kutta-Nystrôm pour la résolution numérique des équations différentielles du premier ordre. Acta Mathematica Universitatis Comenianae, 1, 201-224.

Islam, M. A. (2015). Accurate solutions of initial value problems for ordinary differential equations with the fourth order Runge Kutta method. Journal of Mathematics Research; Canadian Center of Science and Education, 7(3), 41-45.

Jayakumar, T., Kanagarajan, K., & Indrakumar, S. (2012). Numerical solution of nth-order fuzzy differential equation by Runge–Kutta method of order five. International Journal of Mathematical Analysis, 6(58), 2885-2896.

Kanagarajan, K., & Sambath, M. (2010). Runge-Kutta Nystrom method of order three for solving fuzzy differential equations. Computational Methods in Applied Mathematics, 10(2), 195-203.

Kanagarajan, K., Muthukumar, S., & Indrakumar, S. (2014). Numerical solution of fuzzy differential equations by extended Runge-Kutta method and the dependency problem. International Journal of Mathematics Trends and Technology, 6, 113-122.

Kaushik, A. (2019). Numerical study of 2nd incompressible flow in a rectangular domain using chorin’s projection method at high Reynolds number. International Journal of Mathematical, Engineering and Management Sciences, 4(1), 157-169.

Kutta, W. (1901). Beitrag zur näherungsweisen integration totaler differentialgleichungen. Zeitschrift für Mathematik und Physik, 46, 435–453.

Lambert, J. D. (1991). Numerical methods for ordinary differential systems: the initial value problem. John Wiley & Sons, ISBN: 978-0-471-92990-1.

Lee, J. H. J. (2004). Numerical methods for ordinary differential equations: a survey of some standard methods (Doctoral dissertation, ResearchSpace@ Auckland).

Liu, H., & Zou, J. (2006). Some new additive Runge–Kutta methods and their applications. Journal of Computational and Applied Mathematics, 190(1-2), 74-98.

Ma, M., Friedman, M., & Kandel, A. (1999). Numerical solutions of fuzzy differential equations. Fuzzy Sets and Systems, 105(1), 133-138.

Nyström, E. J. (1925). Über die numerische Integration von Differentialgleichungen: (Mitgeteilt am 23. Sept. 1925 von E. Lindelöf und KF Sundman). SocietasscientiarumFennica.

Ostermann, A., & Thalhammer, M. (2002). Convergence of Runge–Kutta methods for nonlinear parabolic equations. Applied Numerical Mathematics, 42(1-3), 367-380.

Parandin, N. (2014). Numerical solution of fuzzy differential equations of 2nd-order by Runge-Kutta method. Journal of Mathematical Extension, 7(3), 47-62.

Pederson, S., & Sambandham, M. (2007). Numerical solution to hybrid fuzzy systems. Mathematical and Computer Modelling, 45(9-10), 1133-1144.

Rabieu, F., & Ismail, F. (2011). Third order improved Runge-Kutta method for solving ordinary differential equation. International Journal of Applied Physics and Mathematics, 1(3), 191-194.

Ramadevi, B., Ramana, R. J. V., & Sugunamma, V. (2019). Influence of thermo diffusion on time dependent casson fluid flow past a wavy surface. International Journal of Mathematical, Engineering and Management Sciences, 3(4), 472-490.

Runge, C. (1895). Über die numerische auflösung von differentialgleichungen. Mathematische Annalen, 46(2), 167-178.

Saveetha, N., & Pandian, S. C. (2012). Numerical solution of fuzzy hybrid differential equation by third order Runge Kutta Nystrom method. Mathematical Theory and Modeling, 2(4), 8-17.

Seikkala, S. (1987). On the fuzzy initial value problem. Fuzzy Sets and Systems, 24(3), 319-330.

Sharmila, R. G., & Amritharaj E.C. H. (2013). Numerical solution of nth order fuzzy initial value problems by fourth order Runge-Kutta method based on centroidal mean. IOSR Journal of Mathematics, 6(3), 47-63.

Srivastava, P. K. (2014). Study of differential equations with their polynomial and nonpolynomial spline based approximation. Acta Technica Corviniensis-Bulletin of Engineering, 7(3), 139.

Srivastava, P. K., & Kumar, M. (2011). Numerical treatment of nonlinear third order boundary value problem. Applied Mathematics, 2(08), 959.

Srivastava, P. K., Kumar, M., & Mohapatra, R. N. (2012). Solution of fourth order boundary value problems by numerical algorithms based on nonpolynomial quintic splines. Journal of Numerical Mathematics and Stochastics, IV, 13-25.

Udwadia, F. E., & Farahani, A.(2008). Accelerated Runge-Kutta methods. Discrete Dynamics in Nature and Society, Article ID 790619, 38 pages. http://dx.doi.org/10.1155/2008/790619.

Wusu, A. S., Akanbi, M. A., & Okunuga, S. A. (2013). A three-stage multiderivative explicit Runge-Kutta method. American Journal of Computational Mathematics, 3(02), 121-126.

Yu, Y., Liu, Z., & Wen, L. (2014). Stability analysis of Runge-Kutta methods for nonlinear functional differential and functional equations. Journal of Applied Mathematics, Article ID 607827, 9 pages http://dx.doi.org/10.1155/2014/607827.

Yuan, H., & Song, C. (2013). Nonlinear stability and convergence of two-step Runge-Kutta methods for Volterra delay integro-differential equations. Abstract and Applied Analysis, Article ID 683137, 14 pages, http://dx.doi.org/10.1155/2013/683137.

Yuan, H., Zhao J., & Xu, Y. (2012). Nonlinear stability and D-convergence of additive Runge-Kutta methods for multidelay-integro-differential equations. Abstract and Applied Analysis, Article ID 854517, 22 pages, http://dx.doi.org/org/10.1155/2012/854517.