Computational Techniques Based on Runge-Kutta Method of Various Order and Type for Solving Differential Equations (original) (raw)

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.

Key takeaways

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  1. The Runge-Kutta method is critical for solving ordinary differential equations (ODEs) with various orders and types.
  2. Recent advancements include third to sixth order methods, enhancing convergence and stability in numerical solutions.
  3. The paper reviews multiple computational techniques and their applications, summarizing findings from the last decade.
  4. Hybrid approaches combining Runge-Kutta methods show potential in improving accuracy for fuzzy differential equations.
  5. Comparative analyses indicate that higher-order Runge-Kutta methods are efficient and accurate for diverse differential equation systems.

Figures (1)

8. Application of Various Other type of R-K Method (iii) R-K method of fourth and fifth order was developed for the solution of dependency problem in fuzzy computation (Kanagarajan et al., 2014. The equation under consideration was Ue Ay pssweewenss UE VY GOR BUD NORE Sy rY NR BN BS AVEC (i) Gottlieb has presented algorithm to solve hyperbolic PDEs through lines approximations using semi-discrete stability of time discretization (Gottlieb, 2005). The author described the connection of the time step limit on SSP schemes with the assumption of contractivity and monotonicity development of SSP techniques and the recently developed theory. He has also considered the optimal explicit SSP algorithms to solve linear problems and nonlinear problems.

8. Application of Various Other type of R-K Method (iii) R-K method of fourth and fifth order was developed for the solution of dependency problem in fuzzy computation (Kanagarajan et al., 2014. The equation under consideration was Ue Ay pssweewenss UE VY GOR BUD NORE Sy rY NR BN BS AVEC (i) Gottlieb has presented algorithm to solve hyperbolic PDEs through lines approximations using semi-discrete stability of time discretization (Gottlieb, 2005). The author described the connection of the time step limit on SSP schemes with the assumption of contractivity and monotonicity development of SSP techniques and the recently developed theory. He has also considered the optimal explicit SSP algorithms to solve linear problems and nonlinear problems.

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