Differential Equation Solver Simulator for Runge-Kutta Methods

Metin HATUN, Fahri VATANSEVER
2.091 358

Abstract


Many of problems in engineering and science is modeled by differential equations mathematically, therefore their solutions have an important role. Many methods have been developed for analytical or numerical solutions of differential equations. In proportion to the development of technology, the numerical solution methods are utilized widely. In particular, the main objectives in real time applications are to reach the correct solution as soon as possible with minimal processing and maximum precision. In the performed study, a simulator that contains Runge-Kutta based 48 methods was developed for numerical solution of differential equations. In the user friendly simulator which can be used also for educational purposes, the solution of defined differential equation under the specified initial condition with given step size or according to the number of points requested within the specified range can be obtained by the selected method. Solutions can be presented to the user both numerical (step values, computation time) and graphically; also the subject explanations about the methods/solutions can be given. Furthermore, the comparative solutions (performance analysis) can be implemented by the simulator. So, the users can realize the numerical solutions of differential equations with different methods by the simulator; the students learn the methods in this field visually with the aid of subject explanation and can implement step by step; the designers can choose the most appropriate method easily, effectively and accurately for their systems by the comparative analysis.

Keywords


Ordinary differential equation; Runge-Kutta methods; Simulator

Full Text:

PDF


DOI: http://dx.doi.org/10.17482/uujfe.70981

References


Ababneh, O.Y. and Rozita, R. (2009) New third order Runge Kutta based on contraharmonic mean for stiff problems, Applied Mathematical Sciences, 3(8), 365-376.

Abraham, O. and Bolarin, G. (2011) On error estimation in Runge-Kutta methods, Leonardo Journal of Sciences, 10(18), 1-10.

Ahmad, R.R. and Yaacob, N. (2005). Third-order composite Runge–Kutta method for stiff problems, International Journal of Computer Mathematics, 82(10), 1221-1226. doi: 10.1080/00207160512331331039

Ahmad, R.R. and Yaacob, N. (2013) Arithmetic-mean Runge-Kutta method and method of lines for solving mildly stiff differential equations, Menemui Matematik (Discovering Mathematics), 35(2), 21-29.

Butcher, J.C. (1964) On Runge-Kutta processes of higher order, Journal of the Australian Mathematical Society, 4(2), 179-197.

Butcher, J.C. (1969) The effective order of Runge-Kutta methods, Lecture Notes in Mathematics, 109, 133-139.

Butcher, J.C. and Johnston, P.B. (1993) Estimating local truncation errors for Runge-Kutta methods, Journal of Computational and Applied Mathematics, 45(1-2), 203-212. doi:10.1016/0377-0427(93)90275-G

Butcher, J.C. (1994) Initial value problems: numerical methods and mathematics, Computers and Mathematics with Applications, 28(10-12), 1-16. doi:10.1016/0898-1221(94)00182-0

Butcher, J.C. (1996) A history of Runge-Kutta methods, Applied Numerical Mathematics, 20(3), 247-260. doi:10.1016/0168-9274(95)00108-5

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. doi:10.1016/S0377-0427(00)00455-6

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. (2010) Trees and numerical methods for ordinary differential equations, Numerical Algorithms, 53(2-3), 153-170. 10.1007/s11075-009-9285-0

Chapra, S.C. and Canale, R.P. (2002) Numerical Methods for Engineers: with Software and Programming Applications, 4th ed., McGraw-Hill, New York.

Derr, L., Outlaw, C., and Sarafyan, D. (1993) A new method for derivation of continuous Runge-Kutta formulas, Computers and Mathematics with Applications, 26(3), 7-13. doi:10.1016/0898-1221(93)90105-5

England, R. (1969) Error estimates for Runge-Kutta type solutions to systems of ordinary differential equations, The Computer Journal, 12(2), 166-170.

Evans, D.J. (1989) New Runge-Kutta methods for initial value problems, Applied Mathematics Letters, 2(1), 25-28. doi:10.1016/0893-9659(89)90109-2

Gill, S. (1951) A process for the step-by-step integration of differential equations in an automatic digital computing machine, Mathematical Proceedings of the Cambridge Philosophical Society, 47(1), 96-108.

http://en.wikipedia.org/wiki/List_of_numerical_analysis_software, Access date: 11.03.2015, Subject: List of numerical analysis software.

http://en.wikipedia.org/wiki/Comparison_of_numerical_analysis_software, Access date: 11.03.2015, Subject: Comparison of numerical analysis software.

http://onsolver.com/diff-equation.php, Access date: 17.06.2015, Subject: Solving of differential equations online.

http://www.wolframalpha.com/widgets/view.jsp?id=e602dcdecb1843943960b5197efd3f2a, Access date: 17.06.2015, Subject: General Differential Equation Solver.

https://www.symbolab.com/solver/ordinary-differential-equation-calculator, Access date: 17.06.2015, Subject: Solver Ordinary Differential Equations Calculator.

http://calculator.tutorvista.com/differential-equation-calculator.html, Access date: 17.06.2015, Subject: Differential Equation Calculator.

http://www.zweigmedia.com/RealWorld/deSystemGrapher/func.html, Access date: 17.06.2015, Subject: Two Dimensional Differential Equation Solver and Grapher V 1.0.

https://www.easycalculation.com/differentiation/first-order-ode-calculator.php, Access date: 17.06.2015, Subject: Homogeneous Differential Equations Calculator.

https://play.google.com/store/apps/details?id=org.krapp.diffequals, Access date: 17.06.2015, Subject: Differential equations.

http://www.appszoom.com/android-app/coupled-differential-equations-mhsxd.html?, Access date: 17.06.2015, Subject: Coupled Differential Equations.

https://play.google.com/store/apps/details?id=com.freddieandbrucie.myalevelmathstutor.firstorderdes, Access date: 17.06.2015, Subject: Differential Equations I.

https://play.google.com/store/apps/details?id=com.freddieandbrucie.myalevelmathstutor.secondorderdes, Access date: 17.06.2015, Subject: Differential Equations II.

http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/, Access date: 17.06.2015, Subject: Differential Equations.

https://www.khanacademy.org/math/differential-equations, Access date: 17.06.2015, Subject: Differential equations.

http://www.sosmath.com/diffeq/diffeq.html, Access date: 17.06.2015, Subject: Differential Equations.

Kopal, Z. (1955) Numerical Analysis, John Wiley & Sons, New-York.

Luther, H.A. and Konen, H.P. (1965) Some fifth-order classical Runge-Kutta formulas, SIAM Review, 7(4), 551-558. doi:10.1137/1007112

Luther, H.A. (1966) Further explicit fifth-order Runge-Kutta formulas, SIAM Review, 8(3), 374-380. doi:10.1137/1008073

Luther, H.A. (1967) An explicit sixth-order Runge-Kutta formula, Mathematics of Computation, 22(102), 434-436. doi:http://dx.doi.org/10.1090/S0025-5718-68-99876-1

MATLAB, The Mathworks Inc., 2007.

Murugesan, K., Dhayabaran, D.P., Amirtharaj, E.C.H. and Evans, D.J. (2001) A comparison of extended Runge-Kutta formulae based on variety of means to solve system of ivps, International Journal of Computer Mathematics, 78(2), 225-252. doi:10.1080/00207160108805108

Ralston, A. (1962) Runge-Kutta methods with minimum error bounds. Mathematics of Computation, 16(80), 431-437. doi:http://dx.doi.org/10.1090/S0025-5718-1962-0150954-0

Ralston, A. and Rabinowitz, P. (1978) A First Course in Numerical Analysis, 2nd ed., McGraw-Hill, New-York.

Vatansever, F. (2006) İleri Programlama Uygulamaları, Seçkin Yayıncılık, Ankara.

Wazwaz, A.M. (1990) A modified third order Runge-Kutta method, Applied Mathematics Letters, 3(3), 123-125. doi:10.1016/0893-9659(90)90154-4

Wazwaz, A.M. (1991) Modified numerical methods based on arithmetic and geometric means, Applied Mathematics Letters, 4(3), 49-52. doi:10.1016/0893-9659(91)90034-S

Wazwaz, A.M. (1994) A comparison of modified Runge-Kutta formulas based on a variety of means, International Journal of Computer Mathematics, 50(1-2), 105-112. doi: 10.1080/00207169408804245




Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.