# zbMATH — the first resource for mathematics

Exponentially fitted and trigonometrically fitted explicit modified Runge-Kutta type methods for solving $$y'''(x)=f(x,y,y')$$. (English) Zbl 1437.65072
Summary: Exponentially fitted and trigonometrically fitted explicit modified Runge-Kutta type (MRKT) methods for solving $$y'''(x)=f(x,y,y')$$ are derived in this paper. These methods are constructed which exactly integrate initial value problems whose solutions are linear combinations of the set functions $$e^{\omega x}$$ and $$e^{-\omega x}$$ for exponentially fitted and $$\sin(\omega x)$$ and $$\cos(\omega x)$$ for trigonometrically fitted with $$\omega \in R$$ being the principal frequency of the problem and the frequency will be used to raise the accuracy of the methods. The new four-stage fifth-order exponentially fitted and trigonometrically fitted explicit MRKT methods are called EFMRKT5 and TFMRKT5, respectively, for solving initial value problems whose solutions involve exponential or trigonometric functions. The numerical results indicate that the new exponentially fitted and trigonometrically fitted explicit modified Runge-Kutta type methods are more efficient than existing methods in the literature.
##### MSC:
 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65L05 Numerical methods for initial value problems involving ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems
Full Text:
##### References:
 [1] Paternoster, B., Runge-Kutta(-Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials, Applied Numerical Mathematics, 28, 2-4, 401-412, (1998) · Zbl 0927.65097 [2] Vanden Berghe, G.; De Meyer, H.; Van Daele, M.; Van Hecke, T., Exponentially fitted Runge-Kutta methods, Journal of Computational and Applied Mathematics, 125, 1-2, 107-115, (2000) · Zbl 0999.65065 [3] Simos, T. E., Exponentially fitted Runge-Kutta methods for the numerical solution of the Schrödinger equation and related problems, Computational Materials Science, 18, 3-4, 315-332, (2000) [4] Kalogiratou, Z.; Simos, T. E., Construction of trigonometrically and exponentially fitted Runge-Kutta-Nyström methods for the numerical solution of the Schrödinger equation and related problems—a method of 8th algebraic order, Journal of Mathematical Chemistry, 31, 2, 211-232, (2002) · Zbl 1002.65077 [5] Kalogiratou, Z.; Monovasilis, T.; Simos, T. E., Computation of the eigenvalues of the Schrödinger equation by exponentially-fitted Runge-Kutta-Nyström methods, Computer Physics Communications, 180, 2, 167-176, (2009) · Zbl 1198.81088 [6] Simos, T. E., Exponentially-fitted Runge-Kutta-Nyström method for the numerical solution of initial-value problems with oscillating solutions, Applied Mathematics Letters, 15, 2, 217-225, (2002) · Zbl 1003.65081 [7] Sakas, D. P.; Simos, T. E., A fifth algebraic order trigonometrically-fitted modified Runge-Kutta zonneveld method for the numerical solution of orbital problems, Mathematical and Computer Modelling, 42, 7-8, 903-920, (2005) · Zbl 1085.65061 [8] Van De Vyver, H., A Runge-Kutta-Nyström pair for the numerical integration of perturbed oscillators, Computer Physics Communications, 167, 2, 129-142, (2005) · Zbl 1196.65116 [9] Yang, H.; Wu, X., Trigonometrically-fitted {ARKN} methods for perturbed oscillators, Applied Numerical Mathematics, 58, 9, 1375-1395, (2008) · Zbl 1153.65073 [10] Demba, M. A.; Senu, N.; Ismail, F., Trigonometrically-fitted explicit four-stage fourth-order runge – kutta – nyström method for the solution of initial value problems with oscillatory behavior, The Global Journal of Pure and Applied Mathematics (GJPAM), 12, 1, 67-80, (2016) [11] Hanan, M., Oscillation criteria for third-order linear differential equations, Pacific Journal of Mathematics, 11, 919-944, (1961) · Zbl 0104.30901 [12] Rovder, J., Oscillation criteria for third-order linear differential equations, Matematický časopis, 25, 3, 231-244, (1975) · Zbl 0309.34028 [13] Lazer, A. C., The behavior of solutions of the differential equation y + p(x)y+q(x)y = 0, Pacific Journal of Mathematics, 17, 435-466, (1966) · Zbl 0143.31501 [14] Jones, G. D., Properties of solutions of a class of third-order differential equations, Journal of Mathematical Analysis and Applications, 48, 165-169, (1974) · Zbl 0289.34046 [15] Fawzi, F. A.; Senu, N.; Ismail, F.; Majid, Z. A., An efficient of direct integrator of Runge-Kutta type method for solving y”’=f(x, y, y’) with application to thin film flow problem, International Journal of Pure and Applied Mathematics, 117, 4 · Zbl 1359.65113 [16] Butcher, J. C., Numerical Methods for Ordinary Differential Equations, (2008), John Wiley & Sons, New York, NY, USA · Zbl 1167.65041 [17] Lambert, J. D., Numerical Methods for Ordinary Differential Systems. The Initial Value Problem, (1993), John Wiley & Sons, New York, NY, USA [18] Anastassi, Z. A.; Simos, T. E., Trigonometrically fitted Runge-Kutta methods for the numerical solution of the Schrödinger equation, Journal of Mathematical Chemistry, 37, 3, 281-293, (2005) · Zbl 1070.81035 [19] Momoniat, E.; Mahomed, F. M., Symmetry reduction and numerical solution of a third-order {ODE} from thin film flow, Mathematical & Computational Applications, 15, 4, 709-719, (2010) · Zbl 1248.76122 [20] Tuck, E. O.; Schwartz, L. W., A numerical and asymptotic study of some third-order ordinary differential equations relevant to draining and coating flows, SIAM Review. A Publication of the Society for Industrial and Applied Mathematics, 32, 3, 453-469, (1990) · Zbl 0705.76062 [21] Biazar, J.; Babolian, E.; Islam, R., Solution of the system of ordinary differential equations by Adomian decomposition method, Applied Mathematics and Computation, 147, 3, 713-719, (2004) · Zbl 1034.65053 [22] Mechee, M.; Senu, N.; Ismail, F.; Nikouravan, B.; Siri, Z., A three-stage fifth-order Runge-Kutta method for directly solving special third-order differential equation with application to thin film flow problem, Mathematical Problems in Engineering, 2013, (2013) · Zbl 1299.76145
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.