Document Type : Original Scientific Paper

**Authors**

Faculty of Mathematical Science, University of Maragheh, Maragheh, I. R. Iran

**Abstract**

In this paper, we present a new high order explicit four-step method of eighth algebraic order for solving second-order linear periodic and oscillatory initial value problems of ordinary differential equations such as undamped Duffing's equation. Numerical stability and phase properties of the new method is analyzed. The main structure of the method is multiderivative, and the combined phases were applied to expand the stability interval and to achieve P-stability. The advantage of the method in comparison with similar methods in terms of efficiency, accuracy, and stability is shown by its implementation in some well-known problems.

**Keywords**

[1] S. Abbas, M. Benchohra, N. Hamidi and J. J. Nieto, Hilfer and Hadamard fractional differential equations in Frchet spaces, TWMS J. Pure Appl. Math. **10** (1) (2019) 102 - 116.

[2] S. D. Achar, Symmetric multistep Obrechkoff methods with zero phase-lag for periodic initial value problems of second order differential equations, J. Appl. Math. Comput. **218** (2011) 2237 - 2248.

[3] A. Ashyralyev, D. Agirseven and R. P. Agarwal, Stability estimates for delay parabolic differential and difference equations, Appl. Comput. Math. **19** (2) (2020) 175-204.

[4] A. Ashyralyev, A. S. Erdogan and S. N. Tekalan, An investigation on finite difference method for the first order partial differential equation with the nonlocal boundary condition, Appl. Comput. Math. **18** (3) (2019) 247 - 260.

[5] J. M. Franco and M. Palacios, High-order P-stable multistep methods, J. Comput. Appl. Math. **30** (1) (1990) 1 - 10.

[6] J. D. Lambert and I. A. Watson, Symmetric multistep methods for periodic initial value problems, J. Inst. Math. Appl. **18** (1976) 189 - 202.

[7] Q. Li and X. Y. Wu, A two-step explicit P-stable method for solving second order initial value problems, Appl. Math. Comput. **138** (2-3) (2003) 435 - 442.

[8] Q. Li and X. Y. Wu, A two-step explicit P-stable method of high phase-lag order for second order IVPs, Appl. Math. Comput. **151** (1) (2004) 17 - 26.

[9] Ch. Lin, Ch. -W. Hsu, T. E. Simos and Ch. Tsitouras, Explicit, semisymmetric, hybrid, six-step, eighth order methods for solving y′′ = f(x; y), Appl. Comput. Math. **18** (3) (2019) 296 - 304.

[10] V. M. Magagula, S. S. Motsa and P. Sibanda, A new bivariate spectral collocation method with quadratic convergence for systems of nonlinear coupled differential equations, Appl. Comput. Math. **18** (2) (2019) 113 - 122.

[11] M. Mehdizadeh Khalsaraei and A. Shokri, A new explicit singularly P-stable four-step method for the numerical solution of second order IVPs, Iranian J. Math. Chem. **11** (1) (2020) 17 - 31.

[12] M. Mehdizadeh Khalsaraei and A. Shokri, The new classes of high order implicit six-step P-stable multiderivative methods for the numerical solution of schrödinger equation, Appl. Comput. Math. **19** (1) (2020) 59 - 86.

[13] M. Mehdizadeh Khalsaraei, A. Shokri and M. Molayi, The new high approximation of stiff systems of first ordinary IVPs arising from chemical reactions by k-step L-stable hybrid methods, Iranian J. Math. Chem. **10** (2) (2019) 181 - 193.

[14] B. Neta, P-stable symmetric super-implicit methods for periodic initial value problems, Comput. Math. Appl. **50** (5-6) (2005) 701 - 705.

[15] S. Harikrishnan, K. Kanagarajan and E. M. Elsayed, Existence and stability results for differential equations with complex order involving Hilfer fractional derivative, TWMS J. Pure Appl. Math. **10 **(1) (2019) 94 - 101.

[16] H. Ramos, Development of a new Runge-Kutta method and its economical implementation, Comput. Math. Methods **1 **(2) (2019) *e*1016.

[17] H. Ramos and J. Vigo-Aguiar, On the frequency choice in trigonometrically fitted methods, J. Appl. Math. Lett. **23 **(11) (2010) 1378 *- *1381.

[18] A. Shokri, A new eight-order symmetric two-step multiderivative method for the numerical solution of second-order IVPs with oscillating solutions, Numer. Algor. **77 **(1) (2018) 95 *- *109.

[19] A. Shokri and M. Mehdizadeh Khalsaraei, A new singularly P-stable multiderivative method for the numerical solution of chemical second-order IVPs, Iranian J. Math. Chem. (2020) DOI: 10.22052/ijmc.2020.237471.1509.

[20] A. Shokri and M. Mehdizadeh Khalsaraei, An efficient high order fourstep multiderivative method for the numerical solution of second-order IVPs with oscillating solutions, Comput. Math. Methods (2020) DOI: 10.1002/cmm4.1116.

[21] A. Shokri, M. Mehdizadeh Khalsaraei and A. Atashyar, A new two-step hybrid singularly P-stable method for the numerical solution of secondorder IVPs with oscillating solutions, Iranian J. Math. Chem. **11** (2) (2020) 113 - 132.

[22] A. Shokri and A. A. Shokri, Implicit one-step L-stable generalized hybrid methods for the numerical solution of first order initial value problems, Iranian J. Math. Chem. **4 **(2) (2013) 201 *- *212.

[23] A. Shokri and M. Tahmourasi, A new two-step Obrechkoff method with vanished phase-lag and some of its derivatives for the numerical solution of radial Schrödinger equation and related IVPs with oscillating solutions, Iranian J. Math. Chem. **8 **(2) (2017) 137 *- *159.

[24] A. Shokri, A. A. Shokri, Sh. Mostafavi and H. Saadat, Trigonometrically fitted two-step Obrechkoff methods for the numerical solution of periodic initial value problems, Iranian J. Math. Chem.**6 **(2) (2015) 145 *- *161.

[25] A. Shokri, J. Vigo-Aguiar, M. Mehdizadeh Khalsaraei and R. Garcia-Rubio, A new class of two-step P-stable TFPL methods for the numerical solution of second order IVPs with oscillating solutions, J. Comput. Appl. Math. **354 **(2019) 551 *- *561.

[26] A. Shokri, J. Vigo-Aguiar, M. Mehdizadeh Khalsaraei and R. Garcia-Rubio, A new four-step P-stable Obrechkoff method with vanished phase-lag and some of its derivatives for the numerical solution of radial Schrödinger equation, J. Comput. Appl. Math. **354** (2019) 569 - 586.

[27] A. Shokri, J. Vigo-Aguiar, M. Mehdizadeh Khalsaraei and R. Garcia-Rubio, A new implicit six-step P-stable method for the numerical solution of Schrödinger equation, Int. J. Comput. Math. **97** (4) (2020) 802 - 817.

[28] T. E. Simos, A P-stable complete in phase Obrechkoff trigonometric fitted method for periodic initial value problems, Proc. R. Soc. **441** (1993) 283-289.

[29] T. E. Simos and Ch. Tsitouras, High phase-lag order, four-step methods for solving y′′ = f(x; y), Appl. Comput. Math. **17** (3) (2018) 307 - 316.

[30] T. E. Simos and P. S. Williams, A finite-difference method for the numerical solution of the Schrödinger equation, J. Comput. Appl. Math. **79** (2) (1997) 189 - 205.

[31] N. H. Sweilam, A. M. Nagy and A. A. El-Sayed, Sinc-Chebyshev collocation method for time-fractional order telegraph equation, Appl. Comput. Math. **19 **(2) (2020) 162 - 174.

[32] T. Gadjiev, S. Aliev and Sh. Galandarova, A priori estimates for solutions to Dirichlet boundary value problems for polyharmonic equations in generalized Morrey spaces, TWMS J. Pure Appl. Math. **9 **(2) (2018) 231 - 242.

[33] M. Van Daele and G. Vanden Berghe, P-stable exponentially fitted Obrechkoff methods of arbitrary order for second order differential equations, Numer. Algor. **46** (2007) 333 - 350.

[34] J. Vigo-Aguiar and H. Ramos, Variable stepsize implementation of multistep methods for y′′ = f(x; y; y′), J. Comput. Appl. Math. **192** (2006) 114 - 131.

[35] Z. Wang, D. Zhao, Y. Dai and D. Wu, An improved trigonometrically fitted P-stable Obrechkoff method for periodic initial value problems, Proc. R. Soc. **461** (2005) 1639 - 1658.

June 2020

Pages 157-172