In this paper, a new family of two-step semi-hybrid schemes of the 12th algebraic order is proposed for the numerical simulation of initial-value problems of second-order ordinary differential equations. The proposed methods are symmetric and belong to the family of multiderivative methods. Each method of the new family appears to be hybrid, but after implementing the hybrid terms, it will continue as a multiderivative method. Therefore, the designation semi-hybrid is used. The consistency, convergence, stability, and periodicity of the methods are investigated and analyzed. In order to show the accuracy, consistency, convergence, and stability of the proposed family, it was tested on some well-known problems, such as the undamped Duffing’s equation. The simulation results demonstrate the efficiency and advantages of the proposed method compared to the currently available methods.
AbbasSBenchohraMHamidiN, et al. Hilfer and Hadamard fractional differential equations in Frchet spaces. TWMS J Pure Appl Math2019; 10: 102–116.
2.
AcharSD.Symmetric multistep Obrechkoff methods with zero phase-lag for periodic initial value problems of second order differential equations. Appl Math Comput2011; 218: 2237-2248.
3.
Anantha KrishnaiahU. P-stable Obrechkoff methods with minimal phase-lag for periodic initial value problems. Math Comp1987; 49: 553-559.
4.
AshyralyevAAgirsevenDAgarwalRP.Stability estimates for delay parabolic differential and difference equations. Appl Comput Math2020; 19: 175-204.
5.
AshyralyevAErdoganASTekalanSN.An investigation on finite difference method for the first order partial differential equation with the nonlocal boundary condition. Appl Comput Math2019; 18: 247-260.
6.
DinarvandSYousefiMChamkhaA.Numerical simulation of unsteady flow toward a stretching/shrinking sheet in porous medium filled with a hybrid nanofluid. J Appl Comput Mech. Epub ahead of print 2019. DOI: 10.22055/jacm.2019.29407.1595.
GautschiW.Numerical integration of ordinary differential equations based on trigonometric polynomials. Numer Math1961; 3: 381–397.
9.
GuliyevVSAkbulutACelikS, et al. Higher order Riesz transforms related to Schrödinger type operator on local generalized Morrey spaces. TWMS J Pure Appl Math2019; 10: 58-75.
10.
HarikrishnanSKanagarajanKElsayedEM.Existence and stability results for differential equations with complex order involving Hilfer fractional derivative. TWMS J Pure Appl Math2019; 10: 94–101.
11.
HenriciP.Discrete variable methods in ordinary differential equations. New York: Wiley, 1962.
12.
LambertJDWatsonIA.Symmetric multistep methods for periodic initial value problems. J Inst Math Appl1976; 18: 189-202.
13.
LinChHsuC-WSimosTE, et al. Explicit, semi-symmetric, hybrid, six-step, eighth order methods for solving . Appl Comput Math2019; 18: 296–304.
14.
MagagulaVMMotsaSSSibandaP.A new bivariate spectral collocation method with quadratic convergence for systems of nonlinear coupled differential equations. Appl Comput Math2019; 18: 113–122.
15.
MatvejchukM.Measure problem on conjugation logics. TWMS J Pure Appl Math2019; 10: 47-57.
16.
Mehdizadeh KhalsaraeiMShokriA. A new explicit singularly P-stable four-step method for the numerical solution of second order IVPs. Iranian J Math Chem2020; 11: 17–31.
17.
Mehdizadeh KhalsaraeiMShokriA. The new classes of high order implicit six-step P-stable multiderivative methods for the numerical solution of Schrödinger equation. Appl Comput Math2020; 19: 59-86.
18.
Mehdizadeh KhalsaraeiMShokriAMolayiM. 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 Chem2019; 10: 181–193.
19.
NajafiASpencerBJr.Validation of model-based real-time hybrid simulation for a lightly-damped and highly-nonlinear structural system. J Appl Comput Mech. Epub ahead of print 2020. DOI: 10.22055/jacm.2020.32584.2039.
20.
NetaB.P-stable symmetric super-implicit methods for periodic initial value problems. Comput Math Appl2005; 50: 701-705.
OdibatZ.Fractional power series solutions of fractional differential equations by using generalized Taylor series. Appl Comput Math2020; 19: 47-58.
23.
QalandarovAAKhaldjigitovAA.Mathematical and numerical modeling of the coupled dynamic thermoelastic problems for isotropic bodies. TWMS J Pure Appl Math2020; 11: 119–126.
24.
ShokriA.An explicit trigonometrically fitted ten-step method with phase-lag of order infinity for the numerical solution of the radial Schrödinger equation. Appl Comput Math2015; 14: 63-74.
25.
ShokriA.The multistep multiderivative methods for the numerical solution of first order initial value problems. TWMS J Pure Appl Math2016; 7: 88-97.
26.
ShokriAMehdizadeh KhalsaraeiM.A new singularly P-stable multiderivative method for the numerical solution of chemical second-order IVPs. Iranian J Math Chem. Epub ahead of print 2020. DOI: 10.22052/ijmc.2020.237471.1509.
27.
ShokriAMehdizadeh KhalsaraeiMAtashyarA.A new two-step hybrid singularly P-stable method for the numerical solution of second-order IVPs with oscillating solutions. Iranian J Math Chem2020; 11: 113–132.
28.
ShokriAMehdizadeh KhalsaraeiM.A new efficient high order four-step multiderivative method for the numerical solution of second-order IVPs with oscillating solutions. Math Interdisc Res2020; 5: 157–172.
29.
ShokriASaadatH.High phase-lag order trigonometrically fitted two-step Obrechkoff methods for the numerical solution of periodic initial value problems. Numer Alg2015; 68: 337–354.
30.
ShokriAShokriAA.Implicit one-step L-stable generalized hybrid methods for the numerical solution of first order initial value problems. Iranian J Math Chem2013; 4: 201-212.
31.
ShokriAShokriAAMostafaviSh, et al. Trigonometrically fitted two-step Obrechkoff methods for the numerical solution of periodic initial value problems. Iranian J Math Chem2015; 6: 145–161.
32.
ShokriATahmourasiM.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 Chem2017; 8: 137–159.
33.
SimosTE.A P-stable complete in phase Obrechkoff trigonometric fitted method for periodic initial value problems. Proc R Soc1993; 441: 283-289.
34.
StiefelEBettisDG.Stabilization of Cowell’s methods. Numer Math1969; 13: 154–175.
35.
SweilamNHNagyAMEl-SayedAA.Sinc-Chebyshev collocation method for time-fractional order telegraph equation. Appl Comput Math2020; 19: 162–174.
36.
TadumadzeTDvalishviliPhShavadzeT.On the representation of solution of the perturbed controlled differential equation with delay and continuous initial condition. Appl Comput Math2019; 18: 305–315.
37.
Ton-ThatH.Finite element analysis of functionally graded skew plates in thermal environment based on the new third-order shear deformation theory. J Appl Comput Mech2020; 6: 1044–1057.
38.
TrochimczukRLukaszewiczAMikolajczykT, et al. Finite element method stiffness analysis of a novel telemanipulator for minimally invasive surgery. Simulation2019; 95: 1015–1025.
39.
TuncTSarikayaMZYaldizH.Fractional hermite Hadamards type inquality for the co-ordinated convex functions. TWMS J Pure Appl Math2020; 11: 3-29.
40.
Van DaeleMVanden BergheG. P-stable exponentially fitted Obrechkoff methods of arbitrary order for second order differential equations. Numer Alg2007; 46: 333–350.
41.
Van der HouwenPJSommeijerBP. Predictor-corrector methods for periodic second-order initial-value problems. IMA J Numer Anal1987; 7: 407–422.
42.
WangZZhaoDDaiY, et al. An improved trigonometrically fitted P-stable Obrechkoff method for periodic initial value problems. Proc R Soc2005; 461: 1639–1658.
