This research paper deals with a numerical algorithm based on the RCW method to solve a set of first-order ordinary differential equations. In this algorithm, the answer of each equation is considered as a polynomial of degree 2. The coefficients used in this polynomial are categorized into groups of free and fixed coefficients. The free coefficients are obtained by optimizing the error function, while the fixed coefficients are computed from the magnitudes of derivative of this polynomial at initial value. To check the correctness of the current algorithm, its results for several case problems are compared with the findings of the Runge-Kutta (RK) method. This comparison shows that the algorithm presented in this paper has a higher accuracy and stability than the RK method even for the nonlinear problems.
Amodio P, Brugnano L, Iavernaro F. Analysis of spectral Hamiltonian boundary value methods (SHBVMs) for the numerical solution of ODE problems. *Numerical Algorithms*, First online. 2019;1–20.
2.
Atashafrooz M, Bahaadini R, Sheibani HR. Nonlocal, strain gradient and surface effects on vibration and instability of nanotubes conveying nanoflow. In: *Mechanics of Advanced Materials and Structures*, First online. 2018. p. 1–13.
3.
Atashafrooz M, Asadi T. The effects of buoyancy force on the irreversibility of three-dimensional step flow in an inclined duct. *Journal of the Serbian Society for Computational Mechanics*. 2019;13(1):1–16.
4.
Atashafrooz M, Sheikholeslami M, Sajjadi H, Delouei AA. Interaction effects of an inclined magnetic field and nanofluid on forced convection heat transfer and flow irreversibility in a duct with an abrupt contraction. *Journal of Magnetism and Magnetic Materials*. 2019;478:216–26.
5.
Blinder SM. *Guide to essential math: A review for physics, chemistry and engineering students*. 2013.
6.
Brugnano L, Magherini C. Recent advances in linear analysis of convergence for splittings for solving ODE problems. *Applied Numerical Mathematics*. 2009;59:542–57.
7.
Charroyer L, Chiello O, Sinou JJ. Self-excited vibrations of a non-smooth contact dynamical system with planar friction based on the shooting method. *International Journal of Mechanical Sciences*. 2018;144:90–101.
8.
Cutolo A, Piccoli B, Rarità L. An upwind-Euler scheme for an ODE-PDE model of supply chains. *SIAM Journal on Scientific Computing*. 2011;33(4):1669–88.
9.
Esmaeilpour M, Ganji DD. Application of He’s homotopy perturbation method to boundary layer flow and convection heat transfer over a flat plate. *Physics Letters A*. 2007;372(1):33–8.
10.
Filipov SM, Gospodinov ID, Faragó I. Replacing the finite difference methods for nonlinear two-point boundary value problems by successive application of the linear shooting method. *Journal of Computational and Applied Mathematics*. 2019;358:46–60.
11.
Franco JM. Exponentially fitted symplectic integrators of RKN type for solving oscillatory problems. *Computer Physics Communications*. 2007;177(6):479–92.
12.
Ha SN. A nonlinear shooting method for two-point boundary value problems. *Computers & Mathematics with Applications*. 2001;42(10–11):1411–20.
13.
Jaskulski M, Atuonwu JC, Tran TTH, Stapley AGF, Tsotsas E. Predictive CFD modeling of whey protein denaturation in skim milk spray drying powder production. *Advanced Powder Technology*. 2017;28(12):3140–7.
14.
Kang J, Zhang D, Zhou F, Li H, Xia T. Numerical modeling and experimental validation of fractional heat transfer induced by gas adsorption in heterogeneous coal matrix. *International Journal of Heat and Mass Transfer*. 2019;128:492–503.
15.
Korkmaz A. Explicit exact solutions to some one-dimensional conformable time fractional equations. *Waves in Random and Complex Media*. 2019;29(1):124–37.
16.
Kumar MS, Sandeep N, Kumar BR, Saleem S. Effect of aligned magnetic field on MHD squeezing flow of Casson fluid between parallel plates. *Defect and Diffusion Forum*. 2018;384:1–11.
17.
Li YW, Wu X. Functionally fitted energy-preserving methods for solving oscillatory nonlinear Hamiltonian systems. *SIAM Journal on Numerical Analysis*. 2016;54(4):2036–59.
18.
Mahmoodabadi MJ, Mahmoodabadi F, Atashafrooz M. Development of the meshless local Petrov-Galerkin method to analyze three-dimensional transient incompressible laminar fluid flow. *Journal of the Serbian Society for Computational Mechanics*. 2018;12(2):128–52.
19.
Mehralian F, Beni YT. Vibration analysis of size-dependent bimorph functionally graded piezoelectric cylindrical shell based on nonlocal strain gradient theory. *Journal of the Brazilian Society of Mechanical Sciences and Engineering*. 2018;40(1):27.
20.
Rahmanzadeh M, Cai L, White RE. A new method for solving initial value problems. *Computers & Chemical Engineering*. 2013;58:33–9.
21.
Rahmanzadeh M, Barfeie M. An explicit time-stepping method based on error minimization for solving stiff system of ordinary differential equations. *Malaysian Journal of Mathematical Sciences*. 2018;12(2):267–83.
22.
Rahmanzadeh M, Asadi T, Atashafrooz M. The development and application of the RCW method for the solution of the Blasius problem. *Journal of Applied and Computational Mechanics*. 2020;6(1):105–11.
23.
Rashid M, Khan MI, Hayat T, Khan MI, Alsaedi A. Entropy generation in flow of ferromagnetic liquid with nonlinear radiation and slip condition. *Journal of Molecular Liquids*. 2019;276:441–52.
24.
Reis M, Kromer JA, Klipp E. General solution of the chemical master equation and modality of marginal distributions for hierarchic first-order reaction networks. *Journal of Mathematical Biology*. 2018;77(2):377–419.
25.
Sajjadi H, Delouei AA, Atashafrooz M, Sheikholeslami M. Double MRT Lattice Boltzmann simulation of 3-D MHD natural convection in a cubic cavity with sinusoidal temperature distribution utilizing nanofluid. *International Journal of Heat and Mass Transfer*. 2018;126:489–503.
26.
Sajjadi H, Delouei AA, Sheikholeslami M, Atashafrooz M, Succi S. Simulation of three-dimensional MHD natural convection using double MRT Lattice Boltzmann method. *Physica A: Statistical Mechanics and its Applications*. 2019;515:474–96.
27.
Shah Z, Gul T, Islam S, Khan MA, Bonyah E, Hussain F, et al. Three-dimensional third grade nanofluid flow in a rotating system between parallel plates with Brownian motion and thermophoresis effects. *Results in Physics*. 2018;10:36–45.
28.
Sheikholeslami M, Sajjadi H, Delouei AA, Atashafrooz M, Li Z. Magnetic force and radiation influences on nanofluid transportation through a permeable media considering Al2O3 nanoparticles. *Journal of Thermal Analysis and Calorimetry*. 2019;136(6):2477–85.
29.
Tang W, Sun Y. Time finite element methods: A unified framework for numerical discretizations of ODEs. *Applied Mathematics and Computation*. 2012;219(4):2158–79.
30.
Wang B, Meng F, Fang Y. Efficient implementation of RKN-type Fourier collocation methods for second-order differential equations. *Applied Numerical Mathematics*. 2017;119:164–78.
31.
Zeeshan A, Majeed A. Heat transfer analysis of Jeffery fluid flow over a stretching sheet with suction/injection and magnetic dipole effect. *Alexandria Engineering Journal*. 2016;55:2171–81.
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