Efficient adaptive step-size formulation of an optimized two-step hybrid block method for directly solving general second-order initial-value problems

• Amat S, Busquier S (2017) After notes on Chebyshev’s iterative method. Appl Math Nonlinear Sci 2(1):1–2

  Article  MathSciNet  Google Scholar 

• Brugano L, Trigiante D (1998) Solving differential problems by multi-step initial and boundary value methods. Gordon and Breach Science Publishers, Amsterdam

• Butcher JC (2008) Numerical methods for ordinary differential equations. Wiley

• Calvo M, Montijano JI, Randez L (2016) A new step-size change technique for Adams methods. Appl Math Nonlinear Sci 1(2):547–558

  Article  MathSciNet  Google Scholar 

• Cordero A, Jaiswal JP, Torregrosa JR (2019) Stability analysis of fourth-order iterative method for finding multiple roots of nonlinear equations. Appl Math Nonlinear Sci 4(1):43–56

  Article  MathSciNet  Google Scholar 

• Fatunla SO (1991) Block methods for second order odes. Int J Comput Math 41:55–63

  Article  Google Scholar 

• Hairer E, Nörsett SP, Wanner G (1993) Solving ordinary differential equations-I. Springer, Berlin

  MATH  Google Scholar 

• Jator SN (2010a) On a class of Hybrid methods for \(y^{\prime \prime }=f(x, y, y^{\prime })\). Int J Pure Appl Math 59(4):381–395

  MathSciNet  MATH  Google Scholar 

• Jator SN (2010b) Solving second order initial value problems by a hybrid multi-step method without predictors. Appl Math Comp 217:4036–4046

  Article  Google Scholar 

• Lambert JD (1991) Numerical methods for ordinary differential systems: the initial value problem. Wiley

• Majid ZA, Azmi NA, Suleiman M (2009) Solving second order ordinary differential equations using two point four step direct implicit block method. Eur J Sci Res 31:29–36

• Milne WE (1953) Numerical solution of differential equations. Wiley

• Monovasils T, Kalogiratou Z, Ramos H, Simos TE (2017) Modified two-step hybrid methods for the numerical integration of oscillatory problems. Math Meth Appl Sci. https://doi.org/10.1002/mma.4386

  Article  MathSciNet  MATH  Google Scholar 

• Omar Z (2015) Developing a single step hybrid block method with generalized three off-step points for the direct solution of second order ordinary differential equations. Int J Math Anal 9(46):2257–2272

  Article  Google Scholar 

• Petkovic MS, Neta B, Petkovic LD, Dzunic J (2013) Multipoint methods for solving nonlinear equations. Elsevier

• Qureshi S, Ramos H (2018) L-stable explicit nonlinear method with constant and variable step-size formulation for solving initial value problems. Int J Nonlinear Sci Numer Simul 19(7–8):741–751

  Article  MathSciNet  Google Scholar 

• Ramos H, Rufai MA (2018) Third derivative modification of k-step block Falkner methods for the numerical solution of second order initial-value problems. Appl Math Comp 333:231–245

  Article  MathSciNet  Google Scholar 

• Ramos H, Singh G (2017a) A note on variable step-size formulation of a Simpson’s-type second derivative block method for solving stiff systems. Appl Math Lett 64:101–107

  Article  MathSciNet  Google Scholar 

• Ramos H, Singh G (2017b) A tenth order A-stable two-step hybrid block method for solving initial value problems of ODEs. Appl Math Comp 310:75–88

  Article  MathSciNet  Google Scholar 

• Ramos H, Vigo Aguiar J (2005) Variable step-size Störmer-Cowell methods. Math Comput Model 42:837–846

  Article  Google Scholar 

• Ramos H, Kalogiratou Z, Monovasilis T, Simos TE (2016a) An optimized two-step hybrid block method for solving general second order initial value problems. Numer Algebra 72(4):1089–1102

  Article  MathSciNet  Google Scholar 

• Ramos H, Singh G, Kanwar V, Bhatia S (2016b) An efficient variable step-size rational Falkner-type method for solving the special second-order IVP. Appl Math Compt 291:39–51

  Article  MathSciNet  Google Scholar 

• Ramos H, Mehta S, Vigo Aguiar J (2017) A unified approach for the development of \(k\)-step block Falkner-type methods for solving general second-order initial-value problems in ODEs. J Comp Appl Math 318:550–564

  Article  MathSciNet  Google Scholar 

• Calvo M, Montijano JI, Randez L (2016) A new step-size change technique for Adams methods. Appl Math Nonlinear Sci 1(2):547–558

• Rosser JB (1967) A Runge-Kutta for all reasons. SIAM Rev 9:417–452

  Article  MathSciNet  Google Scholar 

• Sallam S, Anwar N (2002) Sixth order C 2-spline collocation method for integrating second orderordinary initial value problems. Int J Comput Math 79:625–635

  Article  MathSciNet  Google Scholar 

• Sedgwick AE (1973) An effective variable order variable step Adams method, Report 53, Department of Computer Science, University of Toronto, Toronto

• Shampine LF, Gordon MK (1975) Computer solution of ordinary differential equations: the initial value problem. Freeman, San Francisco

  MATH  Google Scholar 

• Shampine LF, Watts HA (1969) Block implicit one-step methods. Math Comp 23:731–740

  Google Scholar 

• Singh G, Ramos H (2019) An optimized two-step hybrid block method formulated in variable step-size mode for integrating \(y^{\prime \prime } =f (x, y, y^{\prime })\) numerically. Numer Math Theor Meth Appl 12:331–347

  Article  MathSciNet  Google Scholar 

• Vigo Aguiar J, Ramos H (2006) Variable step-size implementation of multi-step methods for \(y^{\prime \prime }=f (x, y, y^{\prime })\). J Comput Appl Math 192:114–131

  Article  MathSciNet  Google Scholar 

• Watts HA (1983) Starting step size for an ODE solver. J Comp Appl Math 9:177–191

  Article  MathSciNet  Google Scholar 

Download references