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Research Article

Trigonometric symmetric boundary value method for oscillating solutions including the sine-Gordon and Poisson equations

| (Reviewing Editor)
Article: 1271269 | Received 10 Oct 2016, Accepted 02 Dec 2016, Published online: 29 Dec 2016

References

  • Amodio, P., Golik, W. L., & Mazzia, F. (1995). Variable-step boundary value methods based on reverse Adams schemes and their grid redistribution. Applied Numerical Mathematics, 18, 5–21.
  • Amodio, P., & Iavernaro, F. (2006). Symmetric Boundary Methods for Second Initial and Boundary value problems. Mediterranean Journal of Mathematics, 3, 383–398.
  • Amodio, P., & Mazzia, F. (1995). Boundary value methods based on Adams. Applied Numerical Mathematics, 18, 23–35.
  • Ananthakrishnaiah, U. (1987). P-stable Obrechkoff methods with minimal phase-lag for periodic initial value problems. Mathematics of Computation, 49, 553–559.
  • Bramble, J. H., & Hubbard, B. E. (1964). On a finite difference analogue of an elliptic boundary problem which is neither diagonally dominant nor of nonnegative type. Journal of Mathematical Physics, 43, 117–132.
  • Brugnano, L., & Trigiante, D. (1998). Solving differential problems by multitep initial and boundary value methods. Amsterdam: Gordon and Breach Science Publishers.
  • Coleman, J. P., & Duxbury, S. C. (2000). Mixed collocation methods for y ″ = f(x, y). Journal of Computational and Applied Mathematics, 126, 47–75.
  • Coleman, J. P., & Ixaru, G. G. (1996). P-stability and exponential-fitting methods for y ″ = f(x, y). IMA Journal of Numerical Analysis, 16, 179–199.
  • Dai, Y., Wang, Z., & Wu, D. (2006). A four-step trigonometric fitted P-stable Obrechkoff method for periodic initial-value problems. Journal of Computational and Applied Mathematics, 225, 192–201.
  • D’Ambrosio, R., Ferro, M., & Paternoster, B. (2009). Two-step hybrid collocation methods for y ″ = f(x, y). Applied Mathematics Letters, 22, 1076–1080.
  • Dehghan, M., & Shokri, A. (2008). A numerical method for one-dimensional nonlinear sine-Gordon equation using collocation and radial basis functions. Numerical Methods for Partial Differential Equations, 24, 687–698.
  • El-Gamel, M. (2012). Sinc-collocation method for solving linear and nonlinear system of second-order boundary value problems. Applied Mathematics, 3, 1627–1633.
  • Fang, Y., Song, Y., & Wu, X. (2009). A robust trigonometrically fitted embedded pair for perturbed oscillators. Journal of Computational and Applied Mathematics, 225, 347–355.
  • Franco, J. M. (1995). An explicit hybrid method of numerov-type for second-order periodic initial value problems. Journal of Computational and Applied Mathematics, 59, 79–90.
  • Franco, J. M. (2002). Runge-Kutta-Nyström methods adapted to the numerical intergration of perturbed oscillators. Computer Physics Communications, 147, 770–787.
  • Franco, J. M., & Gomez, I. (2014). Trigonometrically fitted nonlinear two-step methods for solving second order oscillatory IVPs. Applied Mathematics and Computation, 232, 643–657.
  • Ghelardoni, P., & Marzulli, P. (1995). Stability of some boundary value methods for IVPs. Applied Numerical Mathematics, 18, 141–153.
  • Hairer, E. (1982). A one-step method of order 10 for y ″ = f(x, y). IMA Journal of Numerical Analysis, 2, 83–94.
  • Hairer, E., Nörsett, S. P., & Wanner, G. (1993). Solving ordinary differential equations I, nonstiff problems. Berlin Heidelberg: Springer-Verlag.
  • Ixaru, L., & Berghe, G. V. (2004). Exponential fitting. Dordrecht: Kluwer.
  • Ixaru, L. G., Vanden Berghe, G., & De Meyer, H. (2002). Frequency evaluation in exponential fitting multistep algorithms for ODEs. Journal of Computational and Applied Mathematics, 140, 423–434.
  • Jator, S. N. (2012). A continuous two-step method of order 8 with a block extension for y" = f (x, y, y’). Applied Mathematics and Computation, 219, 781–791.
  • Jator, S. N., Swindle, S., & French, R. (2013). Trigonometrically fitted block numerov type method for y ″ = f(x, y). Numerical Algorithms, 62, 13–26.
  • Lambert, J. D. (1991). Numerical methods for ordinary differential systems. New York, NY: John Wiley.
  • Lambert, J. D., & Watson, A. (1976). Symmetric multistep method for periodic initial value problem. Journal of the Institute of Mathematics and its Applications, 18, 189–202.
  • Nguyen, H. S., Sidje, R. B., & Cong, N. H. (2007). Analysis of trigonometric implicit Runge-Kutta methods. Journal of Computational and Applied Mathematics, 198, 187–207.
  • Ngwane, F. F., & Jator, S. N. (2013). Block hybrid method using trigonometric basis for initial value problems with oscillating solutions. Numerical Algorithms, 63, 713–725.
  • Ozawa, K. (2005). A functionally fitted three-stage explicit singly diagonally implicit Runge-Kutta method. Japan Journal of Industrial and Applied Mathematics, 22, 403–427.
  • Ramos, H., & Vigo-Aguiar, J. (2010). On the frequency choice in trigonometrically fitted methods. Applied Mathematics Letters, 23, 1378–1381.
  • Simos, T. E. (1998). An exponentially-fitted Runge-Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions. Computer Physics Communications, 115, 1–8.
  • Simos, T. E. (2002). Dissipative trigonometrically-fitted methods for second order IVPs with oscillating solution. International Journal of Modern Physics, 13, 1333–1345.
  • Sommeijer, B. P. (1993). Explicit, high-order Runge-Kutta-Nyström methods for parallel computers. Applied Numerical Mathematics, 13, 221–240.
  • Tsitouras, C. H. (2006). Explicit eight order two-step methods with nine stages for integrating oscillatory problems. International Journal of Modern Physics, 17, 861–876.
  • Twizell, E. H., & Khaliq, A. Q. M. (1984). Multiderivative methods for periodic IVPs. SIAM Journal on Numerical Analysis, 21, 111–121.
  • Vanden, G., Ixaru, L. G., & van Daele, M. (2001). Optimal implicit exponentially-fitted Runge-Kutta. Computer Physics Communications, 140, 346–357.
  • Vigo-Aguiar, J., & Ramos, H. (2003). Dissipative Chebyshev exponential-fitted methods for numerical solution of second-order differential equations. Journal of Computational and Applied Mathematics, 158, 187–211.
  • Wang, Z. (2005). P-stable linear symmetric multistep methods for periodic initial-value problems. Computer Physics Communications, 171, 162–174.
  • Wua, J., & Tian, H. (2014). Functionally-fitted block methods for ordinary differential equations. Journal of Computational and Applied Mathematics, 271, 356–368.
  • Xu, Q., & Wang, W. (2011). A new parallel iterative algorithm for solving 2D Poisson equation. Numerical Methods for Partial Differential Equations, 27, 829–853.