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Trigonometrically fitted nonlinear two-step methods for solving second order oscillatory IVPs

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  • Franco, J.M.
  • Gómez, I.

Abstract

The construction of a class of nonlinear two-step methods for solving second order oscillatory IVPs is analyzed. These methods are exact for the linear space generated by the set of functions {sin(ωt),cos(ωt)} (trigonometrically fitted methods) but they do not require a previous knowledge of the frequency ω or a good approximation of it. Some trigonometrically fitted nonlinear two-step schemes with algebraic order up to four are derived and their stability and phase properties are analyzed. It is shown that the new schemes can be efficiently implemented by using a special vector operation (the vector product and quotient). The numerical experiments carried out show that the new nonlinear two-step schemes are more efficient than other standard and nonlinear methods proposed in the scientific literature for solving second order oscillatory differential systems.

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  • Franco, J.M. & Gómez, I., 2014. "Trigonometrically fitted nonlinear two-step methods for solving second order oscillatory IVPs," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 643-657.
    • Handle: RePEc:eee:apmaco:v:232:y:2014:i:c:p:643-657
    DOI: 10.1016/j.amc.2014.01.078
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    Cited by:

    1. Vladislav N. Kovalnogov & Ruslan V. Fedorov & Tamara V. Karpukhina & Theodore E. Simos & Charalampos Tsitouras, 2021. "Sixth Order Numerov-Type Methods with Coefficients Trained to Perform Best on Problems with Oscillating Solutions," Mathematics, MDPI, vol. 9(21), pages 1-12, October.
    2. Vladislav N. Kovalnogov & Ruslan V. Fedorov & Dmitry A. Generalov & Ekaterina V. Tsvetova & Theodore E. Simos & Charalampos Tsitouras, 2022. "On a New Family of Runge–Kutta–Nyström Pairs of Orders 6(4)," Mathematics, MDPI, vol. 10(6), pages 1-15, March.
    3. Houssem Jerbi & Sondess Ben Aoun & Mohamed Omri & Theodore E. Simos & Charalampos Tsitouras, 2022. "A Neural Network Type Approach for Constructing Runge–Kutta Pairs of Orders Six and Five That Perform Best on Problems with Oscillatory Solutions," Mathematics, MDPI, vol. 10(5), pages 1-10, March.
    4. Franco, J.M. & Khiar, Y. & Rández, L., 2015. "Two new embedded pairs of explicit Runge–Kutta methods adapted to the numerical solution of oscillatory problems," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 45-57.
    5. Vladislav N. Kovalnogov & Ruslan V. Fedorov & Andrey V. Chukalin & Theodore E. Simos & Charalampos Tsitouras, 2021. "Evolutionary Derivation of Runge–Kutta Pairs of Orders 5(4) Specially Tuned for Problems with Periodic Solutions," Mathematics, MDPI, vol. 9(18), pages 1-11, September.
    6. Higinio Ramos & Ridwanulahi Abdulganiy & Ruth Olowe & Samuel Jator, 2021. "A Family of Functionally-Fitted Third Derivative Block Falkner Methods for Solving Second-Order Initial-Value Problems with Oscillating Solutions," Mathematics, MDPI, vol. 9(7), pages 1-22, March.

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