Construction of an Optimal Explicit Difference Formula in the Hilbert Space \(W_{2}^{(3,2)} (0,1)\)

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DOI: https://doi.org/10.28924/2291-8639-23-2025-335
Published: Dec 12, 2025
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R.S. Karimov, H.Sh. Rustamov, D.D. Atoev, Z.Z. Shirinov, Construction of an Optimal Explicit Difference Formula in the Hilbert Space \(W_{2}^{(3,2)} (0,1)\), Int. J. Anal. Appl., 23 (2025), 335.

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R.S. Karimov, H.Sh. Rustamov, D.D. Atoev, Z.Z. Shirinov

Abstract

In this paper, an optimal explicit difference formula for any integer k in the Hilbert space W2(3,2)(0, 1) is constructed. Here, an optimal function of the discrete argument is found, which is used to find the optimal error estimate for the explicit difference formula on the class W2(3,2)(0, 1). Furthermore, an optimal error estimate for the explicit difference formula on the class W2(3,2)(0, 1) is calculated. In addition, using the optimal function of the discrete argument, a new formula for the optimal error of explicit difference formulas on the classes W2(m,m−1)(0, 1) is obtained.

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References

  1. R.L. Burden, J.D. Faires, A.M. Burden, Numerical Analysis, Cengage Learning, (2016).
  2. S. Mehrkanoon, Z.A. Majid, M. Suleiman, K.I. Othman, Z.B. Ibrahim, 3-Point Implicit Block Multistep Method for the Solution of First Order ODEs, Bull. Malays. Math. Sci. Soc. (2) 35 (2012), 547–555.
  3. L. Beuken, O. Cheffert, A. Tutueva, D. Butusov, V. Legat, Numerical Stability and Performance of Semi-Explicit and Semi-Implicit Predictor?Corrector Methods, Mathematics 10 (2022), 2015. https://doi.org/10.3390/math10122015.
  4. A. Tutueva, D. Butusov, Stability Analysis and Optimization of Semi-Explicit Predictor?Corrector Methods, Mathematics 9 (2021), 2463. https://doi.org/10.3390/math9192463.
  5. N. Hoang, R. Sidje, On the Equivalence of the Continuous Adams?Bashforth Method and Nordsieck?s Technique for Changing the Step Size, Appl. Math. Lett. 26 (2013), 725–728. https://doi.org/10.1016/j.aml.2013.02.001.
  6. O.Y. Ababneh, New Numerical Methods for Solving Differential Equations, J. Adv. Math. 16 (2019), 8384–8390. https://doi.org/10.24297/jam.v16i0.8280.
  7. M.A. Odunayo, Z. Ogunwobi, Comparison of Adams-Bashforth-Moulton Method and Milne-Simpson Method on Second Order Ordinary Differential Equation, Turk. J. Anal. Number Theory 9 (2021), 1–8. https://doi.org/10.12691/tjant-9-1-1.
  8. S.K. Kar, An Explicit Time-Difference Scheme with an Adams?Bashforth Predictor and a Trapezoidal Corrector, Mon. Weather. Rev. 140 (2012), 307–322. https://doi.org/10.1175/mwr-d-10-05066.1.
  9. E. Vitasek, Approximate Solutions of Abstract Differential Equations, Appl. Math. 52 (2007), 171–183. https://doi.org/10.1007/s10492-007-0008-3.
  10. I. Babushka, E. Vitasek, M. Prager, Numerical Processes in Differential Equations, Mir, 1969.
  11. S.L. Sobolev, Introduction to the Theory of Cubature Formulas, Nauka, (1974).
  12. K.M. Shadimetov, A.R. Hayotov, Construction of Interpolation Splines Minimizing Semi-Norm in $W_{2}^{(m,m-1)}(0,1)$ Space, BIT Numer. Math. 53 (2013), 545–563. https://doi.org/10.1007/s10543-012-0407-z.
  13. K.M. Shadimetov, A.R. Hayotov, Optimal Quadrature Formulas in the Sense of Sard in $W_2^{(m,m-1)}$ Space, Calcolo 51 (2013), 211–243. https://doi.org/10.1007/s10092-013-0076-6.
  14. K. Shadimetov, A. Hayotov, R. Karimov, Optimization of Explicit Difference Methods in the Hilbert Space $W_2^{(2,1)}$, AIP Conf. Proc. 2997 (2023), 020054. https://doi.org/10.1063/5.0144805.
  15. K.M. Shadimetov, R.S. Karimov, Optimization of Adams-Type Difference Formulas in Hilbert Space $ W_2^{(2,1)} (0,1)$, J. Comput. Anal. Appl. 32 (2024), 300–319.
  16. K.M. Shadimetov, R.S. Karimov, Optimal Coefficients of an Implicit Difference Formula in the Hilbert Space, AIP Conf. Proc. 3045 (2024), 060030. https://doi.org/10.1063/5.0200065.
  17. G. Ismatullaev, R. Mirzakabilov, R. Karimov, Cubature Formulas in the Parabolic Domain, AIP Conf. Proc. 3045 (2024), 060040. https://doi.org/10.1063/5.0200066.
  18. K.M. Shadimetov, R.N. Mirzakabilov, Optimal Difference Formulas in the Sobolev Space, J. Math. Sci. 278 (2024), 712–721. https://doi.org/10.1007/s10958-024-06951-2.
  19. K.M. Shadimetov, S. Shonazarov, On an Implicit Optimal Difference Formula, Uzb. Math. J. 68 (2025), 128–136. https://doi.org/10.29229/uzmj.2024-4-15.
  20. K.M. Shadimetov, R.S. Karimov, Wiener-Hopf Type System for Finding Optimal Coefficients of Difference Formulas in the Hilbert Space, Probl. Comput. Appl. Math. 59 (2024), 74–84.
  21. R.W. Hamming, Numerical Methods for Scientists and Engineers, Dover Publications, (2012).
  22. G. Dahlquist, Convergence and Stability in the Numerical Integration of Ordinary Differential Equations, Math. Scand. 4 (1956), 33–53. https://www.jstor.org/stable/24490010.
  23. G. Dahlquist, Stability and Error Bounds in the Numerical Integration of Ordinary Differential Equations, Thesis, University of Stockholm, 1958.