c6eaa9eb-30b1-4896-a26e-b978c30ff62a20240627033701367wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS2224-28801109-276910.37394/23206http://wseas.org/wseas/cms.action?id=40511220241220242310.37394/23206.2024.23https://wseas.com/journals/mathematics/2024.phpM. N.ImanovaScience Development Foundation of the Republic of Azerbaijan, Baku AZ1025, AZERBAIJANV. R.IbrahimovInstitute of Control System named after Academician A. Huseynov, Baku Az1141, AZERBAIJANThe expansion of the application of computational methods for solving many mathematical problems from various fields of natural knowledge does not raise any doubts. One of the promising directions in contemporary sciences is considered to be in areas that are at the intersection of different sciences. Solving such problems is more difficult because different laws from different areas are used. It should be noted that at the intersection of these sciences, there are problems, which can come down to solving ordinary differential equations. Therefore, studies of differential equations have always been considered promising. Based on this, the application of some methods for solving initial problems for first-order ODEs is investigated. For this purpose, scientists studied a numerical solution to the initial problem of the ODE. Here, we have reviewed the study of linear Multistep Methods with constant coefficients. With its help, the order of accuracy of the calculated values is determined. In addition, determines how much accuracy values increase when using Richardson extrapolation methods and also when using linear combinations of various methods. To construct an innovative method is proposed here using advanced methods. It is shown that using these methods it is possible that A-stable methods can be taken as innovative.6272024627202443043745https://wseas.com/journals/mathematics/2024/a905106-024(2024).pdf10.37394/23206.2024.23.45https://wseas.com/journals/mathematics/2024/a905106-024(2024).pdf10.46300/9106.2021.15.8Burova I.G., Application local plynominal and non-polynominal splines of the third order of approximation for the construction of the numerical solution of the Volterra integral, International Journal of Circuits, Systems and Signal Processing, vol. 15, 2021, pp. 63-71, https://doi.org/10.46300/9106.2021.15.8. 10.3390/axioms10020082Juraev D. A., “Solution of the ill-posed Cauchy problem for matrix factorizations of the Helmholtz equation on the plane”, Global and Stochastic Analysis., 8:3 (2021), 1–17. Shura-Bura M.R. Error estimates for numerical integration of ordinary differential equations, Prikl.matem. and mech., 1952, № 5, 575-588, (Russian). 10.7146/math.scand.a-10454Dahlquist G., “Convergence and stability in the numerical integration of ordinary differential equations,” Math. Scand., no. 4, pp. 33-53, 1956, https://doi.org/10.7146/math.scand.a-10454. 10.9734/bjast/2016/22964Mehdiyeva G., Ibrahimov V.R., Imanova M.N., An application of mathematical methods for solving of scientific problems, British journal of applied Science technology, 14(2), 2016, p. 1-15, https://doi.org/10.9734/BJAST/2016/22964. Henrici P., Discrate variable methods in ODE, John Wiley and Sons, Inc, New York. London, 1962. Bakhvalov N.S., Some remarks on the question of numerical intefration of differential equation by the finit-difference method, Academy of Science report, USSA, N3, 1955, 805-808 p., (Russian). 10.1007/s11253-018-1456-5Juraev D.A., Cauchy problem for matrix factorization of the Helmholtz equation, Ukrainian Mathematical Journal, 69, 2018, p.1583-1592, https://doi.org/10.1007/s11253- 018-1456-5. Imanova M.N., On one multistep method of numerical solution for the volterra integral equation, Transactions of NAS of Azerbaijan, 2006, p. 95-104. 10.3390/sym13061087Ibrahimov V. and Imanova M., “Multistep methods of the hybrid type and their application to solve the second kind Volterra integral equation,” Symmetry, vol. 13, no. 6, pp. 1-23, 2021, https://doi.org/10.3390/sym13061087. 10.61485/smcs.27523829/v3n1p2Bulnes J. D. Bulnes, Juraev D. A. Juraev Bonilla, J. L. Bonilla, Travassos, M. A. I. Travassos, “Exact decoupling of a coupled system of two stationary Schrödinger equations”, Stochastic Modelling & Computational Sciences, 3:1 (2023), 23–28. 10.1007/978-81-322-2580-5_95Mehdiyeva G., Ibrahimov V., and Imanova M., “General theory of the applications of multistep methods to calculation of the energy of signals,” In: Zeng, QA. (eds) Wireless Communications, Networking and Applications. Lecture Notes in Electrical Engineering, Springer, New Delhi vol. 348, pp. 1047–1056, 2016, https://doi.org/10.1007/978-81-322-2580-8. 10.1063/1.4992712Mehdiyeva G. Mehdiyeva, Ibrahimov V. Ibrahimov Imanova. M. Imanova, On a Way for Constructing Numerical Methods onthe Joint of Multistep and Hybrid Methods, World Academy of Science, Engineering and Technology, 57 2011, p. 585-588. 10.33773/jum.543320Juraev D. A. Juraev, “On the solution of the Cauchy problem for matrix factorizations of the Helmholtz equation in a multidimensional spatial domain”, Global and Stochastic Analysis, 9:2 (2022), 1–17. 10.1109/cse.2011.34Akinfewa., O.A.Akinfewa., Yao N.M.Yao, Jator S.N.Jator, Implicit two step continuous hybrid block methods with four off steps points for solving stiff ordinary differential equation, WASET, 51, 2011, p.425-428. 10.1145/321250.321261Butcher, J.: A modified multistep method for the numerical integration of ordinary differential equations. J. Assoc. Comput. Math., 12, 124–135 (1965). Ibrahimov, V.R. On a relation between order and degree for stable forward jumping formula, Zh. Vychis. Mat., 1990, p. 1045- 1056. 10.61485/smcs.27523829/v3n1p3Bulnes J. D., Bonilla J. L., Juraev D. A., “Klein-Gordon’s equation for magnons without non-ideal effect on spatial separation of spin waves”, Stochastic Modelling & Computational Sciences, 3:1 (2023), 29–37. 10.5269/bspm.63779Juraev D. A., “The Cauchy problem for matrix factorization of the Helmholtz equation in a multidimensional unbounded domain”, Boletim da Sociedade Paranaense de Matematica, 41 (2023), 1–18. Mehdiyeva, G.Yu. Ibrahimov V.R., Imanova M.N. , On One Application of Hybrid Methods For Solving Volterra Integral Equations, World Academy of Science, Engineering and Technology, 61 2012, p. 809-813. 10.1090/s0025-5718-1979-0537968-6Gupta G., A polynomial representation of hybrid methods for solving ordinary differential equations, Mathematics of comp., 33, 1979, 1251–1256, https://doi.org/10.1090/S0025-5718-1979- 0537968-6. Babushka I., Vitasek E., Prager M., Numerical processes for solving differential equations, Mir 1969, 368. Skvortsov L., Explicit two-step runge-kutta methods, Math. modeling, 21,2009, p.54–65. 10.9790/5728-0722329Awari, Y. Sani, Derivation and Application of six-point linear multistep numerical method for solution of second order initial value problems, IOSR Journal of Mathematics, 20132, p. 23-29. 10.9790/5728-0360814Dachollom Sambo, Chollom, J.P., Oko Nlia, High order hybrid method for the solution of ordinary differential equations, IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278- 5728,. Volume 15, Issue 6 Ser. V (Nov – Dec 2019), pp.31-34. 10.1134/s0012266108100017Bulatov M. V., Ming-Gong Lee, Application of matrix polynomials to the analysis of linear differential-algebraic equations of higher order, Differential Equations, vol. 44, 2008, p. 1353–1360. 10.37394/232011.2022.17.31Burova I. G., Alcybeev G. O., Solution of Integral Equations Using Local Splines of the Second Order, WSEAS Transactions on Applied and Theoretical Mechanics, Vol. 17, 2022, p. 258-262, https://doi.org/10.37394/232011.2022.17.31. 10.37394/23206.2022.21.31Burova I. G., Fredholm Integral Equation and Splines of the Fifth Order of Approximation, WSEAS Transactions on Mathematics, Vol. 21, 2022, pp. 260-270, https://doi.org/10.37394/23206.2022.21.31. 10.37394/23202.2023.22.20Imanova M.N. Ibrahimov V.R., The New Way to Solve Physical Problems Described by ODE of the Second Order with the Special Structure, WSEAS Transactions on Systems, Vol. 22, 2023, p.199-206, https://doi.org/10.37394/23202.2023.22.20. Mehdiyeva G.Yu., Ibrahimov V.R., Imanova, M.N. On the construction test equations and its Applying to solving Volterra integral equation, Methematical methods for information science and economics, Montreux, Switzerland, 2012/12/29, p. 109- 114, https://doi.org/10.1090/S0025-5718- 1979-0537968-6. 10.18187/pjsor.v8i2.294Mehdiyeva,G.Yu. Ibrahimov V.R., Imanova M.N. , Application of a second derivative multi-step method to numerical solution of Volterra integral equation of second kind, Pakistan Journal of Statistics and Operation Research, 28.03.2012, p.245-258, https://doi.org/10.1007/s42452-019-1519-8. Jurayev D.A., Ibrahimov V.R., Agarwal P., Regularization of the Cauchy problem for matrix factorizations of the Helmholtz equation on a two-dimensional bounded domain, Palestine Journal of Mathematics, 12 (1), p. 381-403. 10.17501/biotech.2018.3101Mehdiyeva G.Yu., Ibrahimov V., Imanova M., An application of the hybrid methods to the numerical solution of ordinary diffrential equations of second order, Kazakh National University named after Al-Farabi, Journal of treasury series, mathematics, mechanics, computer science, Almaty, tom 75, No-4, p. 46-54. Imanova M., One the multistep method of numerical solution for Volterra integral equation, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci., 26, vol. 1, p. 95-104. 10.37394/23206.2020.19.19Imanova M., On some comparison of Computing Indefinite integrals with the solution of the initial-value problem for ODE, WSEAS Transactions on Mathematics, Vol. 19, 2020, p. 208-215, https://doi.org/10.37394/23206.2020.19.19. 10.1088/1742-6596/1564/1/012019Imanova M.N., On the comparison of Gauss and Hybrid methods and their application to calculation of definite integrals, Journal of Physics: Conference Series, 1564, vol. 1, 2020, IOP publishing, p. 012019. 10.1063/1.4765535Mehdiyeva G.Yu., Ibrahimov V., Imanova M., Application of the hybrid method with constant coefficients to solving the integrodifferential equations of first order, AIP Conference Proceedings, 2012/11/6, p. 506- 510.