On Some Comparison of the Numerical Methods Applied to Solve ODEs, Volterra Integral and Integro-Differential Equations
DOI:
https://doi.org/10.22105/kmisj.v1i1.41Keywords:
Multistep method with third derivative, stable and degree, the conseption degree and stability, advanced multistep third derivaties methods, explicit and implicit methodsAbstract
The many problems of the different fields of nature are reduce to solve initialvalue problem for the both Ordinary Differential Equation and Volterra integro-differential equation and also to solve Volterra integral equation. And for solving theser problems in usually are used the numerical methods, which are related with the development of computer sciences. In among of them, the multistep multiderivative methods are very developing. Therefor, any result in this area is of interest in always. The ordinary multistep method and multistep second derivative
methods fundamentally investigated by Dahlquist. The Dahlquist theory development by Ibrahimov, who have receive similarly results for the advanced multistep methods and advanced multistep second derivative methods. Here by development of these results, have considered to
investigation of the multistep third derivative methods with constant coefficients. Somebody can be take the results receiving for the estimation of the accuracy for stable and unstable multistep multiderivative methods with third derivative as the simple generalization of the known results.
Here have shown that this is not correct. Noted that multistep multiderivative Methods with third derivative here investigated in fool form, found some connection between degree and order for these methods in the cases: stable and in stable. Constructed some concrete methods of multistep third derivative types and recommended some way for the application of these methods to solve some problems.
References
A. Movahedinejad, A. Abdi, Gh. Hojjati, A hybrid method with optimal stability properties for the numer-ical solution of stiff differential systems,Computational Methods for Differential Equations,4:3, 219–229(2016).
A. Shokri, M.M. Khalsaraei, S. Noeiaghdam, D.A. Juraev, A new divided difference interpolation methodfor two-variable functions,Global and Stochastic Analysis,9:2, 19–26 (2022).
B. Brahim, A novel computational approach for solving fully implicit singular systems of ordinary differential equations, Qeios, X4S2ZL, 1–17 (2023).
D.A. Zhuraev, Cauchy problem for matrix factorizations of the Helmholtz equation, Ukrainian Mathematical Journal, 69:10, 1583–1592 (2018).
D.A. Juraev, S. Noeiaghdam, Regularization of the ill-posed Cauchy problem for matrix factorizations of the Helmholtz equation on the plane, Axioms, 10:2, 1–14 (2021).
D.A. Juraev, S. Noeiaghdam, Modern problems of mathematical physics and their applications, Axioms, 11:2, 1–6 (2022).
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, 1–17 (2022).
D.A. Juraev, The solution of the ill-posed Cauchy problem for matrix factorizations of the Helmholtz equation in a multidimensional bounded domain, Palestine Journal of Mathematics, 11:3, 604–613 (2022).
D.A. Juraev, A. Shokri, D. Marian, Solution of the ill-posed Cauchy problem for systems of elliptic type of the first order, Fractal and Fractional, 6:7, 1–11 (2022).
D.A. Juraev, A. Shokri, D. Marian, On an approximate solution of the Cauchy problem for systems of equations of elliptic type of the first order, Entropy, 24:7, 1–18 (2022).
D.A. Juraev, A. Shokri, D. Marian, On the approximate solution of the Cauchy problem in a multidimensional unbounded domain, Fractal and Fractional, 6:7, 1–14 (2022).
D.A. Juraev, A. Shokri, D. Marian, Regularized solution of the Cauchy problem in an unbounded domain, Symmetry, 14:8, 1–16 (2022).
D.A. Juraev, V. Ibrahimov, P. Agarwal, Regularization of the Cauchy problem for matrix factorizations of the Helmholtz equation on a two-dimensional bounded domain, Palestine Journal of Mathematics, 12:1, 381–403 (2023).
D. Sambo, J.P. Chollom, O.Nlia, High order hybrid method for the solution of ordinary differential equations, IOSR Journal of Mathematics (IOSR-JM), 15:6, 31–34 (2019).
F. Muritala, A.A.K. Jimooh, M.O. Oguniran, A.A. Oyedeji, J.O. Lawal, k-step block hybrid method for numerical approximation of fourth-order ordinary differential equations, Preprint, (2012).
G. Mehdiyeva, M. Imanova, V. Ibrahimov, On a way for constructing numerical methods on the joint of multistep and hybrid methods, World Academy of Science, Engineering and Technology, 9:5, 240–243 (2011).
G.Yu. Mehdiyeva, V.R. Ibrahimov, M.N. Imanova, On one generalization of hybrid methods, Fractional Fourier transform to stability analysis of fractional differential equations with Prabhakar derivatives, Proceedings of the 4th international conference on approximation methods and numerical modelling in environment and natural resources, 543–547 (2011).
G. Mehdiyeva, V. Ibrahimov, M. Imanova, On the construction test equations and its Applying to solving Volterra integral equation, Mathematical methods for information science and economics, Montreux, Switzerland, 109–114 (2012).
G.Yu. Mehdiyeva, M.N. Imanova, V.R. Ibrahimov, An application of the hybrid methods to the numerical solution of ordinary differential equations of second order, Journal of Mathematics, Mechanics and Computer Science, 75:4, 46–54 (2012).
G.Yu. Mehdiyeva, M.N. Imanova, V.R. Ibrahimov, 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, 8:2, 245–258 (2012).
G. Mehdiyeva, V. Ibrahimov, M. Imanova, Application of the hybrid method with constant coefficients to solving the integro-differential equations of first order, AIP Conference Proceedings, 1493:1, 506–510 (2012).
G.Yu. Mehdiyeva, V.R. Ibrahimov, M.N. Imanova, An application of mathematical methods for solving of scientific problems, Current Journal of Applied Science and Technology, 14:2, 1–15 (2016).
G. Mehdiyeva, V. Ibrahimov, M. Imanova, General theory of the application of multistep methods to calculation of the energy of signals, Wireless Communications, Networking and Applications, 1047–1056 (2016).
G.Yu. Mehdiyeva, V.R. Ibrahimov, M.N. Imanova, On the construction of the multistep methods to solving the initial value problem for ODE and the Volterra integro-differential equations, IAPE’19, Oxford, United Kingdom, 1–9 (2019).
G. Mehdiyeva, V. Ibrahimov, M. Imanova, On the construction of the advanced Hybrid Methods and application to solving Volterra integral equation, WSEAS Transactions on Systems and Control, 14, 183–189 (2019).
G. Yu. Mehdiyeva, V. R. Ibrahimov, M. N. Imanova, The application of hybrid methods to solve some problems of mathematical biology, American Journal of Biomedical Science and Research, 18:1, 74–80 (2023).
G. Yu. Mehdiyeva, V. R. Ibrahimov, On the research of multistep methods with constant coefficients, Lambert Academic Publishing, 314 (2023).
G. Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations, Math. Scand., 4, 33–53 (1956).
G. Dahlquist, Stability and error bounds in the numerical integration of ordinary differential equation, Trans. Of the Royal Inst. of Techn., Stockholm, Sweden, 130, 3–87 (1959).
I. G. Burova, G. O. Alcybeev, On the construction of the advanced hybrid methods and application to solving Volterra integral equation, WSEAS Transactions on Systems and Control, 14, 258–262 (2019).
I. G. Burova, Application local polynomial and non-polynomial splines of the third order of approximation for the construction of the numerical solution of the Volterra integral, WSEAS Transactions on Mathematics, 20, 9–23 (2021).
I. G. Burova, Fredholm integral equation and splines of the fifth order of approximation, WSEAS Transactions on Mathematics, 21, 260–270 (2022).
I. G. Burova, G. O. Alcybeev, Solution of integral equations using local splines of the second order, WSEAS Transactions on Applied and Theoretical Mechanics, 17, 258–262 (2022).
J. Kobza, Second derivative methods of Adams type, Applikace Mathematicky, 20, 389–405 (1975).
J. D. Bulnes, D. A. Juraev, J. L. Bonilla, M. A. I. Travassos, Exact decoupling of a coupled system of two stationary Schrödinger equations, Global and Stochastic Analysis, 3:1, 23–28 (2023).
J. D. Bulnes, J. L. Bonilla, D. A. Juraev, Klein-Gordon’s equation for magnons without non-ideal effect on spatial separation of spin waves, Global and Stochastic Analysis, 3:1, 29–37 (2023).
L. Nasirova, G. Mehdiyeva, V. Ibrahimov, Application of multistep methods to the solving second-order ordinary differential equations, The Third International Conference, Baku, Azerbaijan, (2010).
M. Imanova, On some comparison of Computing Indefinite integrals with the solution of the initial-value problem for ODE, WSEAS Transactions on Mathematics, 19, 208–215 (2020).
M. N. Imanova, V. R. Ibrahimov, The new way to solve physical problems described by ODE of the second order with the special structure, WSEAS Transactions on Systems, 22, 199–206 (2023).
M. R. Shura-Bura, Error estimates for numerical integration of ordinary differential equations, Prikl. Matem. and Mech., 5, 575–588 (1952).
N.S. Bakhvalov, Some remarks on the question of numerical integration of differential equations by the finite-difference method, Academy of Science Report, USSR, 3, 805–808 (1955).
M.V. Bulatov, M.G. Lee, Ming-Gong Lee, Application of matrix polynomials to the analysis of linear differential-algebraic equations of higher order, Differential Equations, 44, 1353–1360 (2008).
O.A. Akinfenwa, B. Akinnukawe, S.B. Mudasiru, A Family of Continuous Third Derivative Block Methods for solving stiff systems of first order ordinary differential equations, Journal of the Nigerian Mathematical Society, 34:2, 160–168 (2015).
P. Henrici, Discrete variable methods in ODE, John Wiley and Sons, Inc, New York. London (2008).
S. Deepa, A. Ganesh, V. Ibrahimov, S.S. Santra, V. Govindan, K.M. Khedher, S. Noeiaghdam, Fractional Fourier transform to stability analysis of fractional differential equations with Prabhakar derivatives, Azerbaijan Journal of Mathematics, 12:2, 131–153 (2022).
Sh. Zheng, H. Liu, M. Hafeez, X. Wang, Sh. Fahad, X.G. Yue, Testing the resource curse hypothesis: The dynamic roles of institutional quality, inflation, and growth for Dragon, Resources Policy, 86:A, 103840 (2023).
V.R. Ibrahimov, G.Yu. Mehdiyeva, I.I. Nasirova, On some connections between Runge-Kutta and Adams methods, Transactions of NAS of Azerbaijan, XXV, 183–190 (2005).
V. Ibrahimov, M. Imanova, Multistep methods of the hybrid type and their application to solve the second kind Volterra integral equation, Symmetry, 13:6, 1–23 (2021).
V.R. Ibrahimov, M.N. Imanova, Finite difference methods with improved properties and their application to solving some model problems, 2022 International Conference on Computational Science and Computational Intelligence (CSCI), 463–471 (2022).
V. Ibrahimov, M. Imanova, About some applications of multistep methods with constant coefficients to investigation of some biological problems, American Journal of Biomedical Science and Research, 18, 531–542 (2023).
V.R. Ibrahimov, M.N. Imanova, D.A. Juraev, About the new way for solving some physical problems described by ODE of the second order with the special structure, Global and Stochastic Analysis, 3:1, 99–117 (2023).
V.R. Ibrahimov, M.N. Imanova, D.A. Juraev, Application of the bilateral hybrid methods to solving initial- value problems for the Volterra integro-differential equations,WSEAS Transactions on Mathematics,22,781–791 (2023).
V.R. Ibrahimov, X.G. Yue, D.A. Juraev, On some advantages of the predictor-corrector methods,IETITransactions on Data Analysis and Forecasting (iTDAF),1:4, 79–89 (2023).
X.G. Yue, S. Sahmani, W. Huang, B. Safaei, Three-dimensional isogeometric model for nonlinear vibra-tion analysis of graded inhomogeneous nanocomposite plates with inconstant thickness,Acta Mechanica,234, 5437–5459 (2023).
Y.S. Awari, Derivation and application of six-point linear multistep numerical method for solution ofsecond order initial value problems,IOSR Journal of Mathematics (IOSR-JM),7:2, 23–29 (2013).
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