Attainable Order of Hybrid Methods from Polynomial Nodes for Non-Linear Singularly Perturbed Initial Value Problems
DOI:
https://doi.org/10.62054/ijdm/0201.01Schlagwörter:
Hybrid method, Continuous scheme, Singularly perturbed initial value problem, Stiff oscillatory systemAbstract
We develop hybrid methods utilizing collocation at polynomial nodes for the numerical integration of singularly perturbed non-linear initial-value problems. The present methods are intended for solving nonlinear singularly perturbed initial value problems without linearization and provide third and fourth-order convergence results. We use piecewise-uniform meshes which resolve the difficulties arising from the steep gradient of the solution in the initial layer. Linear stability of these methods are studied. Numerical experiments are carried out to verify the efficiency and accuracy of the methods. The new hybrid integrators are effectively employed to achieve accurate representation of singularly perturbed systems, providing physical interpretation of what they represent in natural phenomena. These are illustrated through phase plots that exhibit unusual and novel behaviors. The resulting surface phase plot curves represent portions of the phase space of a perturbed system, frequently illustrating real world observed phenomena. These are depicted graphically as surface phase plot curves shown in Figures, while numerical values are presented side by side in Tables.
Literaturhinweise
Akinola, R. O., Akoh, A.S. (2023). A seventh-order computational algorithm for the solution of stiff
systems of differential equations, J. Nig. Math. Soc. 42(3), 215 – 242.
Aris, R. (1969). On stability criteria of chemical reaction engineering. Chem. Eng. Sci. 24, 149 - 169.
Burghardt, A., Zaleski, T.(1968), Longitudinal dispersion at small and large Peclet numbers in chemical flow
reactors, Chem. Eng. Sci. 23, 575–591.
Burrage, K., Butcher, J.C.(1979). Stability criteria for implicit Runge-Kutta method, SIAM J. Numer.
Anal., 16, 46 – 57.
Burrage, K., Butcher, J. C. (1980). Nonlinear stability for a general class of differential equations, BIT, Numer.
Math., 20, 185 - 203.
Butcher, J. C. (1987). The Numerical Analysis Of Ordinary Differential Equations: Runge-Kutta and General Linear
Methods. John Wiley and Sons, Ltd.
Butcher, J. C. (2003). Numerical methods for ordinary differential equations, John Wiley & Sons Ltd.
Butcher, J. C. (2008). Numerical Methods for Ordinary Differential Equations, Second Edition, John Wiley & Sons,
Ltd.
Dahlquist, G. (1963). A special stability problem for linear multistep methods, BIT J. Numer. Math.,
, 27 – 43.
Fatunla, S. O. (1991). Block methods for second order ODEs, Intern. J. Comput. Math., 41, 55–63.
Higinio Ramos, Vigo-Aguiar, J., Natesan, S., García-Rubio, R., Queiruga, M. A. (2010). Numerical solution of
nonlinear singularly perturbed problems on nonuniform meshes by using a non-standard algorithm, J. Math.
Chem., 48, 38–54.
Ismail, N. I. N, Majid, Z. A., Senu, N. (2020). Solving neutral delay differential equation ofpantograph type,
Malaysian Journal of Mathematical Sciences 14(S) 107-121.
Kwami, A. M., Kumleng, G.M., Kolo, A. M., Yakubu, D.G. (2016). Block hybrid multistep methods for the
numerical integration of stiff systems of ordinary differential equations arising from chemical reactions,
Abacus, J. Math. Asso. Nig. 42(2) 134 – 164.
Lambert, J. D. (1973). Computational Methods in Ordinary Differential Equations, John Willey and Sons.
New York.
Lambert, J. D. (1991). Numerical Methods for Ordinary Differential Systems. The Initial Value Problem, John
Willey and Sons. New York.
Odekunle, M.R., Oye, N.D., Adee, S.O. (2004). Ademiluyi, R. A., A class of inverse Runge–Kutta schemes for the
numerical integration of singular problems, Appl. Math. Comput. 158, 149–158.
Qureshi, S., Soomro, A., Evren Hıncal, E. (2021). A new family of A- acceptable nonlinear methods with
fixed and variable step-size approach, Compt. Math. Meth. 3(6) e1213.
Ramos, H., Patricio, M. F. (2014). Some new implicit two-step multi-derivative methods for solving special second-
order IVP’s, Appl. Math. Comp., 239, 227- 241.
Ramos, H., Qureshi, S., Soomro, A. (2021), Adaptive step size approach for Simpson’-type block methods with time
efficiency and order stars, Compt. Appl. Math.,40, 219.
Yakubu, D. G., Onumanyi, P., Chollom, J. P. (2004). A new family of general linear methods based on the block
Adams-Moulton multistep methods. J. of Pure and Applied Sciences,7 (1) 98 -106.
Yakubu, D. G., Kumleng, G. M., Markus, S. (2017). Second derivative Runge-Kutta collocation methods based on
Lobbatto nodes for stiff systems, J. Mod. Meth. Numer. Math., 8(1-2)118-138.
Yakubu, D. G., Mohammed, A. Tahiru, A.G., Abubakar, K. S., Adamu, M. Y. (2023). Numerical simulation of nonlinear dynamics of breast cancer models using continuous block implicit hybrid methods, Fractal and Fractional
(3), 237.
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