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Four step hybrid block method for the direct solution of fourth order ordinary differential equations | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 18، دوره 12، شماره 1، مرداد 2021، صفحه 215-229 اصل مقاله (397.21 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2021.4762 | ||
| نویسنده | ||
| Raft Abdelrahim* | ||
| Department of Mathematics, College of Art and Sciences-Tabarjal, Jouf University, Saudi Arabia | ||
| تاریخ دریافت: 16 مهر 1399، تاریخ بازنگری: 08 دی 1399، تاریخ پذیرش: 13 دی 1399 | ||
| چکیده | ||
| This paper proposes a direct four-step implicit hybrid block method for directly solving general fourth-order initial value problems of ordinary differential equations. In deriving this method, the approximate solution in the form of power series is interpolated at four points, i.e $ x_n, x_{n+1},x_{n+2},x_{n+3} $ while its forth derivative is collocated at all grid points, i.e $ x_n,x_{n+\frac{1}{4}},x_{n+1} , x_{n+2}, x_{n+\frac{5}{2}}, x_{n+3},x_{n+\frac{7}{2}} $ and $ x_{n+4} $ to produce the main continuous schemes. In order to verify the applicability of the new method, the properties of the new method such as local truncation error, zero stability, order and convergence are also established. The performance of the newly developed method is then compared with the existing methods in terms of error by solving the same test problems. The numerical results reveal that the proposed method produces better accuracy than several existing methods when solving the same initial value problems (IVPs) of second order ODEs. | ||
| کلیدواژهها | ||
| Hybrid block method؛ Fourth initial value problem؛ Collocation and Interpolation؛ Four step | ||
| مراجع | ||
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[1] R. Abdelrahim and Z. Omar, Direct solution of second-order ordinary differential equation using a single-step hybrid block method of order five, Math. Comput. Appl. 21 (2016) 1–7. [2] R. Abdelrahim and Z. Omar, Solving third order ordinary differential equations using hybrid block method of order five, Int. J. Appl. Engin. Res. 10 (2015) 44307–44310. [3] R. Abdelrahim and Z. Omar, A four-step implicit block method with three generalized off-step points for solving fourth order initial value problems directly, J. King Saud Univ.- Sci. 4 (2017) 401–412. [4] O. Adesanya, M. Momoh, Adamu and A.O. Tahir, Five steps block method for the solution of fourth order ordinary differential equations, Int. J. Engin. Res. Appl. 2 (2012) 991–998. [5] D.O. Awoyemi, Algorithmic collocation approach for direct solution of fourth-order initial-value problems of ordinary differential equations, Int. J. Comput. Math. 47 (2005) 321–329. [6] D.O. Awoyemi, S.J. Kayode and O.A Adoghe, Six-step continuous multistep method for the solution of general fourth-order initial value problems of ordinary differential equations, J. Natural Sci. Res. 5 (2005) 131-138. [7] R. Cortell, Application of the fourth-order Runge-Kutta method for the solution of high-order general initial value problems, Comput. Struct. 49 (1993) 897–900. [8] S.O. Fatunla, Block method for second order ODES, Int. J. Comput. Math. 41 (1991) 55–63. [9] P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 1962. [10] S.N. Jator, Numerical integrators for fourth order initial and boundary value problems, Int. J. Pure Appl. Math. 47 (2008) 563–576. [11] J.D. Lambert, Numerical Methods for Ordinary Differential Systems, Wiley, 1991. [12] Z. Omar and R. Abdelrahim, Application of single step with three generalized hybrid points block method for solving third order ordinary differential equations, Int. J. Appl. Engin. Res. 9 (2016) 2705–2717. [13] E.H. Twizell, A family of numerical methods for the solution of high-order general initial value problems, Comput. Meth. Appl. Mech. Engin. 67 (1988) 15–25. [14] L.K. Yap, F. Ismail and N. Senu, An accurate block hybrid collocation method for third order ordinary differential equations, J. Appl. Math. 2014 (2014) 9. [15] L.K. Yap and F. Ismail, Block hybrid collocation method with application to fourth order differential equations, Math. Probl. Engin. 2015 (2015) 1–6. | ||
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