Use of partial derivatives to derive a convergent numerical scheme with its error estimates

• Autorzy: Qureshi Sania , Adeyeye Oluwaseun , Shaikh Asif Ali

Identyfikatory

DOI

10.17512/jamcm.2019.4.07

• Języki publikacji: EN

Abstrakty

EN

Using the idea of the partial derivative with respect to the ordinate of a given mathematical function, a new numerical scheme having third order convergence has been devised for solving initial value problems in ordinary differential equations. Such problems are deemed to be indispensable in diverse fields of science, medical and engineering and are most often required to be solved by the numerical schemes. In view of this, the proposed numerical scheme is found to be efficient in solving both autonomous and non-autonomous type of problems as supported by some numerical experiments in the present study. Using the Taylor expansion for the slopes involved in the scheme, the leading term of the local truncation error is shown to have contained Ϭ(h⁴) which proves third order accuracy of the scheme. In addition to this, consistency and linear stability analysis of the proposed scheme has extensively been discussed. Numerical experiments show better performance of the proposed numerical scheme when compared with existing numerical schemes of the same order as that of the scheme proposed. CPU time (seconds), maximum absolute relative error and the absolute relative error, computed at the last grid point of the integration interval for the associated initial value problem, are the parameters to test the performance of the proposed numerical scheme. MATLAB Version: 9.4.0.813654 (R2018a) in double-precision on a personal computer equipped with a Processor Intel (R) Core(TM) i3-4500U CPU@ 1.70 GHz running under the Windows 10 operating system has been employed in order to carry out all the required numerical computations.

Słowa kluczowe

EN

multi-derivative; local truncation error; stability; absolute relative errors; consistency; principal term

PL

wielopochodna; pochodna cząstkowa; schemat numeryczny; błąd względny; błąd bezwzględny

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2019
• Tom: Vol. 18, nr 4
• Strony: 73--83
• Opis fizyczny: Bibliogr. 25 poz., rys., tab.

Twórcy

autor

Qureshi Sania

• Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan

autor

Adeyeye Oluwaseun

• Department of Mathematics, School of Quantitative Sciences, Universiti Utara Malaysia, Sintok 06010, Kedah, Malaysia

autor

Shaikh Asif Ali

• Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan

Bibliografia

• [1] Butcher, J.C. (2016). Numerical Methods for Ordinary Differential Equations. John Wiley & Sons.
• [2] Strogatz, S.H. (2018). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. CRC Press.
• [3] Sundnes, J., Lines, G.T., & Tveito, A. (2001). Efficient solution of ordinary differential equations modeling electrical activity in cardiac cells. Mathematical Biosciences, 172(2), 55-72.
• [4] Jordan, D.W., & Smith, P. (1999). Nonlinear ordinary differential equations: an introduction to dynamical systems (Vol. 2). Oxford University Press, USA.
• [5] Tuck, E.O., & Schwartz, L.W. (1990). A numerical and asymptotic study of some third-order ordinary differential equations relevant to draining and coating flows. SIAM Review, 32(3), 453-469.
• [6] Qu, Z., & Garfinkel, A. (1999). An advanced algorithm for solving partial differential equation in cardiac conduction. IEEE Transactions on Biomedical Engineering, 46(9), 1166-1168.
• [7] Zill, D.G. (2012). A First Course in Differential Equations with Modeling Applications. Cengage Learning.
• [8] Hertz, J.A. (2018). Introduction to the Theory of Neural Computation. CRC Press.
• [9] Lobo, J.Z., & Valaulikar, Y.S. (2019). Lie symmetries of first order neutral differential equations. Journal of Applied Mathematics and Computational Mechanics, 18(1), 29-40.
• [10] Fedorenko, R.P. (1994). Stiff systems of ordinary differential equations. Numerical Methods and Applications, 117-154.
• [11] Zill, D., Wright, W.S., & Cullen, M.R. (2011). Advanced Engineering Mathematics. Jones & Bartlett Learning.
• [12] Chapra, S.C. (2012). Applied Numerical Methods. Columbus: McGraw-Hill.
• [13] Yahya, N.A. (2017). Semi-implicit two-step hybrid method with FSAL property for solving second-order ordinary differential equations. International Journal of Advanced and Applied Sciences, 4(6), 169-174.
• [14] Udwadia, F.E., & Farahani, A. (2008). Accelerated Runge-Kutta methods. Discrete Dynamics in Nature and Society, 2008.
• [15] Aliya, T., Shaikh, A.A., & Qureshi, S. (2018). Development of a nonlinear hybrid numerical method. Advances in Differential Equations and Control Processes, 19(3), 275-285.
• [16] Qureshi, S., Memon, Z., & Shaikh, A.A. (2018). Local accuracy and error bounds of the improved Runge-Kutta numerical methods. Journal of Applied Mathematics and Computational Mechanics, 17(4).
• [17] Qureshi, S., & Emmanuel, F.S. (2018). Convergence of a numerical technique via interpolating function to approximate physical dynamical systems. Journal of Advanced Physics, 7(3), 446-450.
• [18] Ramos, H., Kalogiratou, Z., Monovasilis, T., & Simos, T.E. (2016). An optimized two-step hybrid block method for solving general second order initial-value problems. Numerical Algorithms, 72(4), 1089-1102.
• [19] Qureshi, S., & Ramos, H. (2018). L-stable explicit nonlinear method with constant and variable step-size formulation for solving initial value problems. International Journal of Nonlinear Sciences and Numerical Simulation, 19(7-8), 741-751.
• [20] Arqub, O.A., & Al-Smadi, M. (2018). Atangana-Baleanu fractional approach to the solutions of Bagley-Torvik and Painlev´e equations in Hilbert space. Chaos, Solitons & Fractals, 117, 161-167.
• [21] Osman, M.S. (2019). One-soliton shaping and inelastic collision between double solitons in the fifth-order variable-coefficient Sawada-Kotera equation. Nonlinear Dynamics, 96, 1491-1496.
• [22] Abu Arqub, O. (2019). Application of residual power series method for the solution of timefractional Schr¨odinger equations in one-dimensional space. Fundamenta Informaticae, 166(2), 87-110.
• [23] Ali, M.N., Osman, M.S., & Husnine, S.M. (2019). On the analytical solutions of conformable time-fractional extended Zakharov-Kuznetsov equation through (G0=G2)-expansion method and the modified Kudryashov method. SeMA Journal, 76(1), 15-25.
• [24] Abu Arqub, O. (2019). Numerical algorithm for the solutions of fractional order systems of Dirichlet function types with comparative analysis. Fundamenta Informaticae, 166(2), 111-137.
• [25] Lu, D., Osman, M.S., Khater, M.M.A., Attia, R.A.M., & Baleanu, D. (2020). Analytical and numerical simulations for the kinetics of phase separation in iron (Fe–Cr–X (X= Mo, Cu)) based on ternary alloys. Physica A: Statistical Mechanics and its Applications, 537, 122634.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).