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A new nonlinear L-stable scheme with constant and adaptive step-size strategy

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The present study proposes a new explicit nonlinear scheme that solves stiff and nonlinear initial value problems in ordinary differential equations. One of the promising features of this scheme is its fourth-order convergence with strong stability having an unbounded region. A modern approach for relative stability growth analysis is also presented under order stars conditions. The scheme is also good in dealing with singular and stiff type of models. Comparing numerical experiments using various errors, including maximum absolute global error over the integration interval, absolute error at the endpoint, average error, norm of errors, and the CPU times (seconds), shows better performance. An adaptive step-size approach seems to increase the performance of the proposed scheme. The numerical simulations assure us of L -stability, consistency, order, and rapid convergence.
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Bibliogr. 24 poz., rys., tab.
  • Basic Sciences and Related Studies, Mehran University of Engineering and Technology. Jamshoro, 76062, Sindh, Pakistan
  • Basic Sciences and Related Studies, Mehran University of Engineering and Technology. Jamshoro, 76062, Sindh, Pakistan
  • Department of Mathematics, Near East University TRNC, Mersin 10, Turkey
  • Basic Sciences and Related Studies, Mehran University of Engineering and Technology. Jamshoro, 76062, Sindh, Pakistan
  • [1] Shampine, L.F., Gladwell, I., Shampine, L., & Thompson, S. (2003). Solving ODEs with Matlab. Cambridge University Press.
  • [2] Hairer, E., & Wanner, G. (1991). Solving Ordinary Differential Equations II. Berlin-Heidelberg: Springer.
  • [3] Asif, M., Jan, S.U., Haider, N., Al-Mdallal, Q., & Abdeljawad, T. (2020). Numerical modeling of npz and sir models with and without diffusion. Results in Physics, 19, 103512.
  • [4] Stoer, J., & Bulirsch, R. (2013). Introduction to Numerical Analysis. (Vol. 12). Springer Science & Business Media.
  • [5] Qureshi, S., & Yusuf, A. (2020). A new third-order convergent numerical solver for continuous dynamical systems. Journal of King Saud University-Science, 32(2), 1409-1416.
  • [6] Emmanuel, F.S., & Qureshi, S. (2019). Convergent numerical method using the transcendental function of exponential type to solve continuous dynamical systems. Journal of Mathematics, 51(10), 45-56.
  • [7] Gadella, M., & Lara, L.P. (2013). A numerical method for solving ODE by rational approximation. Applied Mathematical Sciences, 7(23), 1119-1130.
  • [8] Sharma, J.R., Arora, H., & Petkovi ́c, M.S. (2014). An efficient derivative-free family of fourth-order methods for solving systems of nonlinear equations. Applied Mathematics and Computation, 235, 383-393.
  • [9] 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).
  • [10] Memon, Z., Qureshi, S., Memon, B.R., & Saleem, M. (2019). An optimized single-step method for integrating Cauchy problems. Journal of Mathematics, 51(9), 33-44.
  • [11] Yaakub, A.R., & Evans, D.J. (1999). A fourth order Runge-Kutta RK (4, 4) method with error control. International Journal of Computer Mathematics, 71(3), 383-411.
  • [12] Awoyemi, D.O. (2005). Algorithmic collocation approach for direct solution of fourth-order initial-value problems of ordinary differential equations. International Journal of Computer Mathematics, 82(3), 321-329 .
  • [13] Qureshi, S., Soomro, A., & Hınc ̧al, E. (2021). A new family of a- acceptable nonlinear methods with fixed and variable stepsize approach. Computational and Mathematical Methods, e1213.
  • [14] Owolabi, K.M. (2011). The 4th-step implicit formula for the solution of initial value problems of second-order ordinary differential equations. African Journal of Mathematics and Computer Science Research, 4(7), 270-272.
  • [15] Ramos, H., Qureshi, S., & Soomro, A. (2021). Adaptive step-size approach for Simpson’s-type block methods with time efficiency and order stars. Computational and Applied Mathematics, 40(6), 1-20.
  • [16] Ramos, H. (2007). A Nonlinear Explicit One-Step Integration Scheme for Singular Autonomous Initial Value Problems. In AIP Conference Proceedings (Vol. 936, No. 1, pp. 448-451). American Institute of Physics.
  • [17] Ramos, H., Singh, G., Kanwar, V., & Bhatia, S. (2015). Solving first-order initial-value problems by using an explicit non-standard A-stable one-step method in the variable step-size formulation.Applied Mathematics and Computation, 268, 796-805.
  • [18] Ying, T.Y. (2014). An explicit two-step rational method for the numerical solution of first order initial value problem. In AIP Conference Proceedings (Vol. 1605, No. 1, pp. 96-100). American Institute of Physics.
  • [19] Qureshi, S., & Ramos, H. (2018). The 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] Ramos, H., Singh, G., Kanwar, V., & Bhatia, S. (2017). An embedded 3(2) pair of nonlinear methods for solving first order initial-value ordinary differential systems. Numerical Algorithms, 75(3), 509-529.
  • [21] Sanugi, B.B., & Evans, D.J. (1994). A new fourth order Runge-Kutta formula based on the harmonic mean. International Journal of Computer Mathematics, 50(1-2), 113-118.
  • [22] Abdeljawad, T., Amin, R., Shah, K., Al-Mdallal, Q., & Jarad, F. (2020). Efficient sustainable algorithm for numerical solutions of systems of fractional order differential equations by Haar avelet collocation method. Alexandria Engineering Journal, 59(4), 2391-2400.
  • [23] Abdeljawad, T., Hajji, M.A., Al-Mdallal, Q.M., & Jarad, F. (2020). Analysis of some generalized abc-fractional logistic models. Alexandria Engineering Journal, 59(4), 2141-2148.
  • [24] Al-Mdallal, Q.M., Hajji, M.A., & Abdeljawad, T. (2021). On the iterative methods for solving fractional initial value problems: new perspective. Journal of Fractional Calculus and Nonlinear Systems, 2(1), 76-81.
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