International Journal of Nonlinear Analysis and Applications
Generalized Euler and Runge-Kutta methods for solving classes of fractional ordinary differential equations
Document Type : Research Paper
Authors
1 Information Technology Research and Development Center (ITRDC), University of Kufa, Najaf, Iraq
2 Department of Mathematics, Faculty of Education - Ibn Haithem, Baghdad University, Iraq
Abstract
A third-order fractional ordinary differential equation (FrODE) is very important in the mathematical modelling of physical problems. Generally, the third-order ODE is solved by converting the differential equation to a system of first-order ODEs. However, it is a lot more efficient in terms of accuracy, a number of function evaluations as well as computational time if the problem can be solved directly using numerical methods. In this paper, we are focused on the derivation of the direct numerical methods which are one, two and three-stage methods for solving third-order FrODEs. The RKD methods with two- and three stages for solving third-order ODEs are adapted for solving special third-order FrDEs. Numerical examples have been evaluated to show the effectiveness of the new methods compared with the analytical method. Numerical experiments are carried out to verify the accuracy and efficiency of the proposed methods. Applications of proposed methods are also presented which yield impressive results for the proposed and modified methods. The numerical solutions of the test problems using proposed methods agree well with the analytical solutions. From the numerical results obtained using proposed methods, we can conclude that the proposed methods in which derived or modified in this paper are very efficient.
Keywords
[1] G. Akram and S.S. Siddiqi, Solution of sixth order boundary value problems using non- polynomial spline technique, Appl. Math. Comput. 181(1) (2006) 708–720.
[2] R. Allogmany and F. Ismail, many and F. Ismail, Direct solution of u ′′ = f(t, u, u′) using three point block method
of order eight with applications, J. King Saud Univ. Sci. 33(2) (2021) 101337.
[3] R. Allogmany, F. Ismail and Z.B. Ibrahim, Implicit two-point block method with third and fourth derivatives for solving general second order odes, Math. Stat. 7(4) (2019) 116–123.
[4] R. Allogmany, F. Ismail, Z. A. Majid and Z. B. Ibrahim, Implicit two-point block method for solving fourth-order initial value problem directly with application, Math. Prob. Engin. 2020 (2020).
[5] M.S. Arshad, D. Baleanu, M.B. Riaz and M. Abbas, A novel 2-stage fractional Runge-Kutta method for a timefractional logistic growth model, Discrete Dyn. Nature Soc. 2020 (2020).
[6] S. Arshad, A. M. Siddiqui, A. Sohail, K. Maqbool and Z. Li, Comparison of optimal homotopy analysis method and fractional homotopy analysis transform method for the dynamical analysis of fractional order optical solitons, Adv. Mech. Engin. 9(3) (2017) 1687814017692946.
[7] S. Arshad, A. Sohail and K. Maqbool, Nonlinear shallow water waves: A fractional order approach, Alexandria Engin. J. 55(1) (2016) 525–532.
[8] A. Boutayeb and E. Twizell, Numerical methods for the solution of special sixth-order boundary-value problems, Int. J. Comput. Math. 45(3-4) (1992) 207–223.
[9] D. Das, P. Ray, R. Bera and P. Sarkar, Solution of nonlinear fractional differential equation (nfde) by homotopy analysis method, Int. J. Sci. Res. Educ. 3(3) (2015) 3084.
[10] S.J. Farlow, Partial Differential Equations for Scientists and Engineers, Courier Dover Publications, 2012.
[11] M.S. Mechee, Direct Integrators of Runge-Kutta Type for Sspecial Third-Order Differential Equations with Their Applications, Thesis, ISM, University of Malaya, 2014.
[12] M.S. Mechee, Generalized RK integrators for solving class of sixth-order ordinary differential equations, J. Interdiscip. Math. 22(8) (2019) 1457–1461.
[13] M.S. Mechee, G.A. Al-Juaifri and A.K. Joohy, Modified homotopy perturbation method for solving generalized linear complex differential equations, Appl. Math. Sci. 11(51) (2017) 2527–2540.
[14] M.S. Mechee, O.I. Al-Shaher and G.A. Al-Juaifri, Haar wavelet technique for solving fractional differential equations with an application, J. Al-Qadisiyah Comput. Sci. Math. 11(1) (2019) 70.
[15] M.S. Mechee, Z.M. Hussain and H.R. Mohammed, On the reliability and stability of direct explicit Runge-Kutta integrators, Global J. Pure Appl. Math. 12(4) (2016) 3959–3975.
[16] M. Mechee, F. Ismail, Z. Hussain and Z. Siri, Direct numerical methods for solving a class of third-order partial differential equations, Appl. Math. Comput. 247 (2014) 663–674.
[17] M.S. Mechee, F. Ismail, N. Senu and Z. Siri, A third-order direct integrators of Runge-Kutta type for special third-order ordinary and delay differential equations, J. Appl. Sci. 2(6) (2014).
[18] M.S. Mechee, F. Ismail, N. Senu and Z. Siri, Directly solving special second order delay differential equations using Runge-Kutta-Nystrom method, Math. Prob. Engin. 2013 (2013)
[19] M.S. Mechee, F. Ismail, Z. Siri and N. Senu, A four stage sixth-order RKD method for directly solving special third order ordinary differential equations, Life Sci. J. 11(3) (2014).
[20] M.S. Mechee and M. Kadhim, Direct explicit integrators of rk type for solving special fourth-order ordinary differential equations with an application, Global J. Pure Appl. Math. 12(6) (2016) 4687–4715.
[2] R. Allogmany and F. Ismail, many and F. Ismail, Direct solution of u ′′ = f(t, u, u′) using three point block method
of order eight with applications, J. King Saud Univ. Sci. 33(2) (2021) 101337.
[3] R. Allogmany, F. Ismail and Z.B. Ibrahim, Implicit two-point block method with third and fourth derivatives for solving general second order odes, Math. Stat. 7(4) (2019) 116–123.
[4] R. Allogmany, F. Ismail, Z. A. Majid and Z. B. Ibrahim, Implicit two-point block method for solving fourth-order initial value problem directly with application, Math. Prob. Engin. 2020 (2020).
[5] M.S. Arshad, D. Baleanu, M.B. Riaz and M. Abbas, A novel 2-stage fractional Runge-Kutta method for a timefractional logistic growth model, Discrete Dyn. Nature Soc. 2020 (2020).
[6] S. Arshad, A. M. Siddiqui, A. Sohail, K. Maqbool and Z. Li, Comparison of optimal homotopy analysis method and fractional homotopy analysis transform method for the dynamical analysis of fractional order optical solitons, Adv. Mech. Engin. 9(3) (2017) 1687814017692946.
[7] S. Arshad, A. Sohail and K. Maqbool, Nonlinear shallow water waves: A fractional order approach, Alexandria Engin. J. 55(1) (2016) 525–532.
[8] A. Boutayeb and E. Twizell, Numerical methods for the solution of special sixth-order boundary-value problems, Int. J. Comput. Math. 45(3-4) (1992) 207–223.
[9] D. Das, P. Ray, R. Bera and P. Sarkar, Solution of nonlinear fractional differential equation (nfde) by homotopy analysis method, Int. J. Sci. Res. Educ. 3(3) (2015) 3084.
[10] S.J. Farlow, Partial Differential Equations for Scientists and Engineers, Courier Dover Publications, 2012.
[11] M.S. Mechee, Direct Integrators of Runge-Kutta Type for Sspecial Third-Order Differential Equations with Their Applications, Thesis, ISM, University of Malaya, 2014.
[12] M.S. Mechee, Generalized RK integrators for solving class of sixth-order ordinary differential equations, J. Interdiscip. Math. 22(8) (2019) 1457–1461.
[13] M.S. Mechee, G.A. Al-Juaifri and A.K. Joohy, Modified homotopy perturbation method for solving generalized linear complex differential equations, Appl. Math. Sci. 11(51) (2017) 2527–2540.
[14] M.S. Mechee, O.I. Al-Shaher and G.A. Al-Juaifri, Haar wavelet technique for solving fractional differential equations with an application, J. Al-Qadisiyah Comput. Sci. Math. 11(1) (2019) 70.
[15] M.S. Mechee, Z.M. Hussain and H.R. Mohammed, On the reliability and stability of direct explicit Runge-Kutta integrators, Global J. Pure Appl. Math. 12(4) (2016) 3959–3975.
[16] M. Mechee, F. Ismail, Z. Hussain and Z. Siri, Direct numerical methods for solving a class of third-order partial differential equations, Appl. Math. Comput. 247 (2014) 663–674.
[17] M.S. Mechee, F. Ismail, N. Senu and Z. Siri, A third-order direct integrators of Runge-Kutta type for special third-order ordinary and delay differential equations, J. Appl. Sci. 2(6) (2014).
[18] M.S. Mechee, F. Ismail, N. Senu and Z. Siri, Directly solving special second order delay differential equations using Runge-Kutta-Nystrom method, Math. Prob. Engin. 2013 (2013)
[19] M.S. Mechee, F. Ismail, Z. Siri and N. Senu, A four stage sixth-order RKD method for directly solving special third order ordinary differential equations, Life Sci. J. 11(3) (2014).
[20] M.S. Mechee and M. Kadhim, Direct explicit integrators of rk type for solving special fourth-order ordinary differential equations with an application, Global J. Pure Appl. Math. 12(6) (2016) 4687–4715.
[21] M.S. Mechee and M.A. Kadhim, Explicit direct integrators of rk type for solving special fifth-order ordinary differential equations, Amer. J. Appl. Sci. 13 (2016) 1452–1460.
[22] M.S. Mechee and J.K. Mshachal, Derivation of direct explicit integrators of rk type for solving class of seventhorder ordinary differential equations, Karbala Int. J. Modern Sci. 5(3) (2019) 8.
[23] M.S. Mechee and J.K. Mshachal, Derivation of embedded explicit rk type methods for directly solving class of seventh-order ordinary differential equations, J. Interdiscip. Math. 22(8) (2019) 1451–1456.
[24] M.S. Mechee and K.B. Mussa, Generalization of RKM integrators for solving a class of eighth-order ordinary differential equations with applications, Adv. Math. Model. Appl. 5(1) (2020) 111–120.
[25] M. S. Mechee and Y. Rajihy, Generalized RK integrators for solving ordinary differential equations: A survey and comparison study, Global J. Pure Appl. Math. 13(7) (2017) 2923–2949.
[26] M.S. Mechee and N. Senu, A new numerical multistep method for solution of second order of ordinary differential equations, Asian Trans. Sci. Technol. 2(2) (2012) 18–22.
[27] M.S. Mechee and N. Senu, Numerical study of fractional differential equations of Lane-Emden type by method of collocation, Appl. Math. 3(8) (2012) 851.
[28] M.S. Mechee and N. Senu, Numerical study of fractional differential equations of Lane-Emden type by the least square method, Int. J. Diff. Equ. Appl. 11(3) (2012) 157–168.
[29] M.S. Mechee, N. Senu, F. Ismail, B. Nikouravan and Z. Siri, A three-stage fifth-order Runge-Kutta method for directly solving special third-order differential equation with application to thin film ow problem, Math. Prob. Engin. 2013 (2013).
[30] M.S. Mechee, H.M. Wali and K.B. Mussa, Developed rkm method for solving ninth-order ordinary differential equations with applications, J. Phys. Conf. Ser. 1664(1) IOP Publishing, 2020, p. 012102.
[31] Z.M. Odibat and S. Momani, An algorithm for the numerical solution of differential equations of fractional order, J. Appl. Math. Inf. 26(1-2) (2008) 15–27.
[32] A. Sohail, S. Arshad, and Z. Ehsan, Numerical analysis of plasma kdv equation: time- fractional approach, Int. J. Appl. Comput. Math. 3(1) (2017) 13251336.
[33] J. Toomre, J.-P. Zahn, J. Latour and E. Spiegel, Stellar convection theory. ii-single- mode study of the second convection zone in an a-type star, Astrophysical J. 207 (1976) 545–563.
[34] M.Y. Turki, Second Derivative Block Methods for Solving First and Higher Order Ordinary Differential Equations, PhD Thesis, UPM, Putra University, 2018.
[35] M.Y. Turki, F. Ismail, N. Senu and Z. Bibi, Two and three point implicit second derivative block methods for solving first order ordinary differential equations, ASM Sci. J. 12 (2019) 10–23.
[36] M.Y. Turki, F. Ismail, N. Senu and Z.B. Ibrahim, Second derivative multistep method for solving first-order ordinary differential equations, AIP Conf. Proc. 1739(1) (2016) 020054.
[37] M.Y. Turki, F. Ismail, N. Senu and Z.B. Ibrahim, Direct integrator of block type methods with additional derivative for general third order initial value problems, Adv. Mech. Engin. 12(10) (2020) 1687814020966188.
[38] E. Twizell, Numerical methods for sixth-order boundary value problems, Numerical Math. Singapore 1988, Springer, 1988, 495–506.
[39] E. Twizell and A. Boutayeb, Numerical methods for the solution of special and general sixth-order boundary-value problems, with applications to benard layer eigenvalue problems, Proc. R. Soc. Lond. A 431(1883) (1990) 433–450.
[40] S.-Q. Wang, Y.-J. Yang and H.K. Jassim, Local fractional function decom- position method for solving inhomogeneous wave equations with local fractional derivative, Abstr. Appl. Anal. 2014 (2014) 1–7.
[41] A.-M. Wazwaz, New (3+ 1)-dimensional nonlinear evolution equation: multiple soliton solutions, Central European J. Engin. 4(4) (2014) 352–356.
[42] Y. Xiao-Jun, H. Srivastava and C. Cattani, Local fractional homotopy perturbation method for solving fractal partial differential equations arising in mathematical physics, Romanian Rep. Phys. 67(4) (2015) 752–761.
[43] X.-J. Yang, J. Tenreiro Machado and H. Srivastava, A new numerical technique for solving the local fractional diffusion equation: Two-dimensional extended differential transform approach, Appl. Math. Comput. 274(C) (2016) 143–151.
[44] X. You and Z. Chen, Direct integrators of Runge-Kutta type for special third-order ordinary differential equations, Appl. Numerical Math. 74 (2013) 128–150.
[22] M.S. Mechee and J.K. Mshachal, Derivation of direct explicit integrators of rk type for solving class of seventhorder ordinary differential equations, Karbala Int. J. Modern Sci. 5(3) (2019) 8.
[23] M.S. Mechee and J.K. Mshachal, Derivation of embedded explicit rk type methods for directly solving class of seventh-order ordinary differential equations, J. Interdiscip. Math. 22(8) (2019) 1451–1456.
[24] M.S. Mechee and K.B. Mussa, Generalization of RKM integrators for solving a class of eighth-order ordinary differential equations with applications, Adv. Math. Model. Appl. 5(1) (2020) 111–120.
[25] M. S. Mechee and Y. Rajihy, Generalized RK integrators for solving ordinary differential equations: A survey and comparison study, Global J. Pure Appl. Math. 13(7) (2017) 2923–2949.
[26] M.S. Mechee and N. Senu, A new numerical multistep method for solution of second order of ordinary differential equations, Asian Trans. Sci. Technol. 2(2) (2012) 18–22.
[27] M.S. Mechee and N. Senu, Numerical study of fractional differential equations of Lane-Emden type by method of collocation, Appl. Math. 3(8) (2012) 851.
[28] M.S. Mechee and N. Senu, Numerical study of fractional differential equations of Lane-Emden type by the least square method, Int. J. Diff. Equ. Appl. 11(3) (2012) 157–168.
[29] M.S. Mechee, N. Senu, F. Ismail, B. Nikouravan and Z. Siri, A three-stage fifth-order Runge-Kutta method for directly solving special third-order differential equation with application to thin film ow problem, Math. Prob. Engin. 2013 (2013).
[30] M.S. Mechee, H.M. Wali and K.B. Mussa, Developed rkm method for solving ninth-order ordinary differential equations with applications, J. Phys. Conf. Ser. 1664(1) IOP Publishing, 2020, p. 012102.
[31] Z.M. Odibat and S. Momani, An algorithm for the numerical solution of differential equations of fractional order, J. Appl. Math. Inf. 26(1-2) (2008) 15–27.
[32] A. Sohail, S. Arshad, and Z. Ehsan, Numerical analysis of plasma kdv equation: time- fractional approach, Int. J. Appl. Comput. Math. 3(1) (2017) 13251336.
[33] J. Toomre, J.-P. Zahn, J. Latour and E. Spiegel, Stellar convection theory. ii-single- mode study of the second convection zone in an a-type star, Astrophysical J. 207 (1976) 545–563.
[34] M.Y. Turki, Second Derivative Block Methods for Solving First and Higher Order Ordinary Differential Equations, PhD Thesis, UPM, Putra University, 2018.
[35] M.Y. Turki, F. Ismail, N. Senu and Z. Bibi, Two and three point implicit second derivative block methods for solving first order ordinary differential equations, ASM Sci. J. 12 (2019) 10–23.
[36] M.Y. Turki, F. Ismail, N. Senu and Z.B. Ibrahim, Second derivative multistep method for solving first-order ordinary differential equations, AIP Conf. Proc. 1739(1) (2016) 020054.
[37] M.Y. Turki, F. Ismail, N. Senu and Z.B. Ibrahim, Direct integrator of block type methods with additional derivative for general third order initial value problems, Adv. Mech. Engin. 12(10) (2020) 1687814020966188.
[38] E. Twizell, Numerical methods for sixth-order boundary value problems, Numerical Math. Singapore 1988, Springer, 1988, 495–506.
[39] E. Twizell and A. Boutayeb, Numerical methods for the solution of special and general sixth-order boundary-value problems, with applications to benard layer eigenvalue problems, Proc. R. Soc. Lond. A 431(1883) (1990) 433–450.
[40] S.-Q. Wang, Y.-J. Yang and H.K. Jassim, Local fractional function decom- position method for solving inhomogeneous wave equations with local fractional derivative, Abstr. Appl. Anal. 2014 (2014) 1–7.
[41] A.-M. Wazwaz, New (3+ 1)-dimensional nonlinear evolution equation: multiple soliton solutions, Central European J. Engin. 4(4) (2014) 352–356.
[42] Y. Xiao-Jun, H. Srivastava and C. Cattani, Local fractional homotopy perturbation method for solving fractal partial differential equations arising in mathematical physics, Romanian Rep. Phys. 67(4) (2015) 752–761.
[43] X.-J. Yang, J. Tenreiro Machado and H. Srivastava, A new numerical technique for solving the local fractional diffusion equation: Two-dimensional extended differential transform approach, Appl. Math. Comput. 274(C) (2016) 143–151.
[44] X. You and Z. Chen, Direct integrators of Runge-Kutta type for special third-order ordinary differential equations, Appl. Numerical Math. 74 (2013) 128–150.
)
Volume 13, Issue 1
March 2022Pages 1737-1745
- Receive Date: 15 March 2021
- Accept Date: 18 May 2021