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Year 2020, Volume: 3 Issue: 2, 153 - 160, 15.12.2020

Elif Nuray Yıldırım Ali Akgul

https://doi.org/10.33401/fujma.795418

Abstract

References

  • [1] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404.
  • [2] S. Bergman, The Kernel Function and Conformal Mapping, American Math. Soc., New York, (1950).
  • [3] M. Cui, Y. Lin, Nonlinear Numerical Analysis in the Reproducing Kernel Space, New York: Nova Sci. Publ., (2009).
  • [4] M. I. Syam, Q. M. Al-Mdallal and M. Al-Refai, A Numerical method for solving a class of fractional Sturm-Liouville eigenvalue problems, Com. in Num. Analy., 2 (2017), 217–232.
  • [5] W. Jiang, T. Tian, Numerical solution of nonlinear Volterra integro-differential equations of fractional order by the reproducing kernel method, App. Math. Mod., 39 (16) (2015), 4871–4876.
  • [6] X. Y. Li, B. Y. Wu, R. T. Wan, Reproducing Kernel Method for Fractional Riccati Differential Equations, Abst. App. Ana., (2014), 1-6.
  • [7] A. Alvandi, M. Paripour, The combined reproducing kernel method and Taylor series to solve nonlinear Abel’s integral equations with weakly singular kernel, Cogent Mathematics, 3 (2016).
  • [8] A. Freihat, R. Abu-Gdairi, H. Khalil, E. Abuteen, M. Al-Smadi, R. A. Khan, Fitted Reproducing Kernel Method for Solving a Class of Third-Order Periodic Boundary Value Problems, American J. App. Sci., 13 (2016), 501–510.
  • [9] G. Akram, H. U. Rehman, Numerical solution of eighth order boundary value problems in reproducing Kernel space, Numer. Algor, 62(3) (2013), 527–540.
  • [10] S. Abbasbandy, Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian’s decomposition method, Appl. Math. Comput, 172 (2006), 485–490.
  • [11] G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, San Diego, (1986).
  • [12] A. Daşcıoğlu, H. Yaslan, The solution of high-order nonlinear ordinary differential equations by Chebshev series, Appl. Math. Comput., 217 (2011), 5658–5666.
  • [13] A.M. Wazwaz, A new method for solving initial value problems in second-order ordinary differential equations, Appl. Math. Comput., 128 (2002), 45–57.
  • [14] M. K. Horn, Fourth- and fifth-order, scaled Runge-Kutta algorithms for treating dense output, SI AM J. Numer. Analysis, 20 (1983), 558-568.
  • [15] L. Fox, D. F. Mayers, Numerical Solution of Ordinary Differential Equations, Chapman and Hall, (1987).
  • [16] A.M. Wazwaz, The numerical solution of fifth-order boundary value problems by the decomposition method, J. Comput. Appl. Math. 136(1–2) (2001), 259–270 .
  • [17] Waeleh et al., Numerical Solution of Higher Order Ordinary Differential Equations by Direct Block Code, J. Math. Sta., 8(1) (2012), 77–81.
  • [18] E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, Tata McGraw-Hill Publishing, (1972).
  • [19] F. Hoppensteadt, Properties of solutions of ordinary differential equations with small parameters, Com. on Pure and App. Math., 24(6) (1971), 807–840.
  • [20] G. R. Sell, On the fundamental theory of ordinary differential equations, Jour. of Diff. Equ. 1 (1965), 370–392.
  • [21] D. Baleanu, A. Fernandez, A. Akgül, On a Fractional Operator Combining Proportional and Classical Differintegrals, Mathematics, 8(3) (2020).
  • [22] A. Akgül, A novel method for a fractional derivative with non-local and non-singular kernel, Chaos, Solitons and Fractals 114, (2020), 478-482.
  • [23] E. K. Akgül, Solutions of the linear and nonlinear differential equations within the generalized fractional derivatives, Chaos: An Inter. J. Nonlin. Sci. 29(2) 023108, (2020).
  • [24] K. M. Owolabi, A. Atangana, A. Akgül, Modelling and analysis of fractal-fractional partial differential equations: Application to reaction-diffusion model, Alexandria Eng. J. 59 (2020), 2477-2490.
  • [25] A. Atangana, A. Akgül, K. M. Owolabi, Analysis of fractal fractional differential equations, Alexandria Eng. J. 59 (2020), 1117-1134.
  • [26] A. Atangana, A. Akgül, Can transfer function and Bode diagram be obtained from Sumudu transform, Alexandria Eng. J. 59 (2020), 1971-1984.

On Solutions of a Higher Order Nonhomogeneous Ordinary Differential Equation

Year 2020, Volume: 3 Issue: 2, 153 - 160, 15.12.2020

Elif Nuray Yıldırım Ali Akgul

https://doi.org/10.33401/fujma.795418

Abstract

Higher order differential equations (ODE) has an important role in the modelling process. It is also much significant which the method is used for the solution. In this study, in order to get the approximate solution of a nonhomogeneous initial value problem, reproducing kernel Hilbert space method is used. Reproducing kernel functions have been obtained and the given problem transformed to the homogeneous form. The results have been presented with the graphics. Absolute errors and relative errors have been given in the tables.

Keywords

Nonhomogeneous ordinary differential equations Approximate solution Initial value problems Reproducing kernel method

References

  • [1] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404.
  • [2] S. Bergman, The Kernel Function and Conformal Mapping, American Math. Soc., New York, (1950).
  • [3] M. Cui, Y. Lin, Nonlinear Numerical Analysis in the Reproducing Kernel Space, New York: Nova Sci. Publ., (2009).
  • [4] M. I. Syam, Q. M. Al-Mdallal and M. Al-Refai, A Numerical method for solving a class of fractional Sturm-Liouville eigenvalue problems, Com. in Num. Analy., 2 (2017), 217–232.
  • [5] W. Jiang, T. Tian, Numerical solution of nonlinear Volterra integro-differential equations of fractional order by the reproducing kernel method, App. Math. Mod., 39 (16) (2015), 4871–4876.
  • [6] X. Y. Li, B. Y. Wu, R. T. Wan, Reproducing Kernel Method for Fractional Riccati Differential Equations, Abst. App. Ana., (2014), 1-6.
  • [7] A. Alvandi, M. Paripour, The combined reproducing kernel method and Taylor series to solve nonlinear Abel’s integral equations with weakly singular kernel, Cogent Mathematics, 3 (2016).
  • [8] A. Freihat, R. Abu-Gdairi, H. Khalil, E. Abuteen, M. Al-Smadi, R. A. Khan, Fitted Reproducing Kernel Method for Solving a Class of Third-Order Periodic Boundary Value Problems, American J. App. Sci., 13 (2016), 501–510.
  • [9] G. Akram, H. U. Rehman, Numerical solution of eighth order boundary value problems in reproducing Kernel space, Numer. Algor, 62(3) (2013), 527–540.
  • [10] S. Abbasbandy, Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian’s decomposition method, Appl. Math. Comput, 172 (2006), 485–490.
  • [11] G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, San Diego, (1986).
  • [12] A. Daşcıoğlu, H. Yaslan, The solution of high-order nonlinear ordinary differential equations by Chebshev series, Appl. Math. Comput., 217 (2011), 5658–5666.
  • [13] A.M. Wazwaz, A new method for solving initial value problems in second-order ordinary differential equations, Appl. Math. Comput., 128 (2002), 45–57.
  • [14] M. K. Horn, Fourth- and fifth-order, scaled Runge-Kutta algorithms for treating dense output, SI AM J. Numer. Analysis, 20 (1983), 558-568.
  • [15] L. Fox, D. F. Mayers, Numerical Solution of Ordinary Differential Equations, Chapman and Hall, (1987).
  • [16] A.M. Wazwaz, The numerical solution of fifth-order boundary value problems by the decomposition method, J. Comput. Appl. Math. 136(1–2) (2001), 259–270 .
  • [17] Waeleh et al., Numerical Solution of Higher Order Ordinary Differential Equations by Direct Block Code, J. Math. Sta., 8(1) (2012), 77–81.
  • [18] E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, Tata McGraw-Hill Publishing, (1972).
  • [19] F. Hoppensteadt, Properties of solutions of ordinary differential equations with small parameters, Com. on Pure and App. Math., 24(6) (1971), 807–840.
  • [20] G. R. Sell, On the fundamental theory of ordinary differential equations, Jour. of Diff. Equ. 1 (1965), 370–392.
  • [21] D. Baleanu, A. Fernandez, A. Akgül, On a Fractional Operator Combining Proportional and Classical Differintegrals, Mathematics, 8(3) (2020).
  • [22] A. Akgül, A novel method for a fractional derivative with non-local and non-singular kernel, Chaos, Solitons and Fractals 114, (2020), 478-482.
  • [23] E. K. Akgül, Solutions of the linear and nonlinear differential equations within the generalized fractional derivatives, Chaos: An Inter. J. Nonlin. Sci. 29(2) 023108, (2020).
  • [24] K. M. Owolabi, A. Atangana, A. Akgül, Modelling and analysis of fractal-fractional partial differential equations: Application to reaction-diffusion model, Alexandria Eng. J. 59 (2020), 2477-2490.
  • [25] A. Atangana, A. Akgül, K. M. Owolabi, Analysis of fractal fractional differential equations, Alexandria Eng. J. 59 (2020), 1117-1134.
  • [26] A. Atangana, A. Akgül, Can transfer function and Bode diagram be obtained from Sumudu transform, Alexandria Eng. J. 59 (2020), 1971-1984.
There are 26 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Elif Nuray Yıldırım 0000-0002-2934-892X

Ali Akgul 0000-0001-9832-1424

Publication Date December 15, 2020
Submission Date September 15, 2020
Acceptance Date November 24, 2020
Published in Issue Year 2020 Volume: 3 Issue: 2

Cite

APA Nuray Yıldırım, E., & Akgul, A. (2020). On Solutions of a Higher Order Nonhomogeneous Ordinary Differential Equation. Fundamental Journal of Mathematics and Applications, 3(2), 153-160. https://doi.org/10.33401/fujma.795418
AMA Nuray Yıldırım E, Akgul A. On Solutions of a Higher Order Nonhomogeneous Ordinary Differential Equation. FUJMA. December 2020;3(2):153-160. doi:10.33401/fujma.795418
Chicago Nuray Yıldırım, Elif, and Ali Akgul. “On Solutions of a Higher Order Nonhomogeneous Ordinary Differential Equation”. Fundamental Journal of Mathematics and Applications 3, no. 2 (December 2020): 153-60. https://doi.org/10.33401/fujma.795418.
EndNote Nuray Yıldırım E, Akgul A (December 1, 2020) On Solutions of a Higher Order Nonhomogeneous Ordinary Differential Equation. Fundamental Journal of Mathematics and Applications 3 2 153–160.
IEEE E. Nuray Yıldırım and A. Akgul, “On Solutions of a Higher Order Nonhomogeneous Ordinary Differential Equation”, FUJMA, vol. 3, no. 2, pp. 153–160, 2020, doi: 10.33401/fujma.795418.
ISNAD Nuray Yıldırım, Elif - Akgul, Ali. “On Solutions of a Higher Order Nonhomogeneous Ordinary Differential Equation”. Fundamental Journal of Mathematics and Applications 3/2 (December 2020), 153-160. https://doi.org/10.33401/fujma.795418.
JAMA Nuray Yıldırım E, Akgul A. On Solutions of a Higher Order Nonhomogeneous Ordinary Differential Equation. FUJMA. 2020;3:153–160.
MLA Nuray Yıldırım, Elif and Ali Akgul. “On Solutions of a Higher Order Nonhomogeneous Ordinary Differential Equation”. Fundamental Journal of Mathematics and Applications, vol. 3, no. 2, 2020, pp. 153-60, doi:10.33401/fujma.795418.
Vancouver Nuray Yıldırım E, Akgul A. On Solutions of a Higher Order Nonhomogeneous Ordinary Differential Equation. FUJMA. 2020;3(2):153-60.
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