Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2020, Cilt: 3 Sayı: 1, 35 - 44, 31.03.2020

Öz

Kaynakça

  • [1] R, Hilfer(Ed), Application of fractional calculus in physics, World scientific publishing Company Singapore, (2000)
  • [2] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, 24, North-Holland Mathematics Studies, Amsterdam, (2006).
  • [3] Anastassiou, GA, On right fractional calculus, Chaos. Solit. Fract. 42(1), 365-376 (2009).
  • [4] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 1999.
  • [5] M.Caputo, Linear model of dissipation whose Q is almost frequency independent-II, Geo. J. Roy. Astr.Soci. 13(5),529-539 1967.
  • [6] K. Shah, H.Khalil and R.A Khan, Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations,Chaos. Solit. Fract. 77, 240-246 (2015).
  • [7] M. Benchohra, J. R. Graef, and S. Hamani, Existence results for boundary value problems with nonlinear fractional differential equations, Appl. Anal. 87, 851-863 (2008).
  • [8] B. Ahmad and J. J. Nieto, Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory, Topol. Method. Nonl. Anal. 35, 295-304 (2010).
  • [9] R. Metzler and J. Klafter, Boundary value problems for fractional diffusion equations, Physics A: Stat. Mech. Appl. 278,107-125 (2005).
  • [10] Y. Hu, ,Y. Luo, and Z. Lu, Analytical solution of the linear fractional differential equation by Adomian decomposition method, J. Comput. Appl. Math. 215, 220 - 229 (2008).
  • [11] Hedrih (Stevanovic) K., Transversal creep vibrations of a beam with fractional derivative constitutive relation order. I - Partial fractional-differential equation. II - Stochastic stability of the beam dynamic shape, under axial bounded noise excitation, Proceedings of Forth International Conference on Nonlinear Mechanics (ICNMIV),Shanghai, P.R. China,, 584-595 (2002).
  • [12] Torvik P.J., Bagley R.L., On the appereance of fractional derivatives in the behaviour of real materials,J. Appl. Mech. 51, 294-298 (1984).
  • [13] R. Metzler and J. Klafter, The random walks guide to anomalous diffusion:a fractional dynamics approach, Physics Reports. 339(1), 1-77 (2000).
  • [14] F. Liu, V. Anh and I. Turner, Numerical solution of the space fractional Fokker-Planck equation, J. Comp. Appl. Math. 66, 209-219 (2005).
  • [15] R. Gorenflo, F. Mainardi, E. Scalas and M. Raberto, Fractional calculus and contiuous time finance III, The diffusion limit, Math. Financ. 171-180 (2000).
  • [16] S. Kumar, A Numerical Study for Solution of Time Fractional Nonlinear Shallow-Water Equation in Oceans, Zeitschrift fur Naturforschung A. 68 a, 1-7 (2013)
  • [17] S. Kumar, Numerical Computation of Time-Fractional Fokker-Planck Equation Arising in Solid State Physics and Circuit theory, Zeitschrift fur Naturforschung. 68a, 1-8 (2013).
  • [18] S. Kumar, A new analytical modelling for telegraph equation via Laplace transform, Appl. Math. Model. 38(13), 3154-3163 (2014)
  • [19] S. Kumar, M. M. Rashidi, New Analytical Method for Gas Dynamics Equation Arising in Shock Fronts, Comp. Phy. Commun. 185 (7), 1947-1954 (2014).
  • [20] H. Bulut, M. Belgacem and H.M. Baskonus, Partial fractional differential equation systems solutions by us- ing Adomian decomposition method, Implementation Proceedings of the Fourth International Conference on Mathematical and Computational Applications, June 11-13, 2013, Manisa, Turkey, pp.138-146.
  • [21] H.Jafari and S. Seifi, Solving a system of nonlinear fractional partial differential equations using homotopy analysis method, Commu. Nonl. Sci. Num. Simul. 14, 1962 - 1969 (2009).
  • [22] K. Shah, A. Ali and R. A. Khan, Numerical solutions of Fractional Order System of Bagley-Torvik Equation Using Operational Matrices, Sindh Univ. Res. Jour. 47 (4), 757-762 (2015)
  • [23] M. X. Yi and Y. M. Chen, Haar wavelet operational matrix method for solving fractional partial differential equations, Comp. Model. Eng. Scie. 88(3), 229-244, 2012.
  • [24] Moaddy, K, Momani, S, Hashim, I, The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics, Comput. Math. Appl. 61, 1209-1216 (2011)
  • [25] M. X. Yi, J. Huang, and J. X. Wei, Block pulse operational matrix method for solving fractional partial differential equations, Appl. Math. Comput. 221, 121- 131, (2013).
  • [26] Z.H.Khan and W.A.Khan, N-transform properties and applications, NUST Jour of Engineering Sciences. 1(1), 127-133 (2008).
  • [27] K. Shah, M. Junaid, and N. Ali, Extraction of Laplace, Sumudu, Fourier and Mellin Transform from the Natural Transform, J. Appl. Environ. Biol. Sci. 5(9)1-10, (2015). [28] K. Shah and R. A. Khan, The Applications of Natural Transform to the Analytical Solutions of Some Fractional Order Ordinary Differential Equations, Sindh Univ. Res. Jour. 47 (4), 683-686 (2015).
  • [29] J. H. He, Coupling method of a homotopy technique and a perturbation technique for non-linear problems, Int. J. Non-Linear Mech. 35( 1) 37-43, 2000.
  • [30] R. Silambarasn and F. B. M. Belgacem, Applications of the Natural Trans-form to Maxwell’s Equations, Progress In Electromagnetic Research Symposium Proceedings, Suzhou, China, Sept, 12(16),(2011).
  • [31] Belgacem, F. B. M. and Silambarasan, R. Theory of Natural Transform, Math, Engg, Sci. Aero. 3 (1), 99--124 (2012)
  • [32] Loonker, Deshna and Banerji, P. K. Applications of Natural transform to di?erential equations, J. Indian Acad. Math. 35 (1), 151 -158 (2013).
  • [33] Madani, M, Fathizadeh, M, Khan, Y, Yildirim, A: On the coupling of the homotopy perturbation method and Laplace transformation, Math. Comput. Model. 53, 1937-1945 (2011).
  • [34] M.S. Rawashdeh and S. Maitama, Solving coupled system of nonlinear PDEs using the Natural transform decomposition method, Int. J. Pure. Appl. Math,. 92(5), 757-776 (2014).
  • [35] A. Elsaid, Adomian polynomials: A powerful tool for iterative method of series solution of nonlinear equations, J. Appl. Anal. Comput. 2(4),381-394 (2012).

Natural decomposition method and coupled systems of nonlinear fractional order partial differential equations

Yıl 2020, Cilt: 3 Sayı: 1, 35 - 44, 31.03.2020

Öz

In this article, we present a method known as the Natural decomposition transform method (NDTM). This method is a couple of Natural transform and Adomian decomposition method. By means of this new method, we successfully handle some coupled systems of nonlinear fractional-order partial differential equations (NFPDEs). We obtain the solutions in the form of
series which rapidly converges to the exact solution. Two test examples are provided for the illustration of our method.

Kaynakça

  • [1] R, Hilfer(Ed), Application of fractional calculus in physics, World scientific publishing Company Singapore, (2000)
  • [2] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, 24, North-Holland Mathematics Studies, Amsterdam, (2006).
  • [3] Anastassiou, GA, On right fractional calculus, Chaos. Solit. Fract. 42(1), 365-376 (2009).
  • [4] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 1999.
  • [5] M.Caputo, Linear model of dissipation whose Q is almost frequency independent-II, Geo. J. Roy. Astr.Soci. 13(5),529-539 1967.
  • [6] K. Shah, H.Khalil and R.A Khan, Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations,Chaos. Solit. Fract. 77, 240-246 (2015).
  • [7] M. Benchohra, J. R. Graef, and S. Hamani, Existence results for boundary value problems with nonlinear fractional differential equations, Appl. Anal. 87, 851-863 (2008).
  • [8] B. Ahmad and J. J. Nieto, Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory, Topol. Method. Nonl. Anal. 35, 295-304 (2010).
  • [9] R. Metzler and J. Klafter, Boundary value problems for fractional diffusion equations, Physics A: Stat. Mech. Appl. 278,107-125 (2005).
  • [10] Y. Hu, ,Y. Luo, and Z. Lu, Analytical solution of the linear fractional differential equation by Adomian decomposition method, J. Comput. Appl. Math. 215, 220 - 229 (2008).
  • [11] Hedrih (Stevanovic) K., Transversal creep vibrations of a beam with fractional derivative constitutive relation order. I - Partial fractional-differential equation. II - Stochastic stability of the beam dynamic shape, under axial bounded noise excitation, Proceedings of Forth International Conference on Nonlinear Mechanics (ICNMIV),Shanghai, P.R. China,, 584-595 (2002).
  • [12] Torvik P.J., Bagley R.L., On the appereance of fractional derivatives in the behaviour of real materials,J. Appl. Mech. 51, 294-298 (1984).
  • [13] R. Metzler and J. Klafter, The random walks guide to anomalous diffusion:a fractional dynamics approach, Physics Reports. 339(1), 1-77 (2000).
  • [14] F. Liu, V. Anh and I. Turner, Numerical solution of the space fractional Fokker-Planck equation, J. Comp. Appl. Math. 66, 209-219 (2005).
  • [15] R. Gorenflo, F. Mainardi, E. Scalas and M. Raberto, Fractional calculus and contiuous time finance III, The diffusion limit, Math. Financ. 171-180 (2000).
  • [16] S. Kumar, A Numerical Study for Solution of Time Fractional Nonlinear Shallow-Water Equation in Oceans, Zeitschrift fur Naturforschung A. 68 a, 1-7 (2013)
  • [17] S. Kumar, Numerical Computation of Time-Fractional Fokker-Planck Equation Arising in Solid State Physics and Circuit theory, Zeitschrift fur Naturforschung. 68a, 1-8 (2013).
  • [18] S. Kumar, A new analytical modelling for telegraph equation via Laplace transform, Appl. Math. Model. 38(13), 3154-3163 (2014)
  • [19] S. Kumar, M. M. Rashidi, New Analytical Method for Gas Dynamics Equation Arising in Shock Fronts, Comp. Phy. Commun. 185 (7), 1947-1954 (2014).
  • [20] H. Bulut, M. Belgacem and H.M. Baskonus, Partial fractional differential equation systems solutions by us- ing Adomian decomposition method, Implementation Proceedings of the Fourth International Conference on Mathematical and Computational Applications, June 11-13, 2013, Manisa, Turkey, pp.138-146.
  • [21] H.Jafari and S. Seifi, Solving a system of nonlinear fractional partial differential equations using homotopy analysis method, Commu. Nonl. Sci. Num. Simul. 14, 1962 - 1969 (2009).
  • [22] K. Shah, A. Ali and R. A. Khan, Numerical solutions of Fractional Order System of Bagley-Torvik Equation Using Operational Matrices, Sindh Univ. Res. Jour. 47 (4), 757-762 (2015)
  • [23] M. X. Yi and Y. M. Chen, Haar wavelet operational matrix method for solving fractional partial differential equations, Comp. Model. Eng. Scie. 88(3), 229-244, 2012.
  • [24] Moaddy, K, Momani, S, Hashim, I, The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics, Comput. Math. Appl. 61, 1209-1216 (2011)
  • [25] M. X. Yi, J. Huang, and J. X. Wei, Block pulse operational matrix method for solving fractional partial differential equations, Appl. Math. Comput. 221, 121- 131, (2013).
  • [26] Z.H.Khan and W.A.Khan, N-transform properties and applications, NUST Jour of Engineering Sciences. 1(1), 127-133 (2008).
  • [27] K. Shah, M. Junaid, and N. Ali, Extraction of Laplace, Sumudu, Fourier and Mellin Transform from the Natural Transform, J. Appl. Environ. Biol. Sci. 5(9)1-10, (2015). [28] K. Shah and R. A. Khan, The Applications of Natural Transform to the Analytical Solutions of Some Fractional Order Ordinary Differential Equations, Sindh Univ. Res. Jour. 47 (4), 683-686 (2015).
  • [29] J. H. He, Coupling method of a homotopy technique and a perturbation technique for non-linear problems, Int. J. Non-Linear Mech. 35( 1) 37-43, 2000.
  • [30] R. Silambarasn and F. B. M. Belgacem, Applications of the Natural Trans-form to Maxwell’s Equations, Progress In Electromagnetic Research Symposium Proceedings, Suzhou, China, Sept, 12(16),(2011).
  • [31] Belgacem, F. B. M. and Silambarasan, R. Theory of Natural Transform, Math, Engg, Sci. Aero. 3 (1), 99--124 (2012)
  • [32] Loonker, Deshna and Banerji, P. K. Applications of Natural transform to di?erential equations, J. Indian Acad. Math. 35 (1), 151 -158 (2013).
  • [33] Madani, M, Fathizadeh, M, Khan, Y, Yildirim, A: On the coupling of the homotopy perturbation method and Laplace transformation, Math. Comput. Model. 53, 1937-1945 (2011).
  • [34] M.S. Rawashdeh and S. Maitama, Solving coupled system of nonlinear PDEs using the Natural transform decomposition method, Int. J. Pure. Appl. Math,. 92(5), 757-776 (2014).
  • [35] A. Elsaid, Adomian polynomials: A powerful tool for iterative method of series solution of nonlinear equations, J. Appl. Anal. Comput. 2(4),381-394 (2012).
Toplam 34 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Articles
Yazarlar

Ibrar Haq Bu kişi benim

Zakir Ullah Bu kişi benim

Yayımlanma Tarihi 31 Mart 2020
Yayımlandığı Sayı Yıl 2020 Cilt: 3 Sayı: 1

Kaynak Göster

APA Haq, I., & Ullah, Z. (2020). Natural decomposition method and coupled systems of nonlinear fractional order partial differential equations. Results in Nonlinear Analysis, 3(1), 35-44.
AMA Haq I, Ullah Z. Natural decomposition method and coupled systems of nonlinear fractional order partial differential equations. RNA. Mart 2020;3(1):35-44.
Chicago Haq, Ibrar, ve Zakir Ullah. “Natural Decomposition Method and Coupled Systems of Nonlinear Fractional Order Partial Differential Equations”. Results in Nonlinear Analysis 3, sy. 1 (Mart 2020): 35-44.
EndNote Haq I, Ullah Z (01 Mart 2020) Natural decomposition method and coupled systems of nonlinear fractional order partial differential equations. Results in Nonlinear Analysis 3 1 35–44.
IEEE I. Haq ve Z. Ullah, “Natural decomposition method and coupled systems of nonlinear fractional order partial differential equations”, RNA, c. 3, sy. 1, ss. 35–44, 2020.
ISNAD Haq, Ibrar - Ullah, Zakir. “Natural Decomposition Method and Coupled Systems of Nonlinear Fractional Order Partial Differential Equations”. Results in Nonlinear Analysis 3/1 (Mart 2020), 35-44.
JAMA Haq I, Ullah Z. Natural decomposition method and coupled systems of nonlinear fractional order partial differential equations. RNA. 2020;3:35–44.
MLA Haq, Ibrar ve Zakir Ullah. “Natural Decomposition Method and Coupled Systems of Nonlinear Fractional Order Partial Differential Equations”. Results in Nonlinear Analysis, c. 3, sy. 1, 2020, ss. 35-44.
Vancouver Haq I, Ullah Z. Natural decomposition method and coupled systems of nonlinear fractional order partial differential equations. RNA. 2020;3(1):35-44.