Dynamical Behaviors of Separated Homotopy Method Defined by Conformable Operator

Abstract

In this paper, we consider some linear/nonlinear differential equations (DEs) containing conformable derivative operator (CDO). We obtain approximate solutions of these mentioned DEs in the form of infinite series which converges swiftly to its exact value by using separated homotopy method (SHM). Using the conformable operator in solutions of different types of DEs makes the solution steps are computable easily. As well as some theoretical results of the conformable operator, it has been used in modelling the DEs and describing certain problems such as engineering, material sciences, economic and other areas of application. In this context, the aim of this study is to apply the mentioned method to some illustrative linear/nonlinear problems and to solve them as mathematically. In addition, comparing the exact solutions with the obtained solutions is considered by the presentation of some plots. Therefore, the results of this study show the reliability and simplicity of the methods constructed with the conformable operator.

Keywords

• Conformable operator,separated homotopy method,approximate solution,nonlinear equation

References

1. [1] J. Hristov, Approximate Solutions to Fractional Subdiffusion Equations, The European Physical Journal Special Topics, 193, 229-243 (2011).
2. [2] D. Baleanu, J.T. Machado, C. Cattani, M.C. Baleanu and X.-J. Yang, Local Fractional Variational Iteration and Decomposition Methods for Wave Equation on Cantor Sets within Local Fractional Operators, Abstract and Applied Analysis, 2014, (2014).
3. [3] Y.Z. Povstenko, Fractional Heat Conduction Equation and Associated Thermal Stress, Journal of Thermal Stresses, 28, 83-102 (2004).
4. [4] V. Turut and N. G¨uzel, On Solving Partial Differential Equations of Fractional Order by Using the Variational Iteration Method and Multivariate Pad´e Approximations, European Journal of Pure and Applied Mathematics, 6, 147-171 (2013).
5. [5] H. Budak, F. Usta, M.Z. Sarikaya and M.E. Ozdemir, On Generalization of Midpoint Type Inequalities with Generalized Fractional Integral Operators, Revista de la Real Academia de Ciencias Exactas, F´ısicas y Naturales. Serie A. Matem´aticas, 1-22 (2017).
6. [6] D. Kumar, J. Singh and D. Baleanu, Analysis of Regularized Long-Wave Equation Associated with a New Fractional Operator with Mittag-Leffler Type Kernel, Physica A: Statistical Mechanics and its Applications, 492, 155-167 (2018).
7. [7] M. Kurulay, A. Secer and M.A. Akinlar, A New Approximate Analytical Solution of Kuramoto-Sivashinsky Equation Using Homotopy Analysis Method, Applied Mathematics and Information Sciences, 7, 267-271 (2013).
8. [8] T.A. Yıldız, S. Arshad and D. Baleanu, New Observations on Optimal Cancer Treatments for a Fractional Tumor Growth Model with and without Singular Kernel, Chaos, Solitons and Fractals, 117, 226-239 (2018).

9. [9] A. Yokus, H.M. Baskonus, T.A. Sulaiman and H. Bulut, Numerical Simulation and Solutions of the Two-Component Second Order Kdv Evolutionary system, Numerical Methods for Partial Differential Equations, 34, 211-227 (2018).
10. [10] N.A. Pirim and F. Ayaz, A New Technique for Solving Fractional Order Systems: Hermite Collocation Method, Applied Mathematics, 7, 2307 (2016).
11. [11] M. Yavuz and N. O¨ zdemir, A Different Approach to the European Option Pricing Model with New Fractional Operator, Mathematical Modelling of Natural Phenomena, 13, 12 (2018).
12. [12] S. Kumar, D. Kumar and J. Singh, Numerical Computation of Fractional Black–Scholes Equation Arising in Financial Market, Egyptian Journal of Basic and Applied Sciences, 1, 177-183 (2014).
13. [13] S. Kumar, A. Yildirim, Y. Khan, H. Jafari, K. Sayevand and L. Wei, Analytical Solution of Fractional Black-Scholes European Option Pricing Equation by Using Laplace Transform, Journal of fractional calculus and Applications, 2, 1-9 (2012).
14. [14] M. Yavuz and N. Ozdemir, A Quantitative Approach to Fractional Option Pricing Problems with Decomposition Series. Konuralp Journal of Mathematics, 6(1), 102-109 (2018).
15. [15] M. Yavuz and N. Ozdemir, European Vanilla Option Pricing Model of Fractional Order without Singular Kernel, Fractal and Fractional, 2, 3 (2018).
16. [16] R.M. Jena and S. Chakraverty, A New Iterative Method Based Solution for Fractional Black–Scholes Option Pricing Equations (BSOPE), SN Applied Sciences, 1, 95 (2019).
17. [17] A. Atangana and B.S.T. Alkahtani, New Model of Groundwater Flowing within a Confine Aquifer: Application of Caputo-Fabrizio Derivative, Arabian Journal of Geosciences, 9, 8 (2016).
18. [18] A. Atangana and D. Baleanu, Caputo-Fabrizio Derivative Applied to Groundwater Flow within Confined Aquifer, Journal of Engineering Mechanics, 143, D4016005 (2017).
19. [19] J. Singh, D. Kumar, Z. Hammouch and A. Atangana, A Fractional Epidemiological Model for Computer Viruses Pertaining to a New Fractional Derivative, Applied Mathematics and Computation, 316, 504-515 (2018).
20. [20] O.A. Arqub and A. El-Ajou, Solution of the Fractional Epidemic Model by Homotopy Analysis Method, Journal of King Saud University-Science, 25, 73-81 (2013).
21. [21] M.A. Dokuyucu, E. Celik, H. Bulut and H.M. Baskonus, Cancer Treatment Model with the Caputo-Fabrizio Fractional Derivative, The European Physical Journal Plus, 133, 92 (2018).
22. [22] Shah, Z., Islam, S., Gul, T., Bonyah, E. and Khan, M. A. The electrical MHD and hall current impact on micropolar nanofluid flow between rotating parallel plates. Results in Physics, 9, 1201-1214 (2018).
23. [23] C. Ionescu, A. Lopes, D. Copot, J.T. Machado and J. Bates, The Role of Fractional Calculus in Modeling Biological Phenomena: A Review, Communications in Nonlinear Science and Numerical Simulation, 51, 141-159 (2017).
24. [24] C.M. Pinto and A.R. Carvalho, A Latency Fractional Order Model for Hiv Dynamics, Journal of Computational and Applied Mathematics, 312, 240-256 (2017).
25. [25] M. Yavuz and E. Bonyah, New Approaches to the Fractional Dynamics of Schistosomiasis Disease Model, Physica A: Statistical Mechanics and its Applications, (2019), In Press.
26. [26] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (Vol. 198), Elsevier, (1998).
27. [27] S. Arshad and V. Lupulescu, On the fractional differential equations with uncertainty, Nonlinear Analysis: Theory, Methods and Applications, 74, 3685-3693 (2011).
28. [28] D. Valerio, JJ. Trujillo, M. Rivero, J.T. Machado and D. Baleanu, Fractional calculus: A survey of useful formulas, The European Physical Journal Special Topics, 222, 1827-1846 (2013).
29. [29] S. Liao, Homotopy Analysis Method in Nonlinear Differential Equations, Verlag Berlin Heidelberg, Springer, (2012).
30. [30] R.Y. Molliq, M.S.M. Noorani, I. Hashim and R.R. Ahmad, Approximate Solutions of Fractional Zakharov–Kuznetsov Equations by Vim, Journal of Computational and Applied Mathematics, 233, 103-108 (2009).
31. [31] M. Zurigat, S. Momani and A. Alawneh, Analytical Approximate Solutions of Systems of Fractional Algebraic–Differential Equations by Homotopy Analysis Method, Computers & Mathematics with Applications, 59, 1227-1235 (2010).
32. [32] V. Gupta and S. Gupta, Application of homotopy analysis method for solving nonlinear Cauchy problem, Surveys in Mathematics and its Applications, 7(1), 105-116 (2012).
33. [33] G. Adomian, A Review of the Decomposition Method in Applied Mathematics, Journal of Mathematical Analysis and Applications, 135, 501-544 (1988).
34. [34] K. Abbaoui and Y. Cherruault, New Ideas for Proving Convergence of Decomposition Methods, Computers & Mathematics with Applications, 29, 103-108 (1995).
35. [35] A.-M. Wazwaz and S.M. El-Sayed, A New Modification of the Adomian Decomposition Method for Linear and Nonlinear Operators, Applied Mathematics and Computation, 122, 393-405 (2001).
36. [36] J.-S. Duan, R. Rach, D. Baleanu and A.-M. Wazwaz, A Review of the Adomian Decomposition Method and Its Applications to Fractional Differential Equations, Communications in Fractional Calculus, 3, 73-99 (2012).
37. [37] V. Turut, E. C¸ elik and M. Yi˘gider, Multivariate Pad´e approximation for solving partial differential equations (PDE). International Journal for Numerical Methods in Fluids, 66(9), 1159-1173 (2011).
38. [38] N. Ozdemir and M. Yavuz, Numerical solution of fractional Black-Scholes equation by using the multivariate Pade´ approximation. Acta Physica Polonica A, 132(3), 1050-1053 (2017).
39. [39] M. Yavuz and B. Yas¸kıran, Conformable Derivative Operator in Modelling Neuronal Dynamics. Applications and Applied Mathematics: An International Journal (AAM), 13, 2, 803-817 (2018).
40. [40] J. He, Homotopy Perturbation Technique, Computer methods in applied mechanics and engineering, 178, 257-262 (1999).
41. [41] J. He, Homotopy Perturbation Method: A New Nonlinear Analytical Technique, Applied Mathematics and Computation, 135, 73-79 (2003).
42. [42] S. Momani and Z. Odibat, Comparison between the Homotopy Perturbation Method and the Variational Iteration Method for Linear Fractional Partial Differential Equations, Computers & Mathematics with Applications, 54, 910-919 (2007).
43. [43] Z. Odibat and S. Momani, Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order. Chaos, Solitons & Fractals, 36(1), 167-174 (2008).
44. [44] O. Abdulaziz, I. Hashim and S. Momani, Application of Homotopy-Perturbation Method to Fractional IVPs, Journal of Computational and Applied Mathematics, 216, 574-584 (2008).
45. [45] H. Jafari and S. Momani, Solving Fractional Diffusion and Wave Equations by Modified Homotopy Perturbation Method, Physics Letters A, 370, 388-396 (2007).
46. [46] M. Yavuz and B. Yas¸kıran, Homotopy methods for fractional linear/nonlinear differential equations with a local derivative operator, Balıkesir U¨ niversitesi Fen Bilimleri Enstit¨us¨u Dergisi, 20(3), 75-89 (2018).
47. [47] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A New Definition of Fractional Derivative, Journal of Computational and Applied Mathematics, 264, 65-70 (2014).
48. [48] H. Batarfi, J. Losada, J.J. Nieto and W. Shammakh, Three-Point Boundary Value Problems for Conformable Fractional Differential Equations, Journal of Function Spaces, 2015, 6 (2015).
49. [49] F. Usta and M.Z. Sarıkaya, Explicit bounds on certain integral inequalities via conformable fractional calculus. Cogent Mathematics and Statistics, 4(1), 1277505 (2017).
50. [50] D. Avcı, B.B.I. Eroglu and N. Ozdemir, Conformable Heat Equation on a Radial Symmetric Plate, Thermal Science, 21, 819-826 (2017).
51. [51] M. Yavuz, Novel Solution Methods for Initial Boundary Value Problems of Fractional Order with Conformable Differentiation, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 8(1), 1-7 (2018).
52. [52] M. Yavuz and N. Ozdemir, New numerical techniques for solving fractional partial differential equations in conformable sense. In: Ostalczyk P., Sankowski D., Nowakowski J. (eds) Non-Integer Order Calculus and its Applications. RRNR 2017. Lecture Notes in Electrical Engineering, vol 496. Springer, Cham (2019).
53. [53] M. Yavuz and B. Yaskıran, Approximate-Analytical Solutions of Cable Equation Using Conformable Fractional Operator, New Trends in Mathematical Sciences 5, 209-219 (2017).
54. [54] F. Evirgen, Conformable Fractional Gradient Based Dynamic System for Constrained Optimization Problem, Acta Physica Polonica A, 132, 1066-1069 (2017).
55. [55] M. Yavuz and N. O¨ zdemir, On the solutions of fractional Cauchy problem featuring conformable derivative. ITM Web of Conferences (Vol. 22, p. 01045). EDP Sciences, (2018).
56. [56] S. Uc¸ar, N.Y. O¨ zgu¨r and B.B.I˙. Erog˘lu, Complex conformable derivative. Arabian Journal of Geosciences, 12(6), 201 (2019).
57. [57] F. Usta and M.Z. Sarıkaya, Some improvements of conformable fractional integral inequalities. International Journal of Analysis and Applications, 14(2), 162-166 (2017).
58. [58] D. Anderson and D. Ulness, Newly Defined Conformable Derivatives, Advances in Dynamical Systems and Applications, 10, 109-137 (2015).
59. [59] K. Moaddy, S. Momani and I. Hashim, The Non-Standard Finite Difference Scheme for Linear Fractional Pdes in Fluid Mechanics, Computers & Mathematics with Applications, 61, 1209-1216 (2011).
60. [60] B. Ghazanfari and F. Veisi, Homotopy Analysis Method for the Fractional Nonlinear Equations, Journal of King Saud University-Science, 23, 389-393 (2011).