Analytical and Qualitative Study of Some Families of FODEs via Differential Transform Method
Abstract
:1. Introduction
2. Fundamental Concepts
3. Study of Problem I
3.1. Qualitative Theory of Problem I
- For any , ∃ a constant with
- Let ∃ constants and , then the given growth condition hold
3.2. Numerical Procedure for Problem I
3.3. Numerical Problems to Verify the Establish Analysis
4. Study of Problem II
4.1. Qualitative Theory of (2)
- For any , there exist a constant such that
- Let there exists constant and ; such that the given growth condition hold:
4.2. Numerical Procedure to Problem II
5. Concluding Remarks
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Miller, K.S.; Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations; Wiley: New York, USA, 1993. [Google Scholar]
- Yang, S.; Xiao, A.; Su, H. Convergence of the variational iteration method for solving multi-order fractional differential equations. Comput. Math. Appl. 2010, 60, 2871–2879. [Google Scholar] [CrossRef] [Green Version]
- Doha, E.H.; Bhrawy, A.H.; Ezz-Eldien, S.S. Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations. Appl. Math. Model. 2011, 35, 5662–5672. [Google Scholar] [CrossRef]
- Doha, E.H.; Bhrawy, A.H.; Ezz-Eldien, S.S. A new Jacobi operational matrix: An application for solving fractional differential equations. Appl. Math. Model. 2012, 36, 4931–4943. [Google Scholar] [CrossRef]
- Goodrich, C. Existence of a positive solution to a class of fractional differential equations. Comput. Math. Appl. 2010, 59, 3889–3999. [Google Scholar] [CrossRef] [Green Version]
- Su, X. Boundary value problem for a coupled system of non linear fractional differential equations. Appl. Math. Lett. 2009, 22, 64–69. [Google Scholar] [CrossRef] [Green Version]
- Ahmad, B.; Nieto, J.J. Existence result for a coupled system of non linear fractional differential equations with three point boundary conditions. Comput. Math. Appl. 2009, 58, 1838–1843. [Google Scholar] [CrossRef] [Green Version]
- Lakshmikantham, V.; Leela, S. Naguma-type uniqueness result for fractional differential equation. Non-Linear Anal. 2009, 71, 2886–2889. [Google Scholar] [CrossRef]
- Lia, Y.; Haq, F.; Shah, K.; Shahzad, M.; Rahman, G. Numerical analysis of fractional order Pine wilt disease model with bilinear incident rate. J. Maths Comput. Sci. 2017, 17, 420–428. [Google Scholar] [CrossRef] [Green Version]
- Haq, F.; Shah, K.; Khan, A.; Shahzad, M.; Rahman, G. Numerical Solution of Fractional Order Epidemic Model of a Vector Born Disease by Laplace Adomian Decomposition Method. Punjab Univ. J. Math. 2007, 49, 13–22. [Google Scholar]
- Ray, S.S.; Bera, R.K. Solution of an extraordinary differential equation by Adomian decomposition method. J. Appl. Math. 2004, 4, 331–338. [Google Scholar] [CrossRef]
- Hashim, I.; Abdulaziz, O.; Momani, S. Homotopy analysis method for fractional IVPs. Commun. Nonlinear Sci. Numer. Simul. 2009, 14, 674–684. [Google Scholar] [CrossRef]
- Bhrawy, A.H.; Alghamdi, M.A. Numerical solutions of odd order linear and nonlinear initial value problems using shifted Jacobi spectral approximations. Abst. Appl. Anal. 2012, 2012, 14. [Google Scholar] [CrossRef]
- Avaji, M.; Hafshejani, J.; Dehcheshmeh, S.; Ghahfarokhi, D. Solution of delay Volterra integral equations using the Variational iteration method. J. Appl. Sci. 2012, 1, 196–200. [Google Scholar] [CrossRef]
- Biazar, J.; Eslami, M. A new homotopy perturbation method for solving systems of partial differential equations. Comput. Math. Appl. 2011, 62, 225–234. [Google Scholar] [CrossRef] [Green Version]
- He, J.H. Homotopy perturbation method for solving boundary value problems. Phys. Lett. A 2006, 350, 87–88. [Google Scholar] [CrossRef]
- Jafari, H.; Daftardar-Gejji, V. Solving a system of nonlinear fractional differential equations using Adomian decomposition. J. Comput. Appl. Math. 2006, 196, 644–651. [Google Scholar] [CrossRef] [Green Version]
- Momani, S.; Odibat, Z. Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method. Appl. Math. Comput. 2006, 177, 488–494. [Google Scholar] [CrossRef]
- Zhou, J.K. Differential Transformation and Its Applications for Electrical Circuits; Huazhong University Press: Wuhan, China, 1986. [Google Scholar]
- Ayaz, F. Solutions of the system of differential equations by differential transform method. Appl. Math. Comput. 2004, 147, 547–567. [Google Scholar] [CrossRef]
- Ayaz, F. Application of differential transform method to differential-algebraic equations. Appl. Math. Comput. 2004, 152, 649–657. [Google Scholar] [CrossRef]
- Arikoglu, A.; Ozkol, I. Solution of boundary value problems for integro-differential equations by using differential transform method. Appl. Math. Comput. 2005, 168, 1145–1158. [Google Scholar] [CrossRef]
- Arikoglu, A.; Ozkol, I. Solution of fractional differential equations by using differential transform method. Chaos Solitons Fractals 2007, 34, 1473–1481. [Google Scholar] [CrossRef]
- Carpinteri, A.; Mainardi, F. Fractals and Fractional Calculus in Continuum Mechanics; Springer: Wien, Austria; New York, NY, USA, 1997. [Google Scholar]
- Ertürka, V.S.; Momani, S. Solving systems of fractional differential equations using differential transform method. J. Comput. Appl. Math. 2008, 215, 142–151. [Google Scholar] [CrossRef] [Green Version]
- Momani, S.; Odibat, Z. Numerical comparison of methods for solving linear differential equations of fractional order. Chaos Solitons Fractals 2007, 31, 1248–1255. [Google Scholar] [CrossRef]
- Akgül, A.; Inc, M.; Hashemi, M.S. Group preserving scheme and reproducing kernel method for the Poisson-Boltzmann equation for semiconductor devices. Nonlinear Dyn. 2017, 88, 2817–2829. [Google Scholar] [CrossRef]
- Hosseini, K.; Bekir, A.; Kaplan, M.; Güner, Ö. On a new technique for solving the nonlinear conformable time-fractional differential equations. Opt. Quantum Electron. 2017, 49, 343. [Google Scholar] [CrossRef]
- Hashemi, M.S.; Inç, M.; Bayram, M. Symmetry properties and exact solutions of the time fractional Kolmogorov-Petrovskii-Piskunov equation. Revista Mexicana Física 2019, 65, 529–535. [Google Scholar] [CrossRef] [Green Version]
- Hashemi, M.S. A novel approach to find exact solutions of fractional evolution equations with non-singular kernel derivative. Chaos Solitons Fractals 2021, 152, 111367. [Google Scholar] [CrossRef]
- Hashemi, M.S.; Baleanu, D. Lie Symmetry Analysis of Fractional Differential Equations; CRC Press: New York, NY, USA, 2020. [Google Scholar]
- Hassan, I.H.A. Application to differential transformation method for solving systems of differential equations. Appl. Math. Model. 2008, 32, 2552–2559. [Google Scholar] [CrossRef]
- Smart, D.R. Fixed Point Theorems; Cambridge University Press: Cambridge, UK, 1980. [Google Scholar]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Neelma; Eiman; Shah, K. Analytical and Qualitative Study of Some Families of FODEs via Differential Transform Method. Foundations 2022, 2, 6-19. https://doi.org/10.3390/foundations2010002
Neelma, Eiman, Shah K. Analytical and Qualitative Study of Some Families of FODEs via Differential Transform Method. Foundations. 2022; 2(1):6-19. https://doi.org/10.3390/foundations2010002
Chicago/Turabian StyleNeelma, Eiman, and Kamal Shah. 2022. "Analytical and Qualitative Study of Some Families of FODEs via Differential Transform Method" Foundations 2, no. 1: 6-19. https://doi.org/10.3390/foundations2010002