Modified Modelling for Heat Like Equations within Caputo Operator
Abstract
:1. Introduction
2. Definitions and Preliminaries
3. The Presentation of the Method
4. Implementation of the Method
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
- Gao, H.; Liu, C.; He, C.; Xu, X.; Wu, S.; Li, Y. Performance analysis and working fluid selection of a supercritical organic Rankine cycle for low grade waste heat recovery. Energies 2012, 5, 3233–3247. [Google Scholar] [CrossRef]
- Glavatskaya, Y.; Podevin, P.; Lemort, V.; Shonda, O.; Descombes, G. Reciprocating expander for an exhaust heat recovery rankine cycle for a passenger car application. Energies 2012, 5, 1751–1765. [Google Scholar] [CrossRef]
- Fu, B.; Ouyang, C.; Li, C.; Wang, J.; Gul, E. An improved mixed integer linear programming approach based on symmetry diminishing for unit commitment of hybrid power system. Energies 2019, 12, 833. [Google Scholar] [CrossRef] [Green Version]
- D’agostino, D.; Zangheri, P.; Castellazzi, L. Towards nearly zero energy buildings in Europe: A focus on retrofit in non-residential buildings. Energies 2017, 10, 117. [Google Scholar] [CrossRef]
- Bornoff, R. Extraction of boundary condition independent dynamic compact thermal models of LEDs—A delphi4LED methodology. Energies 2019, 12, 1628. [Google Scholar] [CrossRef] [Green Version]
- Prince, M.; Vigeant, M.; Nottis, K. Development of the heat and energy concept inventory: Preliminary results on the prevalence and persistence of engineering students’ misconceptions. J. Eng. Educ. 2012, 101, 412–438. [Google Scholar] [CrossRef]
- Khan, H.; Shah, R.; Arif, M.; Bushnaq, S. The Chebyshev Wavelet Method (CWM) for the Numerical Solution of Fractional HIV Infection of CD4+T Cells Model. International. J. Appl. Comput. Math. 2020, 6, 1–17. [Google Scholar]
- Kuo, P.H.; Huang, C.J. A high precision artificial neural networks model for short-term energy load forecasting. Energies 2018, 11, 213. [Google Scholar] [CrossRef] [Green Version]
- Bokhari, A.H.; Mohammad, G.; Mustafa, M.T.; Zaman, F.D. Adomian decomposition method for a nonlinear heat equation with temperature dependent thermal properties. Math. Probl. Eng. 2009, 2009. [Google Scholar] [CrossRef]
- Sarwar, S.; Alkhalaf, S.; Iqbal, S.; Zahid, M.A. A note on optimal homotopy asymptotic method for the solutions of fractional order heat-and wave-like partial differential equations. Comput. Math. Appl. 2015, 70, 942–953. [Google Scholar] [CrossRef]
- Khan, H.; Shah, R.; Kumam, P.; Arif, M. Analytical Solutions of Fractional-Order Heat and Wave Equations by the Natural Transform Decomposition Method. Entropy 2019, 21, 597. [Google Scholar] [CrossRef] [Green Version]
- Secer, A. Approximate analytic solution of fractional heat-like and wave-like equations with variable coefficients using the differential transforms method. Adv. Differ. Equ. 2012, 2012, 198. [Google Scholar] [CrossRef] [Green Version]
- Liu, C.F.; Kong, S.S.; Yuan, S.J. Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem. Therm. Sci. 2013, 17, 715–721. [Google Scholar] [CrossRef]
- Shou, D.H.; He, J.H. Beyond Adomian method: The variational iteration method for solving heat-like and wave-like equations with variable coefficients. Phys. Lett. A 2008, 372, 233–237. [Google Scholar] [CrossRef]
- Batiha, B.; Noorani, M.S.M.; Hashim, I.; Batiha, K. Numerical simulations of systems of PDEs by variational iteration method. Phys. Lett. A 2008, 372, 822–829. [Google Scholar] [CrossRef]
- Wazwaz, A.M. The variational iteration method for solving linear and nonlinear systems of PDEs. Comput. Math. Appl. 2007, 54, 895–902. [Google Scholar] [CrossRef] [Green Version]
- Khan, H.; Shah, R.; Kumam, P.; Baleanu, D.; Arif, M. An efficient analytical technique, for the solution of fractional-order telegraph equations. Mathematics 2019, 7, 426. [Google Scholar] [CrossRef] [Green Version]
- Shah, R.; Khan, H.; Kumam, P.; Arif, M.; Baleanu, D. Natural transform decomposition method for solving fractional-order partial differential equations with proportional delay. Mathematics 2019, 7, 532. [Google Scholar] [CrossRef] [Green Version]
- Jan, R.; Xiao, Y. Effect of partial immunity on transmission dynamics of dengue disease with optimal control. Math. Methods Appl. Sci. 2019, 42, 1967–1983. [Google Scholar] [CrossRef]
- Jan, R.; Xiao, Y. Effect of pulse vaccination on dynamics of dengue with periodic transmission functions. Adv. Differ. Equ. 2019, 1, 368. [Google Scholar] [CrossRef] [Green Version]
- Duangpithak, S. Variational iteration method for special nonlinear partial differential equations. Int. J. Math. Anal. 2012, 6, 1071–1077. [Google Scholar]
- Adomian, G. Solving Frontier Problems of Physics: The Decomposition Method; With a Preface by Yves Cherruault. Fundamental Theories of Physics; Kluwer Academic Publishers Group: Dordrecht, The Netherlands, 1994. [Google Scholar]
- Adomian, G. The diffusion-Brusselator equation. Comput. Math. Appl. 1995, 29, 1–3. [Google Scholar] [CrossRef] [Green Version]
- Eltayeb, H.; Bachar, I.; Kılıçman, A. On conformable double laplace transform and one dimensional fractional coupled burgers’ equation. Symmetry 2019, 11, 417. [Google Scholar] [CrossRef] [Green Version]
- Zhu, J.; Zhang, Y.T.; Newman, S.A.; Alber, M. Application of discontinuous Galerkin methods for reaction-diffusion systems in developmental biology. J. Sci. Comput. 2009, 40, 391–418. [Google Scholar] [CrossRef]
- Abdou, M.A. Approximate solutions of system of PDEEs arising in physics. Int. J. Nonlinear Sci 2011, 12, 305–312. [Google Scholar]
- Quintana-Murillo, J.; Yuste, S.B. A finite difference method with non-uniform timesteps for fractional diffusion and diffusion-wave equations. Eur. Phys. J. Spec. Top. 2013, 222, 1987–1998. [Google Scholar] [CrossRef]
- Zhang, Y.N.; Sun, Z.Z.; Liao, H.L. Finite difference methods for the time fractional diffusion equation on non-uniform meshes. J. Comput. Phys. 2014, 265, 195–210. [Google Scholar] [CrossRef]
- Goufo, E.F.D. Fractal and fractional dynamics for a 3D autonomous and two-wing smooth chaotic system. Alex. Eng. J. 2020. [Google Scholar] [CrossRef]
- Yuste, S.B.; Quintana-Murillo, J. A finite difference method with non-uniform timesteps for fractional diffusion equations. Comput. Phys. Commun. 2012, 183, 2594–2600. [Google Scholar] [CrossRef] [Green Version]
- Priya, G.S.; Prakash, P.; Nieto, J.J.; Kayar, Z. Higher-order numerical scheme for the fractional heat equation with Dirichlet and Neumann boundary conditions. Numer. Heat Transf. Part B Fundam. 2013, 63, 540–559. [Google Scholar] [CrossRef]
- Mustapha, K. An implicit finite-difference time-stepping method for a sub-diffusion equation, with spatial discretization by finite elements. IMA J. Numer. Anal. 2011, 31, 719–739. [Google Scholar] [CrossRef]
- Gao, G.H.; Sun, Z.Z. A compact finite difference scheme for the fractional sub-diffusion equations. J. Comput. Phys. 2011, 230, 586–595. [Google Scholar] [CrossRef]
- Shah, R.; Khan, H.; Arif, M.; Kumam, P. Application of Laplace–Adomian decomposition method for the analytical solution of third-order dispersive fractional partial differential equations. Entropy 2019, 21, 335. [Google Scholar] [CrossRef] [Green Version]
- Mahmood, S.; Shah, R.; Arif, M. Laplace Adomian Decomposition Method for Multi Dimensional Time Fractional Model of Navier-Stokes Equation. Symmetry 2019, 11, 149. [Google Scholar] [CrossRef] [Green Version]
- Shah, R.; Khan, H.; Baleanu, D.; Kumam, P.; Arif, M. A novel method for the analytical solution of fractional Zakharov–Kuznetsov equations. Adv. Differ. Equ. 2019, 2019, 1–14. [Google Scholar] [CrossRef]
- Srivastava, H.M.; Shah, R.; Khan, H.; Arif, M. Some analytical and numerical investigation of a family of fractional-order Helmholtz equations in two space dimensions. Math. Methods Appl. Sci. 2020, 43, 199–212. [Google Scholar] [CrossRef]
- Shah, R.; Khan, H.; Kumam, P.; Arif, M. An analytical technique to solve the system of nonlinear fractional partial differential equations. Mathematics 2019, 7, 505. [Google Scholar] [CrossRef] [Green Version]
- Saad, K.M.; Al-Shomrani, A.A. An application of homotopy analysis transform method for Riccati differential equation of fractional order. J. Fract. Calc. Appl. 2016, 7, 61–72. [Google Scholar]
- Prakash, A.; Verma, V.; Kumar, D.; Singh, J. Analytic study for fractional coupled Burger’s equations via Sumudu transform method. Nonlinear Eng. 2018, 7, 323–332. [Google Scholar] [CrossRef] [Green Version]
- Daftardar-Gejji, V.; Jafari, H. An iterative method for solving nonlinear functional equations. J. Math. Anal. Appl. 2006, 316, 753–763. [Google Scholar] [CrossRef] [Green Version]
- Jafari, H. Iterative Methods for Solving System of Fractional Differential. Ph.D. Thesis, Pune University, Pune City, India, 2006. [Google Scholar]
- Bhalekar, S.; Daftardar-Gejji, V. Solving evolution equations using a new iterative method. Numer. Methods Partial Differ. Equ. Int. J. 2010, 26, 906–916. [Google Scholar] [CrossRef]
- Jafari, H.; Nazari, M.; Baleanu, D.; Khalique, C.M. A new approach for solving a system of fractional partial differential equations. Comput. Math. Appl. 2013, 66, 838–843. [Google Scholar] [CrossRef]
- Yan, L. Numerical solutions of fractional Fokker-Planck equations using iterative Laplace transform method. Abstr. Appl. Anal. 2013, 2013. [Google Scholar] [CrossRef]
- Sharma, S.C.; Bairwa, R.K. Closed Form Solution for the Time-Fractional Schrödinger Equation via Laplace Transform. Int. J. Math. Its Appl. 2015, 3, 53–62. [Google Scholar]
- Sharma, S.C.; Bairwa, R.K. A reliable treatment of Iterative Laplace transform method for fractional Telegraph equations. Annal. Pure Appl. Math 2014, 9, 81–89. [Google Scholar]
- Sharma, S.C.; Bairwa, R.K. Iterative Laplace transform method for solving fractional heat and wave-like equations. Res. J. Math. Stat. Sci. 2015, 3, 4–9. [Google Scholar]
© 2020 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 (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Khan, H.; Khan, A.; Al-Qurashi, M.; Shah, R.; Baleanu, D. Modified Modelling for Heat Like Equations within Caputo Operator. Energies 2020, 13, 2002. https://doi.org/10.3390/en13082002
Khan H, Khan A, Al-Qurashi M, Shah R, Baleanu D. Modified Modelling for Heat Like Equations within Caputo Operator. Energies. 2020; 13(8):2002. https://doi.org/10.3390/en13082002
Chicago/Turabian StyleKhan, Hassan, Adnan Khan, Maysaa Al-Qurashi, Rasool Shah, and Dumitru Baleanu. 2020. "Modified Modelling for Heat Like Equations within Caputo Operator" Energies 13, no. 8: 2002. https://doi.org/10.3390/en13082002