Analytical solutions for conformable fractional Bratu-type equations
Keywords:Adomian Decomposition Method, Fractional Differential Equations, Conformable Fractional Derivative, Fractional Bratu-Type Equations.
Solving fractional differential equations have a prominent function in different science such as physics and engineering. Therefore, are different definitions of the fractional derivative presented in recent years. The aim of the current paper is to solve the fractional differential equation by a semi-analytical method based on conformable fractional derivative. Fractional Bratu-type equations have been solved by the method and to show its capabilities. The obtained results have been compared with the exact solution.
 R. Khalil, M. A. Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264 (2014) 65-70. https://doi.org/10.1016/j.cam.2014.01.002.
 T. Abdeljawad, on conformable fractional calculus, Journal of Computational and Applied Mathematics, 279 (2015) 57-66. https://doi.org/10.1016/j.cam.2014.10.016.
 F.B.M. Duarte, J. A. Tenreiro Machado, Chaotic phenomena and fractional- order dynamics in the trajectory control of redundant manipulators, Nonlinear Dynamics, 29 (2002) 342-362. https://doi.org/10.1023/A:1016559314798.
 O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynamics, 34 (2004) 323-337. https://doi.org/10.1007/s11071-004-3764-6.
 R. L. Magin, Fractional calculus models of complex dynamics in biological tissues, Computer and Mathematics with Applications, 59 (2010) 1586-1593. https://doi.org/10.1016/j.camwa.2009.08.039.
 V. V. Kulish, Jos L. Larg, Application of fractional calculus to fluid mechanics, Journal of Fluids Engineering, 134 (2002).
 V. Gafiychuk, B. Datsko, V. Meleshko, Mathematical modeling of time fractional reaction diffusion systems, Journal of Computational and Applied Mathematics, 220 (2008) 215-225. https://doi.org/10.1016/j.cam.2007.08.011.
 Seadawy A. R, Stability analysis solutions for nonlinear three-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov equation in a magnetized electron-positron plasma, Physica A: Statistical Mechanics and its Applications, 455 (2016) 44-51. https://doi.org/10.1016/j.physa.2016.02.061.
 F. C. Meral, T. J. Royston, R. Magin, Fractional calculus in viscoelasticity:an experimental study, Communications in Nonlinear Science and Numerical Simulation, 15 (2010) 939-945. https://doi.org/10.1016/j.cnsns.2009.05.004.
 F. Mainardi, Fractional calculus: some basic problem in continuum and statistical mechanics, in: A. Carplnteri, F. Mainardi (Eds.), Fractals and Fractional calculus in continuum Mechanics, Springer-Verlag, New York, (1997) 291-348. https://doi.org/10.1007/978-3-7091-2664-67.
 A. R. Seadawy, Nonlinear wave solutions of three-dimensional Zakharov-Kuznetsov-Burgers equation in dusty plasma, Physica A: Statistical Mechanics and its Applications, 439 (2015) 124-131. https://doi.org/10.1016/j.physa.2015.07.025.
 V. Daftardar â€“Gejji, H. Jafari, Solving a multi- order fractional differential equation using Adomian Decomposition, Applied Mathematics and Computation,189 (2007) 541-548. https://doi.org/10.1016/j.amc.2006.11.129.
 B. Ghazanfari, A. Sepahvandzadeh, Adomian decomposition method for solving fractional Bratu-type equations, Journal of mathematics and computer science, 8 (2014) 236-244.
 O. Abdulaziz, I. Hashim, S. Momani, solving systems of fractional differential equations by homotopy perturbation method, Physics Letters A, 372 (2008) 451-459. https://doi.org/10.1016/j.physleta.2007.07.059.
 B. Ghazanfari, A. G. Ghazanfari, M. Fuladvand, Modification of the homotopy perturbation method for numerical solution of Nonlinear Wave and of Nonlinear Wave Equations, Journal of mathematics and computer science, 3 (2011) 212-224.
 M. Mahmoudi, M. V. Kazemi, Solving singular BVPs Ordinary Differential Equations by Modified Homotopy Perturbation Method, Journal of mathematics and computer science, 7 (2013) 138-143.
 M. Rabbani, New Homotopy Perturbation Method to Solve Non-Linear Problems, Journal of mathematics and computer science, 7 (2013) 272-275.
 I. Hashim, O. Abdulaziz, S. Momani, Homotopy Analysis Method for fractional IVPs, Communications in Nonlinear Science and Numerical Simulation, 14 (2009) 674-684. https://doi.org/10.1016/j.cnsns.2007.09.014.
 G. Wu, E. W. M. Lee, Fractional variational iteration method and its application, Physics Letters A, 374 (2010) 2506-2509. https://doi.org/10.1016/j.physleta.2010.04.034.
 Z. Odibat, S. Momani, V. Suat Erturk, Generalized differential transform method: application to differential equations of fractional order, Applied Mathematics and Computation, 197 (2008) 467-477. https://doi.org/10.1016/j.amc.2007.07.068.
 H. Azizi, Gh. Barid Loghmani A numerical method for space fractional diffusion equations using a semi-disrete scheme and Chebyshev collocation method, Journal of mathematics and computer science, 8 (2014) 226â€“235.
 A. Neamaty, B. Agheli, R. Darzi, Solving Fractional Partial Differential Equation by Using Wavelet Operational Method, Journal of mathematics and computer science, 7 (2013) 230â€“240.
 Y. Aksoy, M. Pakdemirli, New perturbationâ€“iteration solutions for Bratu-type equations, Computers and Mathematics with Applications, 59 (8) (2010) 2802-2808. https://doi.org/10.1016/j.camwa.2010.01.050.
 M. Ilie, J. Biazar, Z. Ayati, General solution of Bernoulli and Riccati fractional differential equations based on conformable fractional derivative, International Journal of Applied Mathematical Research, 6(2) (2017) 49-51. https://doi.org/10.14419/ijamr.v6i2.7014.
 M. Ilie, J. Biazar, Z. Ayati, Application of the Lie Symmetry Analysis for second-order fractional differential equations, Iranian Journal of Optimization, 9(2) (2017) 79-83.
 M. Ilie, J. Biazar, Z. Ayati, The first integral method for solving some conformable fractional differential equations, Optical and Quantum Electronics, Vol.50 (2018) No.2, https://doi.org/10.1007/s11082-017-1307-x.
 G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston, (1994). https://doi.org/10.1007/978-94-015-8289-6.
 G. Adomian, A review of the decomposition method in applied mathematics, Journal of Mathematical Analysis and Applications, 135 (1988) 501â€“544. https://doi.org/10.1016/0022-247X(88)90170-9.
 J. Biazar, E. Babolian, A. Nouri, R. Islam, An alternate algorithm for computing Adomian decomposition method in special cases, Applied Mathematics and Computation, 138 (2â€“3) (2003) 523â€“529. https://doi.org/10.1016/S0096-3003(02)00174-1.
 J. Biazar, M. Ilie, A. Khoshkenar, An improvement to an alternate algorithm for computing Adomian polynomials in special cases, Applied Mathematics and Computation, 173(1) (2006) 582-592. https://doi.org/10.1016/j.amc.2005.04.052.
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