A reliable method for the space-time fractional Burgers and time-fractional Cahn-Allen equations via the FRDTM
- Mahmoud S Rawashdeh^{1}Email author
https://doi.org/10.1186/s13662-017-1148-8
© The Author(s) 2017
Received: 9 February 2017
Accepted: 21 March 2017
Published: 31 March 2017
Abstract
We propose a new method called the fractional reduced differential transform method (FRDTM) to solve nonlinear fractional partial differential equations such as the space-time fractional Burgers equations and the time-fractional Cahn-Allen equation. The solutions are given in the form of series with easily computable terms. Numerical solutions are calculated for the fractional Burgers and Cahn-Allen equations to show the nature of solutions as the fractional derivative parameter is changed. The results prove that the proposed method is very effective and simple for obtaining approximate solutions of nonlinear fractional partial differential equations.
Keywords
MSC
1 Introduction
The space-fractional Burgers equation describes the physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe. They are also connected with applications in acoustic phenomena and have been used to model turbulence and certain steady-state viscous flows. Moreover, Burgers equations are used to model the formation and decay of nonplanar shock waves, where the variable x is a coordinate moving with the wave at the speed of sound and the dependent variable u represents the velocity fluctuations. The Burgers equations occur in various areas of applied sciences and physical applications, such as modeling of fluid mechanics and financial mathematics, and the equation has still interesting applications in physics and astrophysics.
The fractional differential equations (FDE) appear more and more frequently in different research areas and engineering applications. There are many physical applications in science and engineering that can be represented by models using fractional differential equations [1–10], which are quite useful for many physical problems. These equations are represented by fractional linear and nonlinear PDEs, and solving such fractional differential equations is very important [11–21].
Many approximation and numerical techniques have been used to solve fractional differential equations [12, 16, 22–25]. Lately, many new approaches to fractional differential equations have been proposed, a few of these methods are as follows: the fractional differential transform method (FDTM) [25–28], the fractional Adomian decomposition method (FADM) [2], the fractional variational iteration method (FVIM) [4], the fractional sub-equation method [23], the fractional natural decomposition method [17, 29] and the fractional homotopy perturbation method (FHPM) [22, 30]. Kurulay [26] found approximate and exact solutions of the space- and time-fractional Burgers equations. Bekir et al. [23] found exact solutions of the time-fractional Cahn-Allen equation. Khan et al. [22] used the generalized differential transform method (GDTM) and the homotopy perturbation method (HPM) to solve the time-fractional Burgers and coupled Burgers equations. Recently, Rawashdeh [16, 31] used the FRDTM to solve nonlinear fractional partial differential equations.
The general response expression contains a parameter describing the order of the fractional derivative that can be varied to obtain various responses. Note that we call Eq. (1.1) the time-fractional Burgers and the space-fractional Burgers equation in the case \(0<\alpha\le1, \eta=0\) and \(0<\beta\le1, \alpha=1\), respectively.
2 Background of fractional calculus
Here are some definitions and facts that we shall use in our work. Some of these basic definitions are due to Liouville [3, 4, 33, 34].
Definition 2.1
A real function \(f(x)\), \(x>0\), is said to be in the space \(C_{\mu }\), \(\mu\in{\mathbb {R}}\), if there exists a real number \(q(>\mu )\) such that \(f(x)=x^{q} g(x)\), where \(g(x)\in C [0,\infty )\), and it is said to be in the space \(C_{\mu}^{m} \) if \(f^{(m)} \in C_{\mu}, m\in{\mathbb {N}}\).
Definition 2.2
Definition 2.3
Lemma 2.4
[6]
We use the Caputo fractional derivative because it allows traditional initial and boundary conditions to be included in the formulation of our work.
3 Analysis of the FRDTM
Definition 3.1
Basic operations of the FRDTM [ 39 ]
Functional form | Transformed form |
---|---|
u(x,t) | \(\frac{1}{\Gamma(k\alpha+1)} [\frac{\partial^{\alpha k } }{\partial t^{\alpha k} } u(x,t) ]_{ t=0} \) |
γu(x,t)±βv(x,t) | \(\gamma U_{k} (x)\pm \beta V_{k} (x)\), where γ and β are constants |
u(x,t).v(x,t) | \(\sum_{i=0}^{k}U_{i} (x)V_{k-i} (x) \) |
u(x,t).v(x,t).w(x,t) | \(\sum_{i=0}^{k}\sum_{j=0}^{i}U_{j} (x) V_{i-j} (x) W_{k-i} (x)\) |
\(\frac{\partial^{n\alpha} }{\partial t^{n\alpha } } u(x,t)\) | \(\frac{\Gamma (k\alpha+n\alpha +1 )}{\Gamma (k\alpha+1 )} U_{k+n} (x)\) |
\(\frac{\partial^{n} }{\partial x^{n} } u(x,t)\) | \(\frac{\partial^{n} }{\partial x^{n} } U_{k}(x)\) |
\(x^{m} t^{n} u(x,t)\) | \(x^{m} U_{k-n} (x )\) |
\(x^{m} t^{n} \) | \(x^{m} \delta (k\alpha-n )\), where \(\delta(k\alpha -n)=\left \{ \begin{array}{l@{\quad}l} 1, & \alpha k=n \\ 0, & \alpha k\ne n \end{array} \right \}\) |
Remark 3.2
In Table 1, Γ represents the gamma function, where \(\Gamma (z+1)=z\Gamma(z), z>0\).
3.1 Methodology
4 Worked examples
We shall employ the FRDTM to three different applications to illustrate the accuracy and efficiency of the method.
4.1 The time-fractional Burgers equation
Remark 4.1
4.2 The space-fractional Burgers equation
4.3 The time-fractional Cahn-Allen equation
Remark 4.3
5 Tables of numerical calculations
The results obtained by the FRDTM for different values of α for Example 4.1
x | t | α = 0.25 | α = 0.5 | α = 0.75 | α = 1 | |
---|---|---|---|---|---|---|
Numerical | Numerical | Numerical | Numerical | Exact | ||
−10 | 2 | 1 | 1 | 1 | 1 | 1 |
4 | 1 | 1 | 1 | 1 | 1 | |
6 | 1 | 1 | 1 | 1 | 1 | |
8 | 1 | 1 | 1 | 1 | 1 | |
−5 | 2 | 1.00014 | 1.00014 | 1 | 1 | 1 |
4 | 1.02059 | 1.00261 | 1.00001 | 1 | 1 | |
6 | 1.37155 | 1.02059 | 1.00014 | 1.00002 | 1 | |
8 | 3.86303 | 1.02059 | 1.00112 | 1.00014 | 1 | |
5 | 2 | 0.200786 | 0.20001 | 0.200001 | 0.2 | 0.2 |
4 | 0.280682 | 0.200786 | 0.200059 | 0.20001 | 0.20004 | |
6 | 1.48828 | 0.211558 | 0.200786 | 0.200123 | 0.204848 | |
8 | 9.53999 | 0.280682 | 0.205256 | 0.200786 | 0.540446 | |
10 | 2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 |
4 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | |
6 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | |
8 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 |
The results obtained by the FRDTM for different values of β for Example 4.2
x | t | β = 0.25 | β = 0.5 | β = 0.75 | β = 1 | |
---|---|---|---|---|---|---|
Numerical | Numerical | Numerical | Numerical | Exact | ||
0.1 | 2 | −0.0764845 | −0.0764173 | −0.0765568 | −0.076893 | −0.073117 |
3 | −0.0224361 | −0.0224164 | −0.0224575 | −0.0225564 | −0.0214477 | |
4 | −0.00609528 | −0.00608992 | −0.00610107 | −0.00612793 | −0.00582667 | |
6 | 0.00309842 | 0.00309573 | 0.00310145 | 0.00311518 | 0.00296201 | |
0.3 | 2 | −0.243822 | −0.24332 | −0.244843 | −0.248451 | −0.21341 |
3 | −0.0721999 | −0.0720561 | −0.0725197 | −0.0736135 | −0.0631537 | |
4 | −0.0195606 | −0.019521 | −0.0196462 | −0.019942 | −0.0171018 | |
6 | 0.0102373 | 0.0102193 | 0.0102888 | 0.0104509 | 0.008961 | |
0.5 | 2 | −0.419375 | −0.418177 | −0.4228 | −0.433774 | −0.336983 |
3 | −0.126549 | −0.126227 | −0.127726 | −0.131255 | −0.101393 | |
4 | −0.034099 | −0.0340076 | −0.0344084 | −0.0353558 | −0.0272609 | |
6 | 0.0188912 | 0.0188622 | 0.0191169 | 0.0197044 | 0.0151832 | |
0.6 | 2 | −0.505904 | −0.504266 | −0.511001 | −0.527077 | −0.389893 |
3 | −0.154611 | −0.154193 | −0.156467 | −0.161845 | −0.118574 | |
4 | −0.0414992 | −0.041377 | −0.0419815 | −0.0434176 | −0.0317049 | |
6 | 0.0238959 | 0.0238714 | 0.0242898 | 0.0252544 | 0.0184231 |
The results obtained by the FRDTM for different values of α for Example 4.3
x | t | α = 0.25 | α = 0.5 | α = 0.75 | α = 1 | |
---|---|---|---|---|---|---|
Numerical | Numerical | Numerical | Numerical | Exact | ||
0.1 | 0.002 | 0.768025 | 0.515858 | 0.487054 | 0.483079 | 0.483079 |
0.003 | 0.79101 | 0.523384 | 0.488734 | 0.483453 | 0.483453 | |
0.004 | 0.807412 | 0.529721 | 0.490276 | 0.483828 | 0.483828 | |
0.006 | 0.830055 | 0.540329 | 0.493102 | 0.484577 | 0.484577 | |
0.3 | 0.002 | 0.7419 | 0.480518 | 0.451845 | 0.447907 | 0.447907 |
0.003 | 0.766709 | 0.488048 | 0.453511 | 0.448278 | 0.448278 | |
0.004 | 0.784528 | 0.494401 | 0.455041 | 0.448649 | 0.448649 | |
0.006 | 0.809302 | 0.505062 | 0.457846 | 0.449391 | 0.449391 | |
0.5 | 0.002 | 0.713958 | 0.445372 | 0.417112 | 0.413248 | 0.413248 |
0.003 | 0.740586 | 0.452832 | 0.418748 | 0.413612 | 0.413612 | |
0.004 | 0.759881 | 0.459137 | 0.420251 | 0.413976 | 0.413976 | |
0.006 | 0.787034 | 0.469743 | 0.423008 | 0.414704 | 0.414704 | |
0.6 | 0.002 | 0.699334 | 0.427979 | 0.400028 | 0.396214 | 0.396214 |
0.003 | 0.726851 | 0.435377 | 0.401642 | 0.396573 | 0.396573 | |
0.004 | 0.746891 | 0.441635 | 0.403127 | 0.396932 | 0.396932 | |
0.006 | 0.775296 | 0.452175 | 0.40585 | 0.397651 | 0.397651 |
6 Conclusion
In this paper, we successfully implemented the FRDTM to find approximate solutions of the space-time fractional Burgers equations and the time-fractional Cahn-Allen equation for different values of α and β and the results we obtained in Examples 4.1, 4.2 and 4.3 were in excellent agreement with the exact solutions. The FRDTM introduces a significant improvement in the field over the existing methods.
Declarations
Acknowledgements
The author would like to thank the editor and the anonymous referees for their comments and suggestions on this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Garg, M, Manohar, P: Numerical solution of fractional diffusion-wave equation with two space variables by matrix method. Fract. Calc. Appl. Anal. 13(2), 191-207 (2010) MathSciNetMATHGoogle Scholar
- Garg, M, Sharma, A: Solution of space-time fractional telegraph equation by Adomian decomposition method. J. Inequal. Spec. Funct. 2(1), 1-7 (2011) MathSciNetMATHGoogle Scholar
- Hilfer, R: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) View ArticleMATHGoogle Scholar
- Inc, M: The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method. J. Math. Anal. Appl. 345, 476-484 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Jafari, H, et al.: A new algorithm for solving dynamic equations on a time scale. J. Comput. Appl. Math. 312, 167-173 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Kilbas, AA, Srivastava, HH, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) MATHGoogle Scholar
- Sakar, GM, et al.: On solutions of fractional Riccati differential equations. Adv. Differ. Equ. 2017, 39 (2017). doi:10.1186/s13662-017-1091-8 MathSciNetView ArticleGoogle Scholar
- Yang, X-J, Gao, F, Srivastava, HM: Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations. Computers & Mathematics with Applications (2016). doi:10.1016/j.camwa.2016.11.012
- Yang, X-J, Tenreiro Machado, JA, Srivastava, HM: A new numerical technique for solving the local fractional diffusion equation: two-dimensional extended differential transform approach. Appl. Comput. Math. 274, 143-151 (2016) MathSciNetGoogle Scholar
- Yang, X-J: Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems. arXiv:1612.03202 (2016)
- Caputo, M: Linear models of dissipation whose Q is almost frequency independent. Part II. Ann. Geophys. 19(4), 383-393 (1966) Google Scholar
- Marin, M, Marinescu, C: Thermoelasticity of initially stressed bodies, asymptotic equipartition of energies. Int. J. Eng. Sci. 36(1), 73-86 (1998) MathSciNetView ArticleMATHGoogle Scholar
- Marin, M: A domain of influence theorem for microstretch elastic materials. Nonlinear Anal., Real World Appl. 11(5), 3446-3452 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Differential Equations. John Wiley, New York (1993) MATHGoogle Scholar
- Podlubny, I: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1998) MATHGoogle Scholar
- Rawashdeh, SM: An efficient approach for time-fractional damped Burger and time-sharma-tasso-Olver equations using the FRDTM. Appl. Math. Inf. Sci. 9(3), 1239-1246 (2015) MathSciNetGoogle Scholar
- Rawashdeh, MS, Al-Jammal, H: Numerical solutions for systems of nonlinear fractional ordinary differential equations using the FNDM. Mediterr. J. Math. 13(6), 4661-4677 (2016) MathSciNetView ArticleMATHGoogle Scholar
- He, JH: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 167, 57-68 (1998) MathSciNetView ArticleMATHGoogle Scholar
- Ray, SS, Bera, RK: Solution of an extraordinary differential equation by Adomian decomposition method. J. Appl. Math. 4, 331-338 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Ray, SS, Bera, RK: An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method. Appl. Math. Comput. 167, 561-571 (2005) MathSciNetMATHGoogle Scholar
- Cascaval, RC, Eckstein, EC, Frota, CL, Goldstein, JA: Fractional telegraph equations. J. Math. Anal. Appl. 276(1), 145-159 (2002) MathSciNetView ArticleMATHGoogle Scholar
- Alam Khan, N, Ara, A, Mahmood, A: Numerical solutions of time-fractional Burgers equations: a comparison between generalized differential transformation technique and homotopy perturbation method. Int. J. Numer. Methods Heat Fluid Flow 22(2), 175-193 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Bekir, A, Aksoy, E, Cevikel, AC: Exact solutions of nonlinear time fractional partial differential equations by sub-equation method. Math. Methods Appl. Sci. 38(13), 2779-2784 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Kumar, D, Singh, J, Kiliman, A: Efficient approach for fractional Harry Dym equation by using Sumudu transform. Abstr. Appl. Anal. 2013, Article ID 608943 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Momani, S, Odibat, Z, Erturk, VS: Generalized differential transform method for solving a space and time fractional diffusion-wave equation. Phys. Lett. A 370, 379-387 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Kurulay, M: The approximate and exact solutions of the space-and time-fractional Burgers equations. Int. J. Res. Rev. Appl. Sci. 3(3), 257-263 (2010) MATHGoogle Scholar
- Momani, S, Odibat, Z: A generalized differential transform method for linear partial differential equations of fractional order. Appl. Math. Lett. 21, 194-199 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Odibat, Z, Momani, S, Erturk, VS: Generalized differential transform method: application to differential equations of fractional order. Appl. Math. Comput. 197, 467-477 (2008) MathSciNetMATHGoogle Scholar
- Rawashdeh, MS, Al-Jammal, H: New approximate solutions to fractional nonlinear systems of partial differential equations using the FNDM. Adv. Differ. Equ. 2016(1), 235 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Momani, S: Non-perturbative analytical solutions of the space- and time-fractional Burgers equations. Chaos Solitons Fractals 28, 930-937 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Rawashdeh, M: A new approach to solve the fractional Harry Dym equation using the FRDTM. Int. J. Pure Appl. Math. 95(4), 553-566 (2014) MathSciNetView ArticleGoogle Scholar
- Esen, A, Yagmurlu, NM, Tasbozan, O: Approximate analytical solution to time-fractional damped Burger and Cahn-Allen equations. Appl. Math. Inf. Sci. 7(5), 1951-1956 (2013) MathSciNetView ArticleGoogle Scholar
- Caputo, M: Elasticitá e Dissipazione (Elasticity and Anelastic Dissipation). Zanichelli, Bologna (1969) Google Scholar
- Caputo, M, Mainardi, F: Linear models of dissipation in anelastic solids. Riv. Nuovo Cimento 1(2), 161-198 (1971) View ArticleGoogle Scholar
- Rawashdeh, M: Improved approximate solutions for nonlinear evolutions equations in mathematical physics using the reduced differential transform method. Journal of Applied Mathematics and Bioinformatics 3(2), 1-14 (2013) MATHGoogle Scholar
- Rawashdeh, M: Using the reduced differential transform method to solve nonlinear PDEs arises in biology and physics. World Appl. Sci. J. 23(8), 1037-1043 (2013) Google Scholar
- Rawashdeh, M, Obeidat, NA: On finding exact and approximate solutions to some PDEs using the reduced differential transform method. Appl. Math. Inf. Sci. 8(5), 2171-2176 (2014) MathSciNetView ArticleGoogle Scholar
- Rawashdeh, M: Approximate solutions for coupled systems of nonlinear PDES using the reduced differential transform method. Math. Comput. Appl. 19(2), 161-171 (2014) MathSciNetGoogle Scholar
- Keskin, KY: Ph.D. Thesis. Selcuk University (2010) (in Turkish) Google Scholar
- Momani, S: Analytic and approximate solutions of the space-and time-fractional telegraph equations. Appl. Math. Comput. 170(2), 1126-1134 (2005) MathSciNetMATHGoogle Scholar