# Analysis of fractional partial differential equations by Taylor series expansion

- Ali Demir
^{1}Email author, - Sertaç Erman
^{1}, - Berrak Özgür
^{1}and - Esra Korkmaz
^{2}

**2013**:68

https://doi.org/10.1186/1687-2770-2013-68

© Demir et al.; licensee Springer 2013

**Received: **19 December 2012

**Accepted: **11 March 2013

**Published: **30 March 2013

## Abstract

We develop a formulation for the analytic or approximate solution of fractional differential equations (FDEs) by using respectively the analytic or approximate solution of the differential equation, obtained by making fractional order of the original problem integer order. It is shown that this method works for FDEs very well. The results reveal that it is very effective and simple in determination of solutions of FDEs.

## 1 Introduction

Fractional differential equations (FDEs) are obtained by generalizing differential equations to an arbitrary order. Since fractional differential equations are used to model complex phenomena, they play a crucial role in engineering, physics and applied mathematics. Therefore they have been generating increasing interest from engineers and scientist in recent years. Since FDEs have memory, nonlocal relations in space and time, complex phenomena can be modeled by using these equations. Due to this fact, materials with memory and hereditary effects, fluid flow, rheology, diffusive transport, electrical networks, electromagnetic theory and probability, signal processing, and many other physical processes are diverse applications of FDEs [1–7].

In [8], the solutions for some nonlinear fractional differential equations are constructed by using symmetry analysis. But, in general, FDEs do not have exact analytic solutions, hence the approximate and numerical solutions of these equations are studied. Analytical approximations of linear and nonlinear FDEs are obtained by the variational iteration method, Adomian’s decomposition method, the homotopy perturbation method and the Lagrange multiplier method [9–24].

In the present paper, we use the Taylor series of an analytical solution for the differential equations which is obtained from FDEs by making the fractional order of the derivative integer, to obtain the analytical or approximate solution of FDEs. We can obtain the exact or approximate solution of FDEs by changing the terms of Taylor series expansion for a solution of a differential equation in such a way that the relationship among the terms of Taylor series expansion in the sense of derivative and fractional derivative remains the same. Applications of this method show that it is easy and effective when applied to any FDEs as long as the differential equation obtained from FDEs has an analytical or approximate solution. We take the fractional derivative in the Caputo sense.

The structure of this article is as follows. In Section 2, we give the construction of analytical or approximate solutions for FDEs including fractional derivative with respect to time. In the same manner, we obtain the analytical or approximate solution of FDEs with fractional derivative with respect to space variable in Section 3. In Section 4, we take the combination of previous two sections, and we get the analytical or approximate solution of FDEs with fractional derivative with respect to time and space variable. Finally, we give some illustrative examples of this method for all cases in Section 5.

## 2 Solution of FDEs including fractional derivative with respect to time

*t*. Then we replace the derivatives with respect to

*t*by fractional derivatives with respect to

*t*in such a way that the relation among the terms of Taylor series is preserved. Moreover, we leave the first

*m*terms of Taylor series fixed since $\alpha =m$. This also allows the solution of the fractional differential equation to satisfy the initial conditions of the problem. Let us assume that the solution of equation (2) is expanded into its Taylor series with respect to

*t*as follows:

*t*as follows:

## 3 Solution of FDEs including fractional derivative with respect to space variable

which is obtained by taking $\alpha =m$.

After finding the analytic or approximate solution of equation (7), we can obtain the exact or approximate solution of equation (6) by changing the terms of Taylor series expansion for the solution of differential equation (7) in such a way that the relationship among the terms of Taylor series expansion in the sense of derivative and fractional derivative with respect to space remains the same. In other words, we expand the exact or approximate solution into its Taylor series with respect to *x*. Then we replace the derivatives with respect to *x* by fractional derivatives with respect to *x* in such a way that the relation among the terms of Taylor series is preserved. Moreover, we leave the first *m* terms of Taylor series fixed since $\alpha =m$. This also allows the solution of the fractional differential equation to satisfy the boundary conditions of the problem.

*x*as follows:

*x*as follows:

## 4 Solution of FDEs including fractional derivative with respect to space variable and time

*x*and ${u}_{1}(t)$ into its Taylor series with respect to

*t*. Then we replace the derivatives with respect to

*x*by fractional derivatives with respect to

*x*in ${u}_{0}(x)$ in such a way that the relation among the terms of Taylor series is preserved. Moreover, we leave the first

*m*terms of Taylor series fixed since $\alpha =m$. Similarly, we do the same thing for ${u}_{1}(t)$. This also allows the solution of the fractional differential equation to satisfy the initial and boundary conditions of the problem. Let us assume that ${u}_{0}(x)$ of equation (12) is expanded into its Taylor series with respect to

*x*as follows:

*t*as follows:

*x*as follows:

## 5 Illustrative applications

**Example 1**

which is exactly the same solution as in [18].

**Example 2**

which is exactly the same solution as in [19].

**Example 3**

which is totaly the same solution as in [20].

**Example 4**

where the boundary conditions are given in fractional terms.

which is exactly the same solution as in [21].

**Example 5**

**Example 6**

**Example 7**

## Declarations

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

The research was supported by parts by the Scientific and Technical Research Council of Turkey (TUBITAK).

## Authors’ Affiliations

## References

- Oldham KB, Spanier J:
*The Fractional Calculus*. Academic Press, New York; 1974.Google Scholar - Podlubny I:
*Fractional Differential Equations*. Academic Press, San Diego; 1999.Google Scholar - Kilbas AA, Srivastava HM, Trujillo JJ:
*Theory and Applications of Fractional Differential Equations*. Elsevier, Amsterdam; 2006.Google Scholar - He JH: Nonlinear oscillation with fractional derivative and its applications.
*International Conference on Vibrating Engineering’98*1998, 288-291.Google Scholar - He JH: Some applications of nonlinear fractional differential equations and their approximations.
*Bull Sci Technol*1999, 15: 86-90.Google Scholar - He JH: Approximate analytical solution for seepage flow with fractional derivatives in porous media.
*Comput Methods Appl Mech Eng*1998, 167: 57-68. 10.1016/S0045-7825(98)00108-XView ArticleGoogle Scholar - Metzler R, Klafter J: The random walk’s guide to anomalous diffusion: a fractional dynamics approach.
*Phys Rep*2000, 339: 1-77. 10.1016/S0370-1573(00)00070-3MathSciNetView ArticleGoogle Scholar - Gazizov RK, Kasatkin AA, Lukashchuk SY: Group-invariant solutions of fractional differential equations. 1. In
*Nonlinear Science and Complexity*. 1st edition. Edited by: Tenreiro Machado JA, Luo ACJ, Barbosa RS, Silva MF, Figueiredo LB. Springer, Dordrecht; 2011:51-61.View ArticleGoogle Scholar - Huang F, Liu F: The time-fractional diffusion equation and fractional advection-dispersion equation.
*ANZIAM J.*2005, 46: 1-14.View ArticleGoogle Scholar - Huang F, Liu F: The fundamental solution of the space-time fractional advection-dispersion equation.
*J. Appl. Math. Comput.*2005, 18(2):339-350. 10.1007/BF02936577MathSciNetView ArticleGoogle Scholar - Momani S: Non-perturbative analytical solutions of the space- and time-fractional Burgers equations.
*Chaos Solitons Fractals*2006, 28(4):930-937. 10.1016/j.chaos.2005.09.002MathSciNetView ArticleGoogle Scholar - Odibat Z, Momani S: Application of variational iteration method to nonlinear differential equations of fractional order.
*Int. J. Nonlinear Sci. Numer. Simul.*2006, 1(7):15-27.Google Scholar - Das S: Solution of fractional vibration equation by the variational iteration method and modified decomposition method.
*Int. J. Nonlinear Sci. Numer. Simul.*2008., 9: Article ID 361Google Scholar - Momani S, Odibat Z: Numerical comparison of methods for solving linear differential equations of fractional order.
*Chaos Solitons Fractals*2007, 31(5):1248-1255. 10.1016/j.chaos.2005.10.068MathSciNetView ArticleGoogle Scholar - Odibat Z, Momani S: Approximate solutions for boundary value problems of time-fractional wave equation.
*Appl. Math. Comput.*2006, 181(1):767-774. 10.1016/j.amc.2006.02.004MathSciNetView ArticleGoogle Scholar - Yildirim A: An algorithm for solving the fractional nonlinear Schrödinger equation by means of the homotopy perturbation method.
*Int. J. Nonlinear Sci. Numer. Simul.*2009, 10: 445-451.Google Scholar - Ganji ZZ, Ganji DD, Jafari H, Rostamian M: Application of the homotopy perturbation method to coupled system of partial differential equations with time fractional derivatives.
*Topol. Methods Nonlinear Anal.*2008., 31: Article ID 341Google Scholar - Hang X, Shi-Yun L, Xiang-Cheng Y: Analysis of nonlinear fractional partial differential equations with the homotopy analysis method.
*Commun. Nonlinear Sci. Numer. Simul.*2009, 14: 1152-1156. 10.1016/j.cnsns.2008.04.008MathSciNetView ArticleGoogle Scholar - Saha RS, Bera RK: The random walk’s guide to anomalous diffusion: a fractional dynamics approach.
*Phys. Rep.*2000, 339: 1-77. 10.1016/S0370-1573(00)00070-3View ArticleGoogle Scholar - Odibat Z, Momani S: Numerical methods for nonlinear partial differential equations of fractional order.
*Appl. Math. Model.*2008, 32: 28-39. 10.1016/j.apm.2006.10.025View ArticleGoogle Scholar - El-Sayed AMA, Gaber M: The Adomian decomposition method for solving partial differential equations of fractal order in finite domains.
*Phys. Lett. A*2006, 359: 175-182. 10.1016/j.physleta.2006.06.024MathSciNetView ArticleGoogle Scholar - Sheu LJ, Tam LM, Lao SK: Parametric analysis and impulsive synchronization of fractional-order Newton-Leipnik systems.
*Int. J. Nonlinear Sci. Numer. Simul.*2009, 10: 33-44.View ArticleGoogle Scholar - Xu C, Wu G, Feng JW, Zhang WQ: Synchronization between two different fractional-order chaotic systems.
*Int. J. Nonlinear Sci. Numer. Simul.*2008, 9: 89-95.Google Scholar - Yildirim A: Homotopy perturbation method for solving the space-time fractional advection-dispersion equation.
*Adv. Water Resour.*2009, 32: 1711-1716. 10.1016/j.advwatres.2009.09.003View ArticleGoogle Scholar

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