Research Article | Open Access

Xiao-Jing Ma, H. M. Srivastava, Dumitru Baleanu, Xiao-Jun Yang, "A New Neumann Series Method for Solving a Family of Local Fractional Fredholm and Volterra Integral Equations", *Mathematical Problems in Engineering*, vol. 2013, Article ID 325121, 6 pages, 2013. https://doi.org/10.1155/2013/325121

# A New Neumann Series Method for Solving a Family of Local Fractional Fredholm and Volterra Integral Equations

**Academic Editor:**J. A. Tenreiro Machado

#### Abstract

We propose a new Neumann series method to solve a family of local fractional Fredholm and Volterra integral equations. The integral operator, which is used in our investigation, is of the local fractional integral operator type. Two illustrative examples show the accuracy and the reliability of the obtained results.

#### 1. Introduction

Many initial- and boundary-value problems associated with ordinary differential equations (ODEs) and partial differential equations (PDEs) can be transformed into problems of solving the corresponding approximate integral equations. However, some initial- and boundary-value domains are fractal curves, which are everywhere continuous, but nowhere differentiable. As a result, we cannot employ the classical calculus, which requires that the defined functions should be differentiable, in order to process various classes of ordinary differential equations (ODEs) and partial differential equations (PDEs). Applications of fractional calculus, in general, and fractional differential equations [1–10], in particular, as well as various transport phenomena in complex and disordered media and fractional systems, have attracted considerable attention during the past two decades or so [11–22].

Recently, local fractional calculus [23–40], processing local fractional continuous non-differential functions, was successfully applied to model the stress-strain relation in fractal elasticity [26, 27], fractal release equation [32], wave equations on Cantor sets [34], fractal heat equation [34], diffusion equation arising in discontinuous heat transfer in fractal media [35], Laplace equation within local fractional operators [36], Schrödinger equation in fractal time-space [37], damped wave equation and dissipative wave equation in fractal strings [38], heat-conduction equation on Cantor sets without heat generation in fractal media [39], and so on. There are some analytical and numerical methods for solving local fractional ODEs and PDEs, such as fractional complex transform method with local fractional operator [35], local fractional variational iteration method [37], Cantor-type cylindrical-coordinate method [38], local fractional Fourier series method [39], local fractional series expansion method [40], Fourier and Laplace transforms with local fractional operator [39], and reference therein.

The Neumann series method was applied to solve the integral equations [41, 42]. Recently, the fractional Neumann series method was considered in [43, 44]. This paper focuses on a new Neumann series method for solving the local fractional Fredholm and Volterra integral equation being here facts in mind. This paper is structured as follows. Section 2 introduces the notations and the basic concepts. Section 3 is devoted to a new Neumann series method via local fractional integral operator. Two illustrative examples are explained in Section 4. Finally, conclusions are reported in Section 5.

#### 2. Preliminaries

In order to investigate the local fractional continuity of non-differential functions, we suggest the result derived from fractal geometry [34, 39].

Let be local fractional continuous on interval ; then we write [34, 35] If is a bi-Lipschitz mapping, then which leads to so that where and .

The result deduced from fractal geometry is related to fractal coarse-grained mass function , which reads [34] as with where is an -dimensional Hausdorff measure.

Notice that we consider that the dimensions of any fractal spaces (e.g., Cantor spaces or the Cantor-like spaces) are a positive numbers. It looks like the Euclidean space because its dimension is also positive number. The detailed results were considered in [34].

For , local fractional integral of of order in the interval is given by [34, 37, 39] where and , , is a partition of the interval .

For any , we have [34] denoted by If , then we have [34] For detailed content of fractal geometrical explanation of local fractional integral, we can see [34, 35]. Some properties of local fractional integral operator were suggested in –.

#### 3. A New Neumann Series Method to Deal with the Local Fractional Fredholm and Volterra Integral Equations

In this section, we consider a new Neumann series method to process the local fractional Fredholm and Volterra integral equations.

A new Neumann series method to deal with the local fractional Fredholm integral equation is written in the following form: It is obtained if we set such that where .

The zeroth approximation can be written as where .

Proceeding in this manner, the final solution can be obtained as where .

Now we structure a new Neumann series method to handle the local fractional Volterra integral equation, which reads as The method is applicable provided that is a local fractional analysis function; that is, have a local fractional Taylor’s expansion around .

can be expressed by a local fractional series expansion; which reads as where the coefficients and are constants that are required to be determined.

We have Thus, using a few terms of the expansion in both sides, we find that We then write the local fractional Taylor’s expansions for and count the first few integrals in (19). After the integration is performed, we equate the coefficients of the same powers of in both sides of (19). By this way, we can determine completely the unknown coefficients and produce solution in a local fractional series form.

#### 4. Examples

*Example 1. *Solve the following local fractional Fredholm integral equation:
Let us consider the zeroth approximation given by
The first approximation can be computed as follows:
Proceeding in this manner, we find the following local fractional series approximation:
Similarly, the third approximation reads as follows:
The fourth approximation yields
In conclusion, we get
Hence,

*Example 2. *Obtain the solution of the following local fractional Volterra equation:
Suppose that there exists the solution in the following local fractional series form:
Then, upon substituting the local fractional series into the equation, we find that
Comparing the coefficients of the same powers of , we get
and so on. Thus, the values of the coefficients can be calculated as follows:
Hence, the local fractional series solution is given by
which are satisfied with the condition given by [34, 39]
where the Mittag-Leffler function defined on fractal set of fractal dimension is suggested by [34, 39]

#### 5. Conclusions

Local fractional differential and integral operators have proven to be useful tools to deal with everywhere continuous (but nowhere differentiable) functions in fractal areas ranging from fundamental science to engineering. In this paper, it is proven that a new Neumann series method can be used for solving the local fractional Fredholm and Volterra integral equations, and their solutions are fractal functions. The proposed method is efficient and leads to accurate, approximately convergent solutions to local fractional Fredholm and Volterra integral equations. It is demonstrated that the solutions of local fractional Fredholm and Volterra integral equations are fractal functions, which are equipped with local fractional continuities. However, the classical and fractional Neumann series methods [41–44] were only applied to continuous functions.

#### Appendix

The following properties of local fractional integral operator are valid [34]. (a)For any, we have local fractional multiple integrals, which are written as [34] (b)If , then [34] (c)The sine and cosine subfunctions can, respectively, be written as follows [34, 39]: (d)Suppose that is local fractional continuous on the interval . Then (e)We have#### References

- K. B. Oldham and J. Spanier,
*The Fractional Calculus*, Academic Press, London, UK, 1974. View at: MathSciNet - K. S. Miller and B. Ross,
*An Introduction to the Fractional Calculus and Fractional Differential Equations*, John Wiley & Sons Inc., New York, NY, USA, 1993. View at: MathSciNet - S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integrals and Derivatives*, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993. View at: MathSciNet - V. Kiryakova,
*Generalized Fractional Calculus and Applications*, vol. 301, Longman Scientific & Technical, Harlow, UK, 1994. View at: MathSciNet - I. Podlubny,
*Fractional Differential Equations*, vol. 198, Academic Press, San Diego, Calif, USA, 1999. View at: MathSciNet - R. Hilfer, Ed.,
*Applications of Fractional Calculus in Physics*, World Scientific Publishing, River Edge, NJ, USA, 2000. View at: Publisher Site | MathSciNet - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, vol. 204, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. View at: MathSciNet - J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado,
*Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering*, Springer, New York, NY, USA, 2007. - F. Mainardi,
*Fractional Calculus and Waves in Linear Viscoelasticity*, Imperial College Press, London, UK, 2010. View at: Publisher Site | MathSciNet - G. M. Zaslavsky, “Chaos, fractional kinetics, and anomalous transport,”
*Physics Reports*, vol. 371, no. 6, pp. 461–580, 2002. View at: Publisher Site | Google Scholar | MathSciNet - J. Hristov, “Heat-balance integral to fractional (half-time) heat diffusion sub-model,”
*Thermal Science*, vol. 14, no. 2, pp. 291–316, 2010. View at: Publisher Site | Google Scholar - B. J. West, M. Bologna, and P. Grigolini,
*Physics of Fractal Operators*, Springer, New York, NY, USA, 2003. View at: Publisher Site | MathSciNet - R. Metzler and J. Klafter, “The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics,”
*Journal of Physics A*, vol. 37, no. 31, pp. R161–R208, 2004. View at: Publisher Site | Google Scholar | MathSciNet - L. M. Zelenyǐ and A. V. Milovanov, “Fractal topology and strange kinetics: from percolation theory to problems in cosmic electrodynamics,”
*Physics-Uspekhi*, vol. 47, no. 8, pp. 749–788, 2004. View at: Publisher Site | Google Scholar - G. M. Zaslavsky,
*Hamiltonian Chaos and Fractional Dynamics*, Oxford University Press, Oxford, UK, 2008. View at: MathSciNet - J. A. T. Machado, “Analysis and design of fractional-order digital control systems,”
*Systems Analysis Modelling Simulation*, vol. 27, no. 2-3, pp. 107–122, 1997. View at: Google Scholar - J. A. T. Machado, “Fractional-order derivative approximations in discrete-time control systems,”
*Systems Analysis Modelling Simulation*, vol. 34, no. 4, pp. 419–434, 1999. View at: Google Scholar - R. Herrmann,
*Fractional Calculus: An Introduction for Physicists*, World Scientific, 2011. - J. Klafter, S. C. Lim, and R. Metzler,
*Fractional Dynamics: Recent Advances*, World Scientific, 2012. - M. D. Ortigueira,
*Fractional Calculus for Scientists and Engineers*, Springer, 2011. - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus Models and Numerical Methods*, Complexity, Nonlinearity and Chaos, World Scientific, Boston, Mass, USA, 2012. - I. Petras,
*Fractional-Order Nonlinear Systems: Modeling. Analysis and Simulation*, Springer, Berlin, Germany, 2011. - K. M. Kolwankar and A. D. Gangal, “Fractional differentiability of nowhere differentiable functions and dimensions,”
*Chaos*, vol. 6, no. 4, pp. 505–513, 1996. View at: Publisher Site | Google Scholar - K. M. Kolwankar and A. D. Gangal, “Local fractional Fokker-Planck equation,”
*Physical Review Letters*, vol. 80, pp. 214–217, 1998. View at: Publisher Site | Google Scholar - F. B. Adda and J. Cresson, “About non-differentiable functions,”
*Journal of Mathematical Analysis and Applications*, vol. 263, no. 2, pp. 721–737, 2001. View at: Google Scholar - A. Carpinteri, B. Chiaia, and P. Cornetti, “Static-kinematic duality and the principle of virtual work in the mechanics of fractal media,”
*Computer Methods in Applied Mechanics and Engineering*, vol. 191, no. 1-2, pp. 3–19, 2001. View at: Google Scholar - A. Carpinteri, B. Chiaia, and P. Cornetti, “The elastic problem for fractal media: basic theory and finite element formulation,”
*Computers and Structures*, vol. 82, no. 6, pp. 499–508, 2004. View at: Google Scholar - Y. Chen, Y. Yan, and K. Zhang, “On the local fractional derivative,”
*Journal of Mathematical Analysis and Applications*, vol. 362, no. 1, pp. 17–33, 2010. View at: Google Scholar - A. Babakhani and V. D. Gejji, “On calculus of local fractional derivatives,”
*Journal of Mathematical Analysis and Applications*, vol. 270, no. 1, pp. 66–79, 2002. View at: Publisher Site | Google Scholar - G. Jumarie, “Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions,”
*Applied Mathematics Letters*, vol. 22, no. 3, pp. 378–385, 2009. View at: Publisher Site | Google Scholar - G. Jumarie, “Probability calculus of fractional order and fractional Taylor's series application to Fokker-Planck equation and information of non-random functions,”
*Chaos, Solitons and Fractals*, vol. 40, no. 3, pp. 1428–1448, 2009. View at: Publisher Site | Google Scholar - W. Chen, H. Sun, X. Zhang, and D. Korošak, “Anomalous diffusion modeling by fractal and fractional derivatives,”
*Computers and Mathematics with Applications*, vol. 59, no. 5, pp. 1754–1758, 2010. View at: Publisher Site | Google Scholar - W. Chen, “Time-space fabric underlying anomalous diffusion,”
*Chaos, Solitons and Fractals*, vol. 28, no. 4, pp. 923–925, 2006. View at: Publisher Site | Google Scholar - X.-J. Yang,
*Advanced Local Fractional Calculus and Its Applications*, World Science, New York, NY, USA, 2012. - M.-S. Hu, D. Baleanu, and X.-J. Yang, “One-phase problems for discontinuous heat transfer in fractal media,”
*Mathematical Problems in Engineering*, vol. 2013, Article ID 358473, 3 pages, 2013. View at: Publisher Site | Google Scholar - A. Liangprom and K. Nonlaopon, “On the convolution equation related to the diamond Klein-Gordon operator,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 908491, 16 pages, 2011. View at: Publisher Site | Google Scholar - W.-H. Su, D. Baleanu, X.-J. Yang, and H. Jafari, “Damped wave equation and dissipative wave equation in fractal strings within the local fractional variational iteration method,”
*Fixed Point Theory and Applications*, vol. 2013, no. 1, pp. 1–11, 2013. View at: Google Scholar - X.-J. Yang, H. M. Srivastava, J.-H. He, and D. Baleanu, “Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives,”
*Physics Letters A*, vol. 377, no. 38–30, pp. 1696–1700. View at: Google Scholar - X.-J. Yang,
*Local Fractional Functional Analysis and Its Applications*, Asian Academic, Hong Kong, China, 2011. - A.-M. Yang, X.-J. Yang, and Z.-B. Li, “Local fractional series expansion method for solving wave and diffusion equations on Cantor sets,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 351057, 5 pages, 2013. View at: Publisher Site | Google Scholar - D. Medková, “On the convergence of Neumann series for noncompact operators,”
*Czechoslovak Mathematical Journal*, vol. 41, no. 2, pp. 312–316, 1991. View at: Google Scholar | MathSciNet - P. Stefanov and G. Uhlmann, “Thermoacoustic tomography with variable sound speed,”
*Inverse Problems*, vol. 25, no. 7, Article ID 075011, 2009. View at: Publisher Site | Google Scholar | MathSciNet - H. M. Srivastava and R. K. Saxena, “Operators of fractional integration and their applications,”
*Applied Mathematics and Computation*, vol. 118, no. 1, pp. 1–52, 2001. View at: Publisher Site | Google Scholar | MathSciNet - A. Ebaid, D. M. M. ElSayed, and M. D. Aljoufi, “Fractional calculus model for damped Mathieu equation: approximate analytical solution,”
*Applied Mathematical Sciences*, vol. 6, no. 81–84, pp. 4075–4080, 2012. View at: Google Scholar | MathSciNet

#### Copyright

Copyright © 2013 Xiao-Jing Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.