### On the Solutions of Linear Fractional Differential Equations of Order \(2q\), Including Small Delay Where \( 0< q<1 \)

#### Abstract

The main goal of this study is to find the solutions of linear fractional differential equations of order \(2q\), including small delay, where \(0<q<1\) which has various applications. The fractional derivatives are taken in the sense of Caputo which is more suitable than Riemann-Liouville sense. We assume that the order \(q\) satisfy the condition \(nq=1\) for some natural number \(n\) which determines the number of the linearly independent solutions. Since the delay term is small, the linear fractional differential equation is expanded in powers series of which reduce the problem to regular or singular perturbation problem for which it is easier to find the solution. The solution is obtained in the form of a series expansion of \(E\). To demonstrate the accuracy and the effectiveness of the proposed approach, some illustrative examples are presented.

#### Keywords

#### Full Text:

PDF#### References

A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Science 20 (2016), 763 – 769, DOI: 10.2298/TSCI160111018A.

A. Atangana and I. Koca, New direction in fractional differentiation, Mathematics in Natural Science 1 (2017), 18 – 25, DOI: 10.22436/mns.01.01.02.

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory, McGraw-Hill Book Company, New York (1978), DOI: 10.1007/978-1-4757-3069-2.

A. Demir and E. Ozbilge, Analysis of the inverse problem in a time fractional parabolic equation with mixed boundary conditions, Boundary Value Problems 2014 (2014), Article number 134, DOI: 10.1186/1687-2770-2014-134.

A. Demir, F. Kanca and E. Ozbilge, Numerical solution and distinguishability in time fractional parabolic equation, Boundary Value Problems 2015 (2015), Article number 142, DOI: 10.1186/s13661-015-0405-6.

A. Demir, S. Erman, B. Özgür and E. Korkmaz, Analysis of fractional partial differential equations by Taylor series expansion, Boundary Value Problems 2013 (2013), Article number 68, DOI: 10.1186/1687-2770-2013-68.

F. Huang and F. Liu, The time-fractional diffusion equation and fractional advection-dispersion equation, The ANZIAM Journal 46 (2005), 317 – 330, DOI: 10.1017/S1446181100008282.

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204 (2006), 1 – 523, https://www.sciencedirect.com/bookseries/north-holland-mathematics-studies/vol/204.

I. Podlubny, Fractional Differential Equations, Academic Press, New York (1999).

B. Sambandham and A.S. Vatsala, Basic results for sequential Caputo fractional differential equations, Mathematics 3 (2015), 76 – 91, DOI: 10.3390/math3010076.

J. G. Simmonds and J. E. Mann Jr., A First Look at Perturbation Theory, 2nd ed., Dover Publications, Mineola, New York (1998).

DOI: http://dx.doi.org/10.26713%2Fcma.v11i1.1294

### Refbacks

- There are currently no refbacks.

eISSN 0975-8607; pISSN 0976-5905