Solutions of Linear Fractional nonHomogeneous Differential Equations with Jumarie Fractional Derivative and Evaluation of Particular Integrals^{1}Department of Mathematics, Nabadwip Vidyasagar College, Nabadwip, Nadia, West Bengal, India
^{2}Department of Applied Mathematics, University of Calcutta, Kolkata, India
^{3}Reactor Control System Design Section Bhabha Atomic Research Centre, Mumbai, India
^{4}Department of Physics, Jadavpur University Kolkata, West Bengal, India
American Journal of Mathematical Analysis. 2015, 3(3), 5464
doi: 10.12691/ajma331
Copyright © 2015 Science and Education PublishingCite this paper: Uttam Ghosh, Susmita Sarkar, Shantanu Das. Solutions of Linear Fractional nonHomogeneous Differential Equations with Jumarie Fractional Derivative and Evaluation of Particular Integrals.
American Journal of Mathematical Analysis. 2015; 3(3):5464. doi: 10.12691/ajma331.
Correspondence to: Uttam Ghosh, Department of Mathematics, Nabadwip Vidyasagar College, Nabadwip, Nadia, West Bengal, India. Email:
uttam_math@yahoo.co.inAbstract
In this paper we describe a method to solve the linear nonhomogeneous fractional differential equations (FDE), composed with Jumarie type fractional derivative, and describe this method developed by us, to find out particular integrals, for several types of forcing functions. The solutions are obtained in terms of MittagLeffler functions, fractional sine and cosine functions. We have used our earlier developed method of finding solution to homogeneous FDE composed via Jumarie fractional derivative, and extended this to nonhomogeneous FDE. We have demonstrated these developed methods with few examples of FDE, and also applied in fractional damped forced differential equation. The short cut rules, that are developed here in this paper to replace the operator D^{a} or operator D^{2}^{a} as were used in classical calculus, gives ease in evaluating particular integrals. Therefore this method proposed by us is useful and advantageous as it is having conjugation with the classical methods of solving nonhomogeneous linear differential equations, and also useful in understanding physical systems described by FDE.
Keywords
References
[1]  S. Das. “Functional Fractional Calculus”, 2^{nd} Edition, SpringerVerlag Germany.(2011). 

[2]  E. Ahmed, A.M.A. ElSayed, H.A.A. ElSaka. “Equilibrium points, stability and numerical solutions of fractionalorder predatorprey and rabies models”. J Math Anal Appl 2007; 325:5453. (2007). 

[3]  A. Alsaedi, S. K Ntouyas, R. P Agarwal, B. Ahmad. “On Caputo type sequential fractional differential equations with nonlocal integral boundary conditions”. Advances in Difference Equations 2015; 33. 112. 

[4]  K. S. Miller, B. Ross. “An Introduction to the Fractional Calculus and Fractional Differential Equations”. John Wiley & Sons, New York, NY, USA; (1993). 

[5]  S. S. Ray, R.K. Bera. “An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method”. Applied Mathematics and Computation. 2005; 167. 561571. (2005). 

Show More References
[6]  I. Podlubny “Fractional Differential Equations”, Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA. 1999; 198. (1999). 

[7]  K. Diethelm. “The analysis of Fractional Differential equations”. SpringerVerlag, Germany. (2010). 

[8]  A. Kilbas, H. M. Srivastava, J.J. Trujillo, “Theory and Applications of Fractional Differential Equations”. NorthHolland Mathematics Studies, Elsevier Science, Amsterdam, the Netherlands. 2006; 204. (2006). 

[9]  B. Ahmad, J.J. Nieto. “RiemannLiouville fractional integrodifferential equations with fractional nonlocal integral boundary conditions”. Bound. Value Probl. 2011; 36. (2011). 

[10]  B. Zheng. “ExpFunction Method for Solving Fractional Partial Differential Equations”. Hindawi Publishing Corporation. The ScientificWorld Journal. 2013; 18. 

[11]  O. Abdulaziz, I. Hashim, S. Momani, “Application of homotopyperturbation method to fractional IVPs”, J. Comput. Appl. Math. 2008; 216. 574584. (2008). 

[12]  G.C. Wu, E.W.M. Lee, “Fractional Variational Iteration Method and its Application”, Phys. Lett. A. 2010; 374 25062509. (2010). 

[13]  D. Nazari, S. Shahmorad. “Application of the fractional differential transform method to fractionalorder integrodifferential equations with nonlocal boundary conditions”. Journal of Computational and Applied Mathematics. 2010; 234(3). 883891. (2010). 

[14]  S. Zhang and H. Q. Zhang, “Fractional subequation method and its applications to nonlinear fractional PDEs”, Phys. Lett. A, 2011; 375. 10691073. (2011). 

[15]  U. Ghosh, S. Sengupta, S. Sarkar and S. Das. “Analytic solution of linear fractional differential equation with Jumarie derivative in term of MittagLeffler function”. American Journal of Mathematical Analysis. 2015; 3(2). 3238. (2015). 

[16]  G. Jumarie. “Modified RiemannLiouville derivative and fractional Taylor series of nondifferentiable functionsFurther results”, Computers and Mathematics with Applications, 2006; 51. 13671376. (2006). 

[17]  U. Ghosh, S. Sengupta, S. Sarkar and S. Das. “Characterization of nondifferentiable points of a function by Fractional derivative of Jumarie type”. European Journal of Academic Essays 2015; 2(2): 7086. (2015). 

[18]  G. M. MittagLeffler. “Sur la nouvelle fonction E_{α} (x)”, C. R. Acad. Sci. Paris, (Ser. II). 1903; 137, 554558. (1903). 

Show Less References