American Journal of Mathematical Analysis
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American Journal of Mathematical Analysis. 2015, 3(3), 54-64
DOI: 10.12691/ajma-3-3-1
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Solutions of Linear Fractional non-Homogeneous Differential Equations with Jumarie Fractional Derivative and Evaluation of Particular Integrals

Uttam Ghosh1, , Susmita Sarkar2 and Shantanu Das3, 4

1Department of Mathematics, Nabadwip Vidyasagar College, Nabadwip, Nadia, West Bengal, India

2Department of Applied Mathematics, University of Calcutta, Kolkata, India

3Reactor Control System Design Section Bhabha Atomic Research Centre, Mumbai, India

4Department of Physics, Jadavpur University Kolkata, West Bengal, India

Pub. Date: September 10, 2015

Cite this paper:
Uttam Ghosh, Susmita Sarkar and Shantanu Das. Solutions of Linear Fractional non-Homogeneous Differential Equations with Jumarie Fractional Derivative and Evaluation of Particular Integrals. American Journal of Mathematical Analysis. 2015; 3(3):54-64. doi: 10.12691/ajma-3-3-1


In this paper we describe a method to solve the linear non-homogeneous 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 Mittag-Leffler 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 non-homogeneous 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 Da or operator D2a 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 non-homogeneous linear differential equations, and also useful in understanding physical systems described by FDE.

Mittag-Leffler functions non-homogeneous fractional differential equations modified riemann-liouville definition

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[1]  S. Das. “Functional Fractional Calculus”, 2nd Edition, Springer-Verlag Germany.(2011).
[2]  E. Ahmed, A.M.A. El-Sayed, H.A.A. El-Saka. “Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models”. J Math Anal Appl 2007; 325:54-53. (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. 1-12.
[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. 561-571. (2005).
[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”. Springer-Verlag, Germany. (2010).
[8]  A. Kilbas, H. M. Srivastava, J.J. Trujillo, “Theory and Applications of Fractional Differential Equations”. North-Holland Mathematics Studies, Elsevier Science, Amsterdam, the Netherlands. 2006; 204. (2006).
[9]  B. Ahmad, J.J. Nieto. “Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions”. Bound. Value Probl. 2011; 36. (2011).
[10]  B. Zheng. “Exp-Function Method for Solving Fractional Partial Differential Equations”. Hindawi Publishing Corporation. The Scientific-World Journal. 2013; 1-8.
[11]  O. Abdulaziz, I. Hashim, S. Momani, “Application of homotopy-perturbation method to fractional IVPs”, J. Comput. Appl. Math. 2008; 216. 574-584. (2008).
[12]  G.C. Wu, E.W.M. Lee, “Fractional Variational Iteration Method and its Application”, Phys. Lett. A. 2010; 374 2506-2509. (2010).
[13]  D. Nazari, S. Shahmorad. “Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions”. Journal of Computational and Applied Mathematics. 2010; 234(3). 883-891. (2010).
[14]  S. Zhang and H. Q. Zhang, “Fractional sub-equation method and its applications to nonlinear fractional PDEs”, Phys. Lett. A, 2011; 375. 1069-1073. (2011).
[15]  U. Ghosh, S. Sengupta, S. Sarkar and S. Das. “Analytic solution of linear fractional differential equation with Jumarie derivative in term of Mittag-Leffler function”. American Journal of Mathematical Analysis. 2015; 3(2). 32-38. (2015).
[16]  G. Jumarie. “Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions-Further results”, Computers and Mathematics with Applications, 2006; 51. 1367-1376. (2006).
[17]  U. Ghosh, S. Sengupta, S. Sarkar and S. Das. “Characterization of non-differentiable points of a function by Fractional derivative of Jumarie type”. European Journal of Academic Essays 2015; 2(2): 70-86. (2015).
[18]  G. M. Mittag-Leffler. “Sur la nouvelle fonction Eα (x)”, C. R. Acad. Sci. Paris, (Ser. II). 1903; 137, 554-558. (1903).