ISSN (Print): 2333-8490

ISSN (Online): 2333-8431

Website: http://www.sciepub.com/journal/ajma

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Currrent Issue: Volume 3, Number 3, 2015

Article

Solution of System of Linear Fractional Differential Equations with Modified Derivative of Jumarie Type

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

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

3Reactor Control Systems Design Section E & I Group B.A.R.C Mumbai India

4Department of Physics, Jadavpur University Kolkata;Department of Appl. Mathematics, University of Calcutta


American Journal of Mathematical Analysis. 2015, 3(3), 72-84
doi: 10.12691/ajma-3-3-3
Copyright © 2015 Science and Education Publishing

Cite this paper:
Uttam Ghosh, Susmita Sarkar, Shantanu Das. Solution of System of Linear Fractional Differential Equations with Modified Derivative of Jumarie Type. American Journal of Mathematical Analysis. 2015; 3(3):72-84. doi: 10.12691/ajma-3-3-3.

Correspondence to: Shantanu  Das, Reactor Control Systems Design Section E & I Group B.A.R.C Mumbai India. Email:

Abstract

Solution of fractional differential equations is an emerging area of present day research because such equations arise in various applied fields. In this paper we have developed analytical method to solve the system of fractional differential equations in-terms of Mittag-Leffler function and generalized Sine and Cosine functions, where the fractional derivative operator is of Jumarie type. The use of Jumarie type fractional derivative, which is modified Rieman-Liouvellie fractional derivative, eases the solution to such fractional order systems. The use of this type of Jumarie fractional derivative gives a conjugation with classical methods of solution of system of linear integer order differential equations, by usage of Mittag-Leffler and generalized trigonometric functions. The ease of this method and its conjugation to classical method to solve system of linear fractional differential equation is appealing to researchers in fractional dynamic systems. Here after developing the method, the algorithm is applied in physical system of fractional differential equation. The analytical results obtained are then graphically plotted for several examples for system of linear fractional differential equation.

Keywords

References

[1]  K. S. Miller, B. Ross. An Introduction to the Fractional Calculus and Fractional Differential Equations.John Wiley & Sons, New York, NY, USA; 1993.
 
[2]  S. Das. Functional Fractional Calculus 2nd Edition, Springer-Verlag 2011.
 
[3]  I. Podlubny. Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA. 1999; 198.
 
[4]  K. Diethelm. The analysis of Fractional Differential equations. Springer-Verlag, 2010.
 
[5]  A. Kilbas, H. M. Srivastava, J.J. Trujillo. Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Elsevier Science, Amsterdam, the Netherlands, 204. 2006.
 
Show More References
[6]  M. Caputo, “Linear models of dissipation whose q is almost frequency independent,” Geophysical Journal of the Royal Astronomical Society, 1967. vol. 13, no. 5, pp. 529-539.
 
[7]  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.
 
[8]  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 2(2): 70-86, 2015.
 
[9]  K. Diethelm, N. J. Ford and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynamics, 29, 3-22, 2002.
 
[10]  Y. Hua, Y. Luoa, Z. Lu, Analytical solution of the linear fractional differential equation by Adomian decomposition method. Journal of Computational and Applied Mathematics. 215 (2008) 220-229.
 
[11]  S. S. Ray, R.K. Bera. An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method. Applied Mathematics and Computation. 167 (2005) 561-571.
 
[12]  O. Abdulaziz, I. Hashim, S. Momani, Application of homotopy-perturbation method to fractional IVPs, J. Comput. Appl. Math. 216 (2008) 574-584.
 
[13]  R. Yulita Molliq, M.S.M. Noorani, I. Hashim, R.R. Ahmad. Approximate solutions of fractional Zakharov–Kuznetsov equations by VIM. Journal of Computational and Applied Mathematics. 233(2). 2009. 103-108.
 
[14]  V.S. Ertürk, S. Momani. Solving systems of fractional differential equations using differential transform method. Journal of Computational and Applied Mathematics 215 (2008) 142-151.
 
[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 3(2). 2015. 32-38, 2015.
 
[16]  G. Jumarie. Fourier’s Transformation of fractional order via Mittag-Leffler function and modified Riemann-Liouville derivatives. J. Appl. Math. & informatics. 2008. 26. 1101-1121.
 
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Article

On Geometrical Methods that Provide a Short Proof of Four Color Theorem

1La Playa Street. 304. San Francisco.CA 94109, US


American Journal of Mathematical Analysis. 2015, 3(3), 65-71
doi: 10.12691/ajma-3-3-2
Copyright © 2015 Science and Education Publishing

Cite this paper:
BAHMAN MASHOOD. On Geometrical Methods that Provide a Short Proof of Four Color Theorem. American Journal of Mathematical Analysis. 2015; 3(3):65-71. doi: 10.12691/ajma-3-3-2.

Correspondence to: BAHMAN  MASHOOD, La Playa Street. 304. San Francisco.CA 94109, US. Email: b mashood@hotmail:com

Abstract

In this article we introduce a short and comprehensive proof of four color theorem based on geometrical methods. At the end of the article we will provide a short proof of the De Bruijn Erdos theorem for locally finite infinite graphs.

Keywords

References

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Article

Solutions of Linear Fractional non-Homogeneous Differential Equations with Jumarie Fractional Derivative and Evaluation of Particular Integrals

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


American Journal of Mathematical Analysis. 2015, 3(3), 54-64
doi: 10.12691/ajma-3-3-1
Copyright © 2015 Science and Education Publishing

Cite this paper:
Uttam Ghosh, Susmita Sarkar, 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.

Correspondence to: Uttam  Ghosh, Department of Mathematics, Nabadwip Vidyasagar College, Nabadwip, Nadia, West Bengal, India. Email: uttam_math@yahoo.co.in

Abstract

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.

Keywords

References

[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).
 
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”. 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).
 
Show Less References