American Journal of Mathematical and Computer Modelling
Volume 2, Issue 1, February 2017, Pages: 13-23
Received: Oct. 30, 2016;
Accepted: Nov. 25, 2016;
Published: Jan. 13, 2017
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Kirtiwant Parshuram Ghadle, Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, India
Khan Firdous, Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, India
Khan Arshiya Anjum, Department of Computer Science and Information Technology, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, India
By handling the one dimensional partial differential equation with three methods i.e. Adomain decomposition method(ADM), Variation iteration method(VIM) and the New iterative method(NIM) and applied logarithmic and exponential functions as initial condition. A general framework of these methods is presented for analytical treatment of fractional partial differential equation arises in fluid mechanics. The fractional derivatives are described in the Caputo sense. The equation used in this paper is fractional wave equation, fractional burgers equation and fractional Klein-Gordon equation. After comparison of the results, the series of solution are found which is very helpful. The basic idea described in this paper is accepted to be further in use to solve other similar linear problems in fractional calculus.
Kirtiwant Parshuram Ghadle,
Khan Arshiya Anjum,
Solution of FPDE in Fluid Mechanics by ADM, VIM and NIM, American Journal of Mathematical and Computer Modelling.
Vol. 2, No. 1,
2017, pp. 13-23.
J. H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media”, Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57-68, 1998.
I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
V. Daftardar-Gejji and H. Jafari, “An iterative method for solving nonlinear functional equations”, Journal of Mathematical Analysis and Applications, vol. 316, no. 2, pp. 753-763, 2006.
A. A. Hemeda, “New iterative method: application to the nthorder integro-differential equations”, Information B, vol. 16, no. 6, pp. 3841-3852, 2013.
S. Bhalekar and V. Daftardar-Gejji, “New iterative method: application to partial differential equations”, Applied Mathematics and Computation, vol. 203, no. 2, pp. 778-783, 2008.
G. Adomian, “Solving Frontier Problems of Physics: The Decomposition Method”, Kluwer, 1994.
J. H. He, “Homotopy perturbation technique”, Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257-262, 1999.
A. A. Hemeda, “Variational iteration method for solving nonlinear coupled equations in 2-dimensional space in fluid mechanics”, International Journal of Contemporary Mathematical Sciences, vol. 7, no. 37, pp. 1839-1852, 2012.
J. Henderson, R. Luca, “Positive solution for a system of fractional differential equations with coupled integral boundary conditions”, Applied Mathematics and Computation, vol. 249, pp. 182-197, 2014.
A. Bekri, E. Aksoy and A. C. Cevikel, “Exact solutions of nonlinear time fractional partial differential equations by sub-equation method”, Math. Meth. Appl. Sci. vol. 38, pp. 2779-2784, 2015.
X-J. Yang, H. M. Srivastava and C. Cattani, “Local fractional homotopy perturbation method for solving fractional partial differential equations arising in mathematical physics”, Romanian Reports in Physics, vol. 67, no. 3, pp. 752-761, 2015.
A. H. Bhrawy, E. H. Doha and S. S. Ezz-Eldien, “A Spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations”, Journal of Computational Physics, vol. 293, pp. 142-156, 2015.
M. Caputo, “Linear models of dissipation whose Q is almost frequency independent part II”, Geophysical Journal International, vol. 13, no. 5, pp. 529-539, 1967.
I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation”, Fractional Calculus and Applied Analysis, vol. 5, no. 4, pp. 367-386, 2002.
Z. Odibat and S. Momani, “The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics”, Computers and Mathematics with Applications, vol. 58, no. 11-12, pp. 2199-2208, 2009.
A. A. Hemeda, “Solution of fractional partial differential equations in fluid mechanics by extension of some iterative method”, Abstract and Applied Analysis, vol. 2013, pp. 9.