Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Numerical Inversion Technique for the One and Two-Dimensional L2-Transform Using the Fourier Series and Its Application to Fractional Partial Differential Equations

  • Aghili, Arman (Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan) ;
  • Ansari, Alireza (Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahrekord University)
  • Received : 2010.08.13
  • Accepted : 2012.06.04
  • Published : 2012.12.23
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we use a computational algorithm for the inversion of the one and two-dimensional $\mathcal{L}_2$-transform based on the Bromwich's integral and the Fourier series. The new inversion formula can evaluate the inverse of the $\mathcal{L}_2$-transform with considerable accuracy over a wide range of values of the independent variable and can be devised for the functions which are not Laplace transformable and have damping motion in small interval near origin.

Keywords

References

  1. A. Aghili, A. Ansari and A. Sedghi, An inversion technique for the L2-transform with applications, Inter. J. Contemp. Math. Sci., 2(28)(2007), 1387-1394. https://doi.org/10.12988/ijcms.2007.07146
  2. A. Aghili, A. Ansari, Complex inversion formula for stieltjes and widder transforms with applications, Inter. J. Contemp. Math. Sci., 3(16)(2008), 761-770.
  3. A. Aghili, A. Ansari, Solving partial fractional differential equations using the LA- transform, Asia-Euro. J. Math., 3(2)(2010), 209-220. https://doi.org/10.1142/S1793557110000143
  4. A. Aghili, F. Safarian, An Inversion technique for ${\varepsilon}_{2,1}$-transform with applications, Inter. J. Contemp. Math. Sci., 3(16)(2008), 771-780.
  5. K. Crump, Numerical inversion of Laplace transforms using a Fourier series approximation, Appl. Numer. Math., 23(1)(1976), 89-96.
  6. M. Dehghan, J. Manafian and A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Meth. Part. Diff. Equ., 26(2010), 448-479.
  7. M. Dehghan, J. Manafian and A. Saadatmandi, The solution of the linear fractional partial differential equations using the homotopy analysis method, Z. Naturforsch, 65a(2010), 935-549.
  8. H. Dubner, J. Abate, Numerical inversion of Laplace transform by relating them to the finite Fourier cosine transform, Appl. Math. Comput., 15(1)(1968), 115-123.
  9. M. Lakestani, M. Dehghan and S. Irandoust-Pakchin, The construction of operational matrix of fractional derivatives using B-spline functions, Commun. Nonlinear Sci. Numer. Simul., 17(3)(2012), 1149-1162. https://doi.org/10.1016/j.cnsns.2011.07.018
  10. J. R. Macdonald, Accelerated convergence,divergence, iteration, extrapolation and curve-fitting, J. Appl. Phys., 17(1964), 3034-3041.
  11. M. V. Moorthy, Numerical inversion of two dimensional Laplace tranforms-fourier series representation, Appl. Numer. Math., 17(1995), 119-127. https://doi.org/10.1016/0168-9274(95)00015-M
  12. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  13. A. Saadatmandi, M. Dehghan, A new operational matrix for solving fractionalorder differential equations, Comput. Math. Appl., 59(2010), 1326-1336. https://doi.org/10.1016/j.camwa.2009.07.006
  14. A. Saadatmandi, M. Dehghan, A tau approach for solution of the space fractional diffusion equation, Comput Math Appl., 62(2011), 1135-1142. https://doi.org/10.1016/j.camwa.2011.04.014
  15. P. N. Shankar, On the evolution of disturbances at an inviscid interface, J. Fluid. Mech., 108(1981), 159-170. https://doi.org/10.1017/S002211208100205X
  16. O. Yurekli, I. Sadek, A Parseval-Goldstein type theorem on the Widder potential transform and its applications, Inter. J. Math. Math. Sci., 14(1991), 517-524. https://doi.org/10.1155/S0161171291000704
  17. O. Yurekli, New identities involving the Laplace and the L2-transforms and their applications, Appl. Math. Comput., 99(1999), 141-151. https://doi.org/10.1016/S0096-3003(98)00002-2