KYUNGPOOK Math. J. 52(2012), 383-395 http://dx.doi.org/10.5666/KMJ.2012.52.4.383 Numerical Inversion Technique for the One and Two-Dimen- sional L 2 -Transform Using the Fourier Series and Its Applica- tion to Fractional Partial Differential Equations Arman Aghili * Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, P. O. Box 1914, Rasht, Iran e-mail : armanaghili@yahoo.com Alireza Ansari Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahrekord University, P. O. Box 115, Shahrekord, Iran e-mail : alireza_1038@yahoo.com Abstract. In this paper, we use a computational algorithm for the inversion of the one and two-dimensional L 2 -transform based on the Bromwich’s integral and the Fourier series. The new inversion formula can evaluate the inverse of the L2-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. 1. Introduction The Laplace-type integral transform called the L 2 -transform was introduced by Yurekli and Sadek [16] and is denoted as follows (1.1) L 2 {f (t); s} = ∫ ∞ 0 te -s 2 t 2 f (t)dt, where f (t) is piecewise continuous and of the exponential order α (i.e. | f (t) |≤ Me αt 2 for real number α and positive constant M ) and s is complex parameter. Authors [1-4] generalized definition (1-1) for the two-dimensional L 2 -transform of the function f (t 1 ,t 2 ) by the following relation (1.2) L (s 1 ,s 2 ) 2 {f (t 1 ,t 2 )} = ∫ ∞ 0 ∫ ∞ 0 t 1 t 2 e -s 2 1 t 2 1 -s 2 2 t 2 2 f (t 1 ,t 2 )dt 1 dt 2 , * Corresponding Author. Received August 13, 2010; revised May 22, 2012; accepted June 4, 2012. 2010 Mathematics Subject Classification: 26A33, 44A10, 44A35. Key words and phrases: Laplace transform, L 2 -transform, Fourier series, Numerical inver- sion method. 383