AN ALTERNATIVE TECHNIQUE FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS

Abstract

In this paper, a new method for solving ordinary di erential equations is given by using the generalized Laplace transform Ln. Firstly, the authors introduce a di erential operator  that is called the -derivative. A relation between the Ln-transform of the -derivative of a function and the Ln- transform of the function itself are derived. Then, the convolution theorem is proven. Using obtained theorems, a few initial-value problems for ordinary di erential equations are solved as illustrations.

Keywords

• The Laplace transform,The Ln-transform,The L􀀀1 n -transform and Linear ordinary dierential equations

References

1. [1] A.,Aghili, A.,Ansari, A.,Sedghi, An inversion technique for the L2-transform with applica- tions, Int. J. Contemp. Math. Sciences, (2007), 2.28, 1387-1394.
2. [2] A.,Aghili, A.,Ansari, A new approach to solving SIEs and system of PFDEs using the L2- transform, Dierential Equations and Control Processes, (2010), N3, 1817-2172.
3. [3] L.,Debnath, The double Laplace Transforms and their properties with applications to func- tional, integral and partial dierential equations, International Journal of Applied and Computational Mathematics, (2015), 1-19.
4. [4] N.,Dernek, F.,Aylkc, Identities for the Ln-transform, The L2n-transform and the P2n- transform and their applications, Journal of Inequality and Special Functions, (2014), 5.4, 1-16.
5. [5] N.,Dernek, F.,Aylkc, Laplace ve L2 dnsmleriyle ksmi trevli denklemlerin czmleri, Marmara University, (2014), Master Thesis.
6. [6] D.G.,Duy, Transform methods for solving partial dierential equations, Symbolic and Numeric Computation, (2004).
7. [7] A.,Erdelyi, W.,Magnus, F.,Oberhettinger, F.G.,Tricomi, Tables of integral transforms Vol. 1, (1954), New York,NY,USA, McGraw-Hill.
8. [8] A.,Erdelyi, W.,Magnus, F.,Oberhettinger, F.G.,Tricomi, Tables of integral transforms Vol. 2, (1954), New York,NY,USA, McGraw-Hill.

9. [9] O.,Yurekli, Theorems on L2-transform and its applications, (1983), Vol.13 of Physical Sciences Data, Elsevier Scientic Publishing Co., Amsterdam.
10. [10] O.,Yurekli, I.,Sadek, A Parseval-Goldstein type theorem on the Widder potential trans- form and its applications, International Journal of Mathematics and Mathematical Sciences, (1991), 14.3, 517-524.
11. [11] O.,Yurekli, S.,Wilson, A new method of solving Bessel's dierential equation using the L2- transform, Applied Mathematics and Computation, (2002), 130.2, 587-591.
12. [12] O.,Yurekli, S.,Wilson, A new method of solving Hermite's dierential equation using the L2-transform, Applied Mathematics and Computation, (2003), 145.2, 495-500.
13. [13] E.,Zauderer, Partial dierential equations of applied mathematics, (2004), John Wiley and Sons, Inc.