2855 Fawzi Abdelwahid 1 , IJMCR Volume 10 Issue 08 August 2022 Volume 10 Issue 08 August 2022, Page no. – 2855-2859 Index Copernicus ICV: 57.55, Impact Factor: 7.362 DOI: 10.47191/ijmcr/v10i8.06 A New Approach on the Two-Dimensional Differential Transform Fadwa A. M. Madi 1 , Fawzi Abdelwahid 2 1,2 Department of Mathematics, Faculty of Science, University of Benghazi, Benghazi, Libya ARTICLE INFO ABSTRACT Published Online: 22 August 2022 Corresponding Authors: Fawzi Abdelwahid In recent years, new formulas of the two-dimensional differential transform have been proven by using the definition of the transform. In this work, we use a new approach based on the definition of the transform and the summation properties to prove the two-dimensional differential transform of the product of two functions, then we used this result to establish other useful formulas. This study shows that this procedure can be used to find formulas for many complicated terms. This enables us to apply the differential transform method on many types of partial differential equations. To demonstrate this approach, we applied the dimensional differential transform method on selected equations and compared our results with analytical solutions obtained by other methods KEYWORDS: Two-dimensional differential transform; Partial differential equations; Differential transform method I. INTRODUCTION The concept of differential transform was first introduced by Zhou [1]. Then Chen and Ho in [2] developed the deferential transform on partial deferential equations to obtained closed form series solutions for linear and nonlinear initial value problems. Ref. [3] introduced new useful formulas for one- dimensional differential transform and applied the differential transform method on selected ordinary differential equations. In [4], we reviewed the two- dimensional differential transform (2-DT) and applied the differential transform method on selected nonlinear partial differential equations. The main aim of this work is to use the summation properties to prove the two-dimensional differential transform of the product of two functions and then use the product formula to find other formulas. This procedure as we will see in the numerical examples will help us to apply the differential transform method on many types of partial differential equations. To do that, we introduce in the next section, the two- dimensional differential transform and the review the proofs of some basic formulas which based on the definition of differential transform [5, 6]. II. Basic Definitions and Formulas To introduce the 2-DT, we assume that ( , ) w xy be a ( ) C   function and 0 0 ( , ) x y be any point of  , where  is an open domain of 2 R , and then we defined the Taylor series of the function (, ) wxy about 0 0 ( , ) x y as    0 0 ( ,) 0 0 , 0 ( , ) , ( )( ) !! kh k h kh x x y w x wxy x x y y kh              (1) Then the two-dimensional differential transform, which denoted by T D , is defined as following: Definition: (2.1): Let   , wxy be an analytic function about 0 0 ( , ) x y , then the 2-DT, of   , wxy is defined as:       0 0 ( , ) ( , ) , : ( , ) , , ! ! T x y kh D w x y W kh w xy kh           (2) Hence, if we chose 0 0 ( , ) (0, 0) x y  , then the definition (2) reduces to