Using the differential transformation method, the differential transform version of equation (5) is ( k + 1 ) U ( k + 1 ) = 1 2 ∑ l = 0 k 1 2 l l ! 1 2 k − l U ( k − l ) + 1 2 U ( k ), k ≥ 0 (6).
Saeed and Rahman [31] also solved equation (8) using the differential transformation method, but they transformed equation (8) into a system of three differential equations, which is a uselessly complicated approach to solving equation (8).
In this study, generalized Hirota Satsuma coupled KdV equation is solved using by two recent semi-analytic methods, differential transform method (DTM) and reduced form of differential transformation method (so called RDTM).
To deal with seasonality, the ARIMA model is extended to a general multiplicative seasonal ( {text{ARIMA}};(p, d, q) times (P, D, Q)^{s} ), where the time sequence demonstrates both trend and seasonal trend; non-stationary sequence is transformed into a smooth one via differential transformation.
The operators in two-dimensional differential transformation method [24].
Jacobian matrix is deduced by differential transformation method.
Theorem 3 Assume that W ( k ), U ( k ) and U i ( k ) are the differential transformations of the functions w ( t ), u ( t ) and u i ( t ), respectively, and r, q, q i ∈ ( 0, 1 ), i = 1, 2. Then: ( I ) If w ( t ) = ∫ 0 r t u ( q s ) d s, then W ( k ) = 1 k r k q k − 1 U ( k − 1 ).
From definitions (3), (4), we can derive the following: Theorem 1 Assume that F ( k ), G ( k ), H ( k ) and U i ( k ), i = 1, …, n, are the differential transformations of the functions f ( x ), g ( x ), h ( x ) and u i ( x ), i = 1, …, n, respectively, then.
Theorem 2 Assume that W ( k ), U ( k ) and U i ( k ) are the differential transformations of the functions w ( t ), u ( t ) and u i ( t ), respectively, and q, q i ∈ ( 0, 1 ), i = 1, 2. Then: ( i ) If w ( t ) = u ( q t ), then W ( k ) = q k U ( k ).
Theorem 1 Assume that F ( k ), G ( k ), H ( k ) and U i ( k ), i = 1, …, n, are the differential transformations of the functions f ( t ), g ( t ), h ( t ) and u i ( t ), i = 1, …, n, respectively, then: If f ( t ) = d n g ( t ) d t n, then F ( k ) = ( k + n ) ! k !
The temperature distribution has been determined by solving the highly non-linear governing equations using a semi-analytical transformation technique called Differential Transform Method.
