the Sumudu transform for functions of two variables

摘要

The Sumudu transform, defined earlier by Watugala (1993), is extended to functions of two variables. Using this extended definition, a function of two variables such as f(x, y) is transformed to a function such as F(u, v). In the case of f(x, y) which can be expressed as a double series in the form ∑∑a_ijx~iy~j, the Sumudu transform F(u, v) is simply ∑∑I! j!a_iju~iv~j. The extended Sumudu transform has many interesting properties like its predecessor: (i)F(0, 0) is Equal to f(0, 0); (ii) Sumudu transforms of f (x, 0) and f(0, y) are F(u, v), respectively; (iii) f (bx, cy) is transformed to F(bu, cv), where b and c are constants; (iv) partial derivative of f(x, y) with respect to x is transformed to the sumudu transform of [f(x, y)-f(0, y)] divided by u; (v) the consistency of units of measurements in the terms of a partial differential equation governing a physical system is unaltered by the Sumudu transform. Extended Sumudu transform can be used to solve partial differential equations with known initial Conditions as shown by an example.