Proceedings in pdf format. - Sociotechnical Systems Engineering ...

After conversions and summing of similar items:1 2 1 4 1 6u( x, y) = a0,0(1 − x + x − x + ...) +2 24 7201 2 1 4 1 6+ a0,1y(1 − x + x − x + ...) +2 24 7201 2 1 4 1 6+ a1,0x(1 − x + x − x + ...) +6 120 50402 1 2 1 4 1 6 2 1 2 1 4+ a0,2[ y (1 − x + x − x + ...) − x (1 − x + x − ...)] +2 24 720 6 1201 2 1 4 1 6+ a1,1xy(1 − x + x − x + ...) +6 120 50403 1 2 1 4 2 1 2 1 4+ a0,3[ y (1 − x + x −...) − 3 x y(1 − x + x − ...)] +2 24 6 1202 1 2 1 4 1 3 1 2 1 4 1 6+ a1,2[ xy (1 − x + x −...) − x (1 − x + x − x + ...)] +6 120 3 10 280 151204 1 2 2 2 1 2 4 1 2+ a0,4[ y (1 − x + ...) − 6 x y (1 − x + ...) + x (1 − x + ...)] +2 6 103 1 2 1 4 3 1 2 1 4+ a1,3[ xy (1 − x + x −...) − x y(1 − x + x − ...)] + ...6 120 10 280The final form of the solution is:∞∑( 0, k 0, k 1, k 1, k )( , ) = ⋅ ( , ) + ⋅ ( , )u x y a Mag x y a Mag x yk = 0(13), where a0,kand a1,kfree variables.Then formulas for the functionsMag0, k ( x,y)and Mag1, k ( x,y)aremade for more general equation (10), and not (11).Next for the certainty, it will be agreed to name theMag0, k ( x,y)type of function andMag1, k ( , )x yas maternal functions or as thefunctions of Mary, and one-dimensional functions of2the type Ge( n,wx )as paternal functions or asthe functions of Herman. For example: the function ofMary for the three-dimensional equation of thermalconductivity, function of Mary for the fourdimensionalequation of Klein-Gordon, etc.The functions Mag0, k ( x,y)and( )Mag1, kx,y are determined from thefollowing expressions, which almost completelycorrespond to the analogous expressions for Laplace'sequation:( )Mag x y Ge wx20,0, = (0, ),Annual Proceedings of Vidzeme University College “ICTE in Regional Development”, 2006136