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

aaaaa6= −aa2,1 = − a0,3226= − a a3,1 = − a1,36626= − a a4,1 = − a2,3121226= − a a5,1 = − a3,3202026= − a a6,1 = − a4,330302,0 0,23,0 1,24,0 2,25,0 3,26,0 4,2- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -aaaaa1220= − a a2,3 = − a0,5221220= − a a3,3 = − a1,5661220= − a a4,3 = − a2,512121220= − a a5,3 = − a3,520201220= − a a6,3 = − a4,530302,2 0,43,2 1,44,2 2,45,2 3,46,2 4,4- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -Let us replace the known variables (known - in thesense of the existing expression for them through thefree variables) in the equations, where they are usedrepeatedly, to their expressions.2 12a4,0 = − ( − a0,4)= a0,412 22 12 1a5,0 = − ( − a1,4 ) == a1,420 6 52 12 30a = − ( − ( − a )) = −a30 12 26,0 0,6 0,6- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -a6 20= − ( − a12 2) = 5aa12 30= − ( − a12 2) = 15a4,1 0,5 0,54,2 0,6 0,6- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -6 20a = − ( − a ) = a20 65,1 1,5 1,5- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -Thus all coefficients ai + 2 , j can be expressedthrough the free coefficients a 0 , j and a 1 , j .Further the second substitution will be made - let ussubstitute the obtained expressions for the coefficientsai + 2 , j into solution of (2)∑∑∞ ∞i ju ( x,y)= ai,jxy.i=0 j=0Afterwards the following solution of the equation isobtained:u( x, y)= a + a x + a y − a x + a xy + a y −2 20,0 1,0 0,1 0,2 1,1 0,21 3 2 2 3 4 3 2 2− a1,2 x − 3a0,3x y + a1,2 xy + a0,3 y + a0,4x − a1,3 x y − 6a0,4x y +33 4 15 4 3 2+ a1 , 3x y + a0 , 4y + a1 , 4x + 5 a0 , 5x y − 2 a1 , 4x y −52 3 4 5 6 5− 1 0 a x y + a x y + a y − a x + a x y +0 , 5 1 , 4 0 , 5 0 , 6 1 , 54 2 1 03 3 2 4 5 6+ 1 5 a0 , 6x y − a1 , 5x y − 1 5 a0 , 6x y + a1 , 5x y + a0 , 6y + . . .3Annual Proceedings of Vidzeme University College “ICTE in Regional Development”, 2006131