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DUAL SUBSTITUTION METHOD OF SOLVING PARTIAL DIFFERENTIAL EQUATIONSValerijs StepuchevsLatvian Intelligent Systems, Ltd., Sigulda, LatviaE-mail: valerijs@lis.lvKEYWORDSPartial differential equations, Dual substitution,Special functions.ABSTRACTAlready for more than 300 years greatmathematicians find and study different specialfunctions. The majorities of these special functions arethe solutions of the differential equations, which areobtained as a result of separation of variables in theequations of mathematical physics. In the article thenew simple method of solving the partial differentialequations is proposed, and as a result, another methodof the solution of new special functions from theequations of mathematical physics. In this method,there is no stage of separation of variables during thesolution of equation, and this means that thedetermined one-dimensional special functions havenature that is more general. In the article, Helmholtz'sequation is described as an example, and it is knownthat this equation is widely used for solutions of tasksconnected with the steady fluctuations (mechanical,acoustic, thermal, electromagnetic, etc.). It is a currentissue to find the optimal solution of all these tasks,which can be find the way allowed for traditional ICTEresources.SOLUTION OF TWO-DIMENSIONALDIFFERENTIAL EQUATION OF LAPLACELet us examine equation:2 2∂ u ∂ u+ = 02 2 (1)∂x∂ ywhere x and y – the independent variables,u( x, y ) – the analytic function from x and y .In (Полянин 2001) the quite general method ofconstructing the exact solutions through the analyticcomplex variable functions is given. In this article, asimplest method will be shown for the obtaining of thecomplete set of the analytic functions.An equation is given (1) and it is given, that intosome region the solution G takes form∑∑∞ ∞i ju ( x,y)= ai,jx y(2).i=0 j=0Let us make substitution of this solution (2) intodifferential equation (1) (this is the first substitution).Comparing coefficients with the identical degrees, theinfinite system of equations will be obtained for theacoefficients i,j (this is a familiar procedure, but thisis only the first step):⎛ 2a2,0 + 2a0,2= 0⎞⎜⎟⎜6a3,0 + 2a1,2= 0⎟⎜...⎟ .⎜⎟⎜( i + 1)( i + 2) ai+ 2, j+ ( j + 1)( j + 2) ai, j+2= 0⎟⎜...⎟⎝⎠Then the system can be written as follows (thegroup of the coefficients of those corresponding for thedifferential equation ai + 2 , j is expressed throughthe group of free coefficients a 0 , j and a 1 , j ):⎛ a⎜⎜ a⎜⎜⎜...⎜⎜ a⎜⎝...= −a2,0 0,21= − a33,0 1,2( j + 1)( j + 2)= −a( i + 1)( i + 2)i + 2, j i, j + 2Several equations will be written to understandfurther actions better:⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠.Annual Proceedings of Vidzeme University College “ICTE in Regional Development”, 2006130