SOLUTION OF A NONLINEAR TRICOMI EQUATION BY TAYLOR SERIES

Abstract

A recursive relation between derivative terms can be derived from any given nonlinear PDE. This recursive relation can be used to construct a two-dimensional Taylor series which satisfies both the PDE and the boundary and/or initial conditions. The solution of a nonlinear PDE contains simple poles, or singularities, in the complex planes of the independent variables. The construction of the two-dimentional Taylor series must proceed with careful analysis of the positions of these simple poles, or singularities. Once their positions are known (or have been estimated), analytic continuation is used to avoid them in the nonlinear solution. We illustrate this solution method by solving a nonlinear Tricomi equation, y.u xx + u yy + u 2 = 0, with boundary conditions:- u(x,-1) = u(x,1) = x, u(0,y) = f(y), and u(1,y) = g(y). The functions f(y) and g(y) are eigen-functions compatible with the ramp-function boundary conditions. At the present stage of its development, the Taylor series method is capable of solving problems with arbitrary boundary conditions on only two sides. We shall also discuss the solution for the same Tricomi equation with boundary conditions u(x,-1) = u(x,1) = x–x 2 .