Fundamental Solution Of The Tricomi Equations With Pole At The Elliptic Region

The second-order differential equation in two independent variables x and y,Tu=yu_xx+u_yy=0 known as the Tricomi equation, is a classical example of a partial differential equation of mixed type. T is called the Tricomi operator. The equation is elliptic in the upper half plane y?0, parabolic along the x-axis y=0, and hyperbolic in the down half plane y?0.The mixed type equation is one of the remarkable representatives of the partial differential equation. It has many applications in mathematics, physics and gas dynamics etc. For example, the smooth tran-sonic steady flow satisfies a mixed type equation. Unlike the uniform elliptic and uniform hyperbolic equations the mixed type equation need further reserch.The Tricomi operator is invariant under the?3,2?-dilation. Thus, proceeding as physicists usually do, it has homogeneous solutions relative to this dilation.Barros-Neto and Gelfand [6] apply the method of characteristics and the homogenous property of the Tricomi operator to obtain the fundamental solutions of Tricomi operator with pole at the degenerate line. In the first charpter of this article we introduce the preliminaries and main ideas of their work. Barros-Neto and Gelfand [7]also use the method of characteristics to deduce the fundamental solutions of Tricomi operator with pole at the elliptic region. In particular, if the pole tends to the degenerate line, the limit of the fundamental solution is a combination ofF+ andF-.For the real part, the sum of coefficients equal to 1, while for the imaginary part, the coefficients cancellate. So the limit is still a fundamental solution of the Tricomi operator. In the third charpter of this article, we apply the method of series development of Barros-Neto and Cardoso [3] and got a fundamental solution of the Tricomi operator with pole at the elliptic region. In our case, when the pole tends to the degenerate line, the limit of the fundamental solution is still a combination ofF+andF-. But, for the real part, the sum of coefficients does not equal to 1, and, for the imaginary part, the coefficients does not cancellate. So the limit is not a fundamental solution of the Tricomi operator.The work of this paper set a foundation for further study about boundary value problem and Cauchy problem of the mixed type equation.