Research On Chaos Of One Class Of Nonlinear Singular Integral Equation

In this article, we research one class of nonlinear singular integral equation as follows here L denotes a closed smooth curve in complex plan, cp(/) is an unknown function, a(t)、b(t)、c(t) are all given polynomials, the integral exits with the Cauchy principle value.First of all, we discuss it in the condition that a(t)、b(t)、c(t) are constants, i.e. we want to find its solution (?)(t) in Holder continuous function space and L donates an unit circle. We discretize the above equation through Lagrange rational interpolation method and collocation methods, then transfer the discretized equation into ternary quadratic nonlinear equations, and solve the equations by programming, that is, the approximate solution of the above nonlinear singular integral equation, we analyse the shape of the solutions through images at the same time, trying to find the connections between the solutions and the coefficients.Similarly, in the interval [-1,1], the form of the first equation is as followsBy Lagrange interpolation method, we find that in some cases, the number of solutions by program is less than its actual number in the process of assigning values to a(t)、b(t)、c(t) to solve the nonlinear equations, when we seek the solutions through iterative method, bifurcation on some point appears, we assume that there exists some connection between nonlinear singular integral equation and chaos phenomenon, which is meaningful to our research.