The Research On The Solution Of A Kind Of Nonlinear Singular Integral Equation

The nonlinear singular integral equationwhere b₀,b₁,d₀,d₁ are constants and L is a smooth contour,is soloved in Holder continuous space by transforming it to a Riemann boundary value problem with square roots.It is found that, in general,it has other solutions besides the trivial ones.The expression of such solutions as well as the conditions of its solvability is obtained.This paper is composed of five parts.In the first part,we introduce briefly history and context and some known results about the nonlinear singular integral equation.In the second part,we introduce some concepts and lemmas about the nonlinear singular integral equation.In the third part,we maily study how to transform the nonlinear of singular integral equationto a Riemann boundary value problem with square roots.we will discuss the solution of the equation and the conditions of its solvability when L is a smooth closed contour.In the four part,we study the nonlinear singular integral equationwhen L is an open arc.At last,we analyse some special conditions about the nonlinear singular integral equation.