| Conformal mapping is an important research topic in complex analysis and geometric function theory,and also has important applications in fluid mechanics and electronic mathematics.In this paper,we study the computation of conformal invariants,such as interior and exterior modulus of circular arc polygon,the hyperbolic capacity and elliptic capacity of compact sets in the unit disk by using the boundary integral equation methods with generalized Neumann kernel,then give some numerical experiment results and error estimates and make comparison with other different methods.In addition,we also study some related conjectures,such as the side sliding conjecture,isoperimetric conjectures of hyperbolic capacity and elliptic capacity of compact sets by computing conformal invariants.The value was validated numerically.The main content of this article is as follows: First,by introducing the Riemann-Hilbert problem,the integral equation method for computing numerical conformal mapping and conformal invariants is introduced,which is applied to circular arc polygons.Then,we study the problem of computing interior and exterior modulus for bounded quadrilaterals.We compute the interior modulus of classic polygonal-shaped quadrilaterals and circular polygonal-shaped quadrilaterals,and provide numerical results and error estimates,respectively.The exterior modulus of rectangular quadrilaterals and trapezoidal quadri-laterals are computed and the side sliding conjecture is verified; The interior and exterior modulus of the computed quadrilaterals are compared with other methods.Finally,the computation problem of hyperbolic capacity and elliptic capacity of compact sets within the unit disk is studied.Two computing functions are given,and the feasibility and accuracy of the computing functions are verified by computing the corresponding capacity of compact sets including circular domain,the equilateral triangle domain and the parallelogram domain.For applications,the isoperimetric properties of hyperbolic capacity and elliptic capacity are studied and conjectures are given. |
