The unified transform for mixed boundary condition problems in unbounded domains

Area of Science:

• Applied Mathematics
• Mathematical Physics
• Computational Science

Background:

• Many problems in acoustics and electrostatics involve unbounded domains and solutions with corner singularities.
• Traditional methods like the Wiener-Hopf technique can be complex and limited in scope.
• Mixed boundary condition problems are common in scattering and field calculations.

Purpose of the Study:

• To implement the unified transform method for solving boundary value problems with corner singularities in unbounded domains.
• To demonstrate the method's ability to handle mixed boundary conditions without resorting to the Wiener-Hopf technique.
• To provide an alternative, efficient approach for problems in acoustic scattering and electric field calculations.

Main Methods:

• The unified transform method is applied to construct a global relation.
• This relation connects known boundary data (e.g., scattered normal velocity) to unknown boundary values (e.g., pressure jump).
• Approximation of data and evaluation at collocation points yield expansion coefficients for unknown values.

Main Results:

• The unified transform method successfully solves problems with corner singularities and mixed boundary conditions.
• The approach is validated for the modified Helmholtz and Helmholtz equations.
• Comparisons show favorable results against traditional Wiener-Hopf and other spectral/boundary methods.

Conclusions:

• The unified transform offers a powerful and versatile alternative for solving a wide range of boundary value problems.
• It simplifies the analysis of problems involving acoustic scattering and electric fields with complex geometries.
• The method provides accurate solutions and is applicable to various equations, including Helmholtz types.