Physics

Nonlinear system

Collocation method

Volterra integral equation

Spectral method

Mathematics

Economic growth

Collocation (remote sensing)

Gene

Statistics

Curse of dimensionality

Machine learning

Mathematical analysis

Singularity

Economics

Integral equation

Chemistry

Transformation (genetics)

Convergence (economics)

Computer science

Quantum mechanics

Differential equation

Applied mathematics

Singular integral

Smoothing

Ordinary differential equation

Biochemistry

Engineering With Computers

The main purpose of this work is to develop a spectrally accurate collocation method for solving weakly singular integral equations of the second kind with nonsmooth solutions in high dimensions. The proposed spectral collocation method is based on a multivariate Jacobi approximation in the frequency space. The essential idea is to adopt a smoothing transformation for the spectral collocation method to circumvent the curse of singularity at the beginning of time. As such, the singularity of the numerical approximation can be tailored to that of the singular solutions. A rigorous convergence analysis is provided and confirmed by numerical tests with nonsmooth solutions in two dimensions. The results in this paper seem to be the first spectral approach with a theoretical justification for high-dimensional nonlinear weakly singular Volterra type equations with nonsmooth solutions.

A novel Jacob spectral method for multi-dimensional weakly singular nonlinear Volterra integral equations with nonsmooth solutions