A fast sparse spectral method for nonlinear integro-differential Volterra equations with general kernels

Journal article

In this thesis, we introduce new numerical approaches to two important types of integral equation problems using sparse spectral methods. First, linear as well as nonlinear Volterra integral and integro-differential equations and second, power-law integral equations on d-dimensional balls involved in the solution of equilibrium measure problems. These methods are based on ultraspherical spectral methods and share key properties and advantages as a result of their joint starting point: By working in appropriately weighted orthogonal Jacobi polynomial bases, we obtain recursively generated banded operators allowing us to obtain high precision solutions at low computational cost. This thesis consists of three chapters in which the background of the above-mentioned problems and methods are respectively introduced in the context of their mathematical theory and applications, the necessary results to construct the operators and obtain solutions are proved and the method's applicability and efficiency are showcased by comparing them with current state-of-the-art approaches and analytic results where available. The first chapter gives a general scope introduction to sparse spectral methods using Jacobi polynomials in one and higher dimensions. The second chapter concerns the numerical solution of Volterra integral equations. The introduced method achieves exponential convergence and works for general kernels, a major advantage over comparable methods which are limited to convolution kernels. The third chapter introduces an approximately banded method to solve power law kernel equilibrium measures in arbitrary dimensional balls. This choice of domain is suggested by the radial symmetry of the problem and analytic results on the supports of the resulting measures. For our method, we obtain the crucial property of computational cost independent of the dimension of the domain, a major contrast to particle simulations which are the current standard approach to these problems and scale extremely poorly with both the dimension and the number of particles.Open Acces

Authors: Gutleb, TS

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer
• Journal: Advances in Computational Mathematics
• Volume: 47
• Issue: 3
• Pages: 42
• Article number: 42
• Publication date: 2021-05-07
• DOI: 10.1007/s10444-021-09866-7
• EISSN: 1572-9044
• ISSN: 1019-7168
• Language: English
• Keywords: Polynomial; Nonlinear system; Exponential function; Volterra equations; Computational Science and Engineering; Volterra integral equation; Spectral method