Global analytic expansion of solution for a class of linear parabolic systems with coupling of first order derivatives terms

TL;DR

This paper develops global analytic solutions for a class of coupled linear parabolic systems with variable coefficients, enabling analytical and numerical advances in solving related PDEs and applications in finance and physics.

Contribution

It provides the first global analytic expansion of fundamental solutions for coupled parabolic systems with space-time dependent coefficients, including convergence proof.

Findings

Pointwise convergence of the analytic expansion is established.

Analytic representations facilitate defining generalized stochastic processes.

Numerical schemes based on these representations are accurate and stable.

Abstract

We derive global analytic representations of fundamental solutions for a class of linear parabolic systems with full coupling of first order derivative terms where coefficient may depend on space and time. Pointwise convergence of the global analytic expansion is proved. This leads to analytic representations of solutions of initial-boundary problems of first and second type in terms of convolution integrals or convolution integrals and linear integral equations. The results have both analytical and numerical impact. Analytically, our representations of fundamental solutions of coupled parabolic systems may be used to define generalized stochastic processes. Moreover, some classical analytical results based on a priori estimates of elliptic equations are a simple corollary of our main result. Numerically, accurate, stable and efficient schemes for computation and error estimates in strong norms can be obtained for a considerable class of Cauchy- and initial-boundary problems of parabolic type. Furthermore, there are obvious and less obvious applications to finance and physics. Warning: The argument given in the current version is only valid in special cases (essentially the scalar case). A more involved argument is needed for systems and will be communicated soon in a replacement,