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

We derive global analytic representations of fundamental solutions for a class of linear parabolic systems with full coupling of first order derivative terms where coefficients 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 Cauchyand initialboundary problems of parabolic type. Important instances of application are representations of solutions of multidimensional Burgers equations with forcing and potential initial conditions and Pauli equation describing the non-relativistic limit of Dirac theory for electrons in a magnetic field. 2000 Mathematics Subject Classification. 35K40.