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

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