A general method is presented for finding asymptotic solutions of problems posed for linear partial differential equations containing lower order (dispersive) terms. Powers of a large parameter λ appear in these equations multiplying the lower order terms. The expansion procedure is a “ray method”, i.e. all of the functions that appear in the expansion satisfy ordinary differential equations along certain curves called rays. Special attention is paid to the equation of heat conduction. In this case, 1/λ is related to the thermal conductivity of the medium. For this equation several problems are considered in which the parameter λ enters the data or the inhomogeneous (source) term in various ways.