The Trefftz polynomials consist of polynomial type solutions of heat conduction equation,which provide us a boundary-type meshless method to solve heat conduction equation.In order to stably solve backward heat conduction problem(BHCP)and inverse heat source problem(IHSP),which both known to be highly ill-posed,by a polynomial method we employ a new post-conditioner in the resultant linear system obtained by the polynomial Trefftz method,and then we use the conjugate gradient method(CGM)to solve the post-conditioned linear system to determine the unknown coeffcients used in the expansion by the multiple scale polynomial Trefftz method(MSPTM),of which the scales are determined a priori by the collocation points.The MSPTM can retrieve the missing initial data and unknown heat source rather well.Several numerical examples of the BHCP and IHSP demonstrate that the present method is e_ective,even for those of strongly ill-posed ones under large noises.