This work examines steady-state heat conduction in a stochastic, heterogeneous medium where the thermal conductivity varies linearly along one direction and its slope consists of a constant plus a zero-mean random part. As a first step, the governing Laplace's equation is solved using a coordinate transformation of the independent spatial variables and the exact Green's functions in both two and three dimensions are obtained for a linearly varying conductivity profile. In addition, a boundary integral equation statement in which the Green's functions appear as kernels is concurrently obtained. Next, material stochasticity is introduced and the perturbation approach is employed for deriving the mean value and covariance of the Green's functions using up to second order terms. Perturbations are also used in conjuction with the discretized boundary integral equation statement so that a mean vector and a covariance matrix for the response (temperature, heat flux) are also obtained. An example involving steady-state temperature distribution in a block along the direction where conductivity varies on the horizontal plane due to a buried heat source serves to illustrate the method. Finally, comparisons are made with Monte Carlo simulations.
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