Present study proposes a methodology and provides computational results for a conjugate heat transfer problem, where one would like to level out the temperature along the solid-fluid interface. Such constraints or needs may emerges in industrial processes such as mold casting and efficient heat exchanger design. In order to describe the methodology, we consider a benchmark problem of heat and mass transfer through a backward facing step duct with a finite thickness of solid wall. We performed computations at two different Reynolds numbers, 200 and 400 by utilizing our in-house lattice Boltzmann method solver. We validated our in-house solver by comparing our results with those obtained from a conventional Navier-Stokes solver and analytical solution. The solver is able to run on graphical processing unit and model conjugate heat transfer problem without any specific treatment at the solid-fluid interface. Utilizing the optimization algorithm based on trust region method and the advantageous of being able to run our solver on graphical processing unit, we reach constant temperature along the solid fluid interface within 1.55% maximum deviation. The methodology we propose here takes the thermal conductivity of the solid as a variable instead of wall thickness. Thereby, the problem turns into a feasible optimization problem since it does not require remeshing. After reaching the goal, one can easily determine required solid wall thickness along the flow direction by taking the reciprocal of the determined local thermal conductivity according to Fouriers Law.