Numerical solutions have been obtained for a diffusion e ame in the two-dimensional, laminar, steady, viscous, multicomponent, compressible mixing layer in the presence of a pressure gradient by using the boundary-layer approximationsand solvingthe x-momentum,energy,and speciesconservation equations. Thenumericalsolutions have been validated against similarity solutions and then are extended to cases where no similarity solution exists. The numerical solutions are used to study the ignition process and the e ame structure in an accelerating transonic mixing layer. It is shown how ignition length depends on initial temperature, initial pressure, initial velocity, viscosity, and pressure gradient. Ignition is found to occur on the high-temperature-air side. Oxidation kinetics and transport are both controlling in the upstream ignition region. Farther downstream, transport is controlling in the fully established e ame. The boundary-layer approximation is found to be valid everywhere including the upstream ignition region.