Could you write down the equation here please, so you can get the best help.

Is your time discretisation explicit or implicit?

Where are the nonlinear terms? Is it the nonlinear thermal diffusivity or is it a nonlinear source term?

Dear Anton

My time discretisation is implicit.

Non-linear terms include in both thermal diffusion and source terms.

I think i should solve them numerically but which method should be followed?

Are dLs and dLv differential terms. If they are differential terms you need to integrate them and then evaluate

∂/∂x [⌊1-∅]k (∂T_s)/∂x]

I assume ∅ is a constant?

Then you need to discretise the PDE using finite differences perhaps or other more complicated methods. Then you will have to code it for whatever software you are using, and be aware of instability problems and accuracy.

LS