The fixed method solves the equation $.$ Given the $$and $$at the j$-$th step of the backward Euler method, what are the functions $$for the equation

At each step of the backward Euler method, we know the step size h$,$ the time $$for the current step and $$from the last step, from which we need to figure out all the information needed by the numerical methods for nonlinear equations. The tolerance and the maximum number of iteration steps can be chosen based on the accuracy requirement, $,,$ and our experience. But it is tricky to choose a proper initial point to start the iteration of numerical methods for nonlinear equations. For the fixed$-$point method and the Newton$$s method, one easy way is to choose the initial point $$to be $$in order to start their iterations. This often gives a fast convergence because

$$is usually close to$$when the mesh size $$is small.