Discretisation of the boundary conditions

Considering the boundary conditions in finite difference methods we will distinguish three different cases  

If the temperature at the boundary x = xR is given, then the grid divisions should be chosen such that the boundary coincides with a grid line x - const. The left hand boundary is then xR = x0 with the right hand boundary xR = xn+1 = x0 + (n + 1) Ax. The given temperature r) (xR,tk) is used as the temperature value or Jj+1 in the difference equation (2.240). 

The local derivative which appears in (2.247) is replaced by the rather exact central difference quotient. It holds at the left hand boundary that 

Elimination of i9q from both these equations yields 

This equation replaces (2.250), when the boundary condition (2.247) is to be satisfied at the left hand boundary. 

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