Partial Differential Equations in Time and One Space Dimension

Another option for solving boundary value problems is to treat them like initial value problems. Since a second-order equation can be reduced to two first-order equations, two initial conditions are necessary. One condition will be known at a boundary. Simply assume a value for the other dependent variable at that same boundary, integrate to the other side and check if the required boundary condition is satisfied. If not, change the initial value and repeat the integration. The success of this method depends upon the skill with which you program the iterations from one trial to the next. 

It is possible to program the finite difference method in Excel and use the Calculation feature to handle the circular reference. Turn off the iteration, prepare the spreadsheet, and then turn the calculation back on. Whether this converges depends upon the initial guess. 

Next copy this equation over a series of cells that correspond to the number of grid points. For the first and last cell (grid point) use a different equation, appropriate to the boundary condition. Then turn on the iteration feature to get the solution. 

The next step is to re-solve the problem using more grid points, and a smaller Ax. This means you have to program the spreadsheet again, but it does give you an indication of whether the answer changes much as the mesh is refined, which is what insures you that you have solved the problem accurately enough. 

The same thing can be done using the finite element method, but the details are more complicated and beyond the scope of this book. 

---

Partial differential equations in time and one space dimension ... 

PARTIAL DIFFERENTIAL EQUATIONS IN TIME AND ONE SPACE DIMENSION 317... 

---