7.5 Two-Dimensional Problems

7.5 Two-Dimensional Problems

• The boundary was formed by the removal of the small rectangle.

• Solutions to two-dimensional problems can be produced by separation of variables and other analytical methods.
• Simple numerical methods work well on two-dimensional problems.
• The heat and wave equations on two-dimensional regions that fit on graph paper will be limited in this exposition.

• We will use the space position with one or two subscripts and time level as an approximation to the solution of a problem.
• The Laplacian operator will need to be replaced due to heat and wave problems.

• The replacement equations are obtained.

• The rules of thumb are still valid, and the stability considerations of Section 7.2 are still important.
• It makes the equations simpler.

All temperatures are given as 1

• Boundary temperatures are represented by the 0's in these equations.

• The numbering of the points is shown.
• The replacement equation is the same as in Eqs.

• A typical replacement for the wave equation is constructed.

• The stability rules are still in effect.
• In order to get a sensible solution.

• Assume that the initial data from Eqs.

• The equation is running.

• To find the starting equation we have to solve it.

• If they fit neatly on a rectangular grid, nonrectangular regions can be handled.
• These points are illustrated by several exercises.

• Set up replacement equations using the given space mesh and the number shown in the figure cited.
• If there is a disagreement, boundary conditions should be used.

• The region is a cross.

• The square was taken from the upper right corner.
• Assume zero displacement on the boundary and 1 in the lower right corner.