1. Numerical Solutions of Partial Differential Equations CHAPTER 16

2. Ch16_2 Contents  16.1 Laplace’s Equation 16.1 Laplace’s Equation  16.2 The Heat Equation 16.2 The Heat Equation  16.3 The Wave Equation 16.3 The Wave Equation

3. Ch16_3 16.1 Laplace’s Equation  Difference Equation Replacement Suppose we are seeking a solution u(x, t) of Laplace’s equation in a planar region R that is bounded by C. See Fig 16.1.

4. Ch16_4 Fig 16.1

5. Ch16_5  From (6) of Sec 6.5, using central differences (1) (2) Now by adding (1) and (2) we obtain a five-point approximation to the Laplacian:

6. Ch16_6  Hence

7. Ch16_7 16.2 Heat Equation  Difference Equation Replacement Recall that the heat equation: (1) Using central difference approximation:

8. Ch16_8  (1) becomes (2) If we let = ck/h 2 and

9. Ch16_9 16.3 The Wave equation  Difference Equation Replacement Recall that the wave equation: (1) Using central difference approximation:

10. Ch16_10  Then we have (2) and (3)