Solving Partial Differential Equation Numerically Pertemuan 13 Matakuliah: S0262-Analisis Numerik Tahun: 2010

Material Outline Partial differential equation –2 nd order partial differential equations –Linear 2 nd order PDE –Elliptic equations: Laplace equations

4  PARTIAL DIFFERENTIAL EQUATIONS BASIC CONCEPTS -A PDE is an equation involving one or more partial derivatives of dependent variables u. -The order of PDE is the highest partial derivative at the PDE -The degree of the highest partial derivative is the degree of the PDE -If all dependent and independent variables as well as their partial derivatives in 1 st degree then PDE called linear PDE -The most important PDE in application is 2 nd order PDE

5  PARTIAL DIFFERENTIAL EQUATIONS 2 ND ORDER LINEAR PDE For dependent variable u with 2 independent variables x and y: PDE can be divided in to 3 categories: B 2 – 4 ACCategory <0Elliptic =0Parabolic >0Hyperbolic

6 IMPORTANT 2 nd ORDER PDE

7 TWO DIMENSIONAL LAPLACE EQUATION (ELLIPTIC EQUATION) Elliptic equation (Laplace equation) in engineering field typically used to characterize steady state, boundary value problems. Steady state flow of heat in heated plate can be written as

8 Solution Technique The solution will be based on finite difference technique that is PDE is transformed into difference equation. For this, the plate is treated as a grid of discrete points

9 Solution Technique For this, the plate is treated as a grid of discrete points x y 0,0 m+1,0 0,n+1 i,j i-1,j i,j+1 i,j-1 i+1,j

10 Solution Technique Central Finite Difference:

11 Solution Technique Central Finite Difference:

12 Solution Technique Central Finite Difference Equation:

13 TWO DIMENSIONAL LAPLACE EQUATION Example: A 4 × 4 heated plate, where all sides are kept at constant temperature as given in the following figure: How the temperature distributed? T= 100 o C T= 75 o C T= 50 o C T= 0 o C

14 TWO DIMENSIONAL LAPLACE EQUATION Solution: T= 100 o C T= 75 o C T= 50 o C T= 0 o C T 0,0 T 1,0 T 4,0 T 0,1 T 0,4 T 4,4 Boundary conditions:

15 TWO DIMENSIONAL LAPLACE EQUATION Solution: T= 100 o C T= 75 o C T= 50 o C T= 0 o C T 0,0 T 1,0 T 4,0 T 0,1 T 0,4 T 4,4 The above equation will be used to solve the temperature of inner points