Solving Boundary Value Problems Using Numerical Analysis
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Solving Boundary Value Problems using Numerical AnalysisDecember 4, 2003Potential TheorySolution of a Two dimensional Boundary Value Problem using Numerical AnalysisNumerical analysis will be used to solve a two dimensional boundary-valued problem. We will solve for steady-state temperatures of a slab. All edges are kept at 0 degrees, except one side which is 100 degrees as shown in the figure. The Laplacian equation governs this situation and is given by [pic 1]. We will use the central-difference approximation for the second derivative to solve for the Laplace Equation. The central-difference approximation is with an error approximation depending on the value of h. The approximations for [pic 3]becomes [pic 4]and [pic 5]. Add the two equations and set equal to zero. The equations lead to the difference equation [pic 6] ,which requires each [pic 7]value to be the average of its four nearest neighbors. We focus our attention on a square lattice of points with horizontal and vertical separation h. Our difference equation can be abbreviated to [pic 8] with points labeled as in figure 2.[pic 2] [pic 9] [pic 10][pic 11][pic 12][pic 13] [pic 14] [pic 15] L [pic 16][pic 17] [pic 18][pic 19] L Figure 1 [pic 20][pic 21][pic 22][pic 23][pic 24][pic 25][pic 26][pic 27][pic 28][pic 29][pic 30][pic 31][pic 32][pic 33][pic 34]
Figure 2Writing such an equation for each interior point E, we have a linear system in which each equation involves five unknowns, except when a known boundary value reduces the number. We choose h so that there are only nine interior points, as in Figure 2. Numbering these points from left to right, top row first, our nine equations for the boundary conditions become:[pic 35][pic 36][pic 37][pic 38][pic 39][pic 40][pic 41][pic 42][pic 43]Solving the nine equations leads to the following solution:[pic 44] 7.1429, [pic 45]9.8214, [pic 46] 7.1429[pic 47]18.7500, [pic 48]25.000, [pic 49]18.7500[pic 50]42.8571, [pic 51]52.6786, [pic 52]42.8571 This problem is homework #1 problem #3 and the analytical solution is [pic 53], with [pic 54]25 which is equivalent to [pic 55].We will now solve for a steady state temperature of a rectangle slab. We will assign dimensions to the variable L. Assume that the slab is 20 cm wide and 10 cm high with all edges at 0 degrees except the right edge, which is at 100 degrees. The boundary condition is shown in Figure 3. [pic 56]Figure 3ADEBCFigure 4Numbering these points from left to right, top row first, our twenty-one equations for the boundary conditions become:[pic 57][pic 58][pic 59][pic 60][pic 61][pic 62][pic 63][pic 64][pic 65][pic 66][pic 67][pic 68][pic 69][pic 70][pic 71][pic 72][pic 73][pic 74][pic 75][pic 76][pic 77]Solving the twenty-one solutions leads to the following solution:ColumnRow 1Row 2Row 310.3500.49890.35020.91321.28940.913232.01032.83242.010344.29576.01944.295759.153212.65389.1532619.663226.289419.6632743.210153.177443.2101