hương pháp tiếp cận của chúng tôi để giải quyết những vấn đề này sẽ được chia nhỏ các khu vực quan tâm thành một mạng hxk kích thước mắt lưới và viết các phương trình khác nhau tương ứng với các điểm mạng, để có được giá trị của hàm tại mỗi điểm mạng. X điểm chung mạng, y Ithe phái sinh thể hiện là | 266 EXCEL NUMERICAL METHODS Thus for example Laplace s equation 12-2 is rewritten as F x h y 2F x y F x -h y F x y k - 2F x ỳ F x y-k 12-13 Our approach for solving these problems will be to subdivide the region of interest into a lattice of mesh size h X k and write the difference equations that correspond to the lattice points to obtain values of the function at each lattice point. For the general lattice point x yt the derivative expression is F xi i y 2F X yi Fjx y F Xị yi i - 2F x . y F xt y i h2 k2 12-14 If h k equation 12-14 simplifies to F xi 1 z yi i - 4F x yt F xt y F x _ y 0 12-15 from which we obtain equation 12-16 F x y F XM z F xi 0 F x X -1 f W1 12-16 4 For the case where h k an expression for F x y can readily be obtained from equation 12-14. Note that four lattice points are involved in the calculation of F x y by equation 12-16 as represented in Figure 12-1. This representation is sometimes referred to as the stencil of the method. Figure 12-1. Stencil of the finite difference method for the solution of an elliptic PDE. The points shown as solid squares represent previously calculated values of the function the open square represents the value to be calculated. CHAPTER 12 PARTIAL DIFFERENTIAL EQUATIONS 267 Methods for the solution of equation 12-16 can best be illustrated by reference to a concrete example. An Example Temperature Distribution in a Heated Metal Plate A typical example of an elliptic partial differential equation involves the solution of a steady-state heat-flow problem. For example if a thin steel plate 10x10 cm has one of the edges held at 100 C and the other three edges at o c what are the steady-state temperatures within the plate For simplicity we assume that heat is not lost through the faces of the plate. We subdivide the plate by means of a grid with h k cm thus creating a lattice of size 20 X 20. At equilibrium heat flows in the x-axis direction into a lattice element at a rate proportional to the temperature of the .
