6.3 Finite Difference approximations to partial derivatives

In the chapter 5 various finite difference approximations to ordinary differential equations have been generated by making use of Taylor series expansion of functions at some point say x0. The same can be extended to higher dimensions in the following manner.

Descritisation of a rectangular domain

Consider a rectangular domain D: and .

Draw straight lines parallel to x-axis and y-axis as shown in the figure 6.3.1 such that for i = 1, 2, 3 ......n-1 and for j = 1, 2, 3..........m-1 where and are small positive steplengths obtained by =a/n and =b/m.

Figure 6.3.1

Let be any point in the region D then the co-ordinates xi and yj can be obtained by



where (x0,y0) is the coordinates of the left bottom most point of the rectangle, that is (0, 0) in the present problem . If u(x,y) is any continuous function with all necessary derivatives exisit in D then



From (6.3.3) partial derivatives can be approximated as


Making use of these approximations to replace partial derivatives, the partial differential equations are converted into difference equations and the resultant system of algebraic equations are solved using any direct or iterative methods. Since the analytical methods for finding solution of second order partial differential equations depend on the type of PDE, the numerical schemes also depend on the type of PDE.

Solution of Transcendental Equations | Solution of Linear System of Algebraic Equations | Interpolation & Curve Fitting
Numerical Differentiation & Integration | Numerical Solution of Ordinary Differential Equations
Numerical Solution of Partial Differential Equations