Finite difference method for BVPs

The basic principle in using the finite difference method to solve BVPs is to replace all the derivatives in the differential equation with difference-quotient approximations. First, the interval a x b is discretized into n equally spaced intervals (an unequal spaced interval may also be used)  

From the Taylor theorem, we can derive the difference approximation for the first- and second-order derivatives by expanding y about x,.When using central differences, the first-order derivative is approximated by 

This means there are n - 1 unknowns, i.e. 1,72, -.Tn-uandn - 1 algebraic equations that should be solved for. This can be written in the matrix form. Ay = f where 

Consequently, the approximative solution to the BVP is calculated on a computational mesh, and it results in a system of algebraic equations. In this example, die problem is linear and the Gaussian elimination can be used to solve the equation system (n — 1 algebraic equations). A Dirichlet boundary condition was specified in dus problem. Note that the boundary conditions are accounted for in two of diese equations, i.e. 

As shown here, the implementation of Diiichlet boundary conditions is straightforward. In contrast, ifthe problem contains aNeumannboundary condition, i.e. a derivative on the boundary, a little more work will be needed. 

---

Finite difference method for BVP with nonlinear DE require"odebvfd" Use odelfdO function... 

---