Numerical Solutions To Second Order One Dimensional Boundary Value Problems
Usually boundary value problems (BVPs) are numerically solved by solving an approximately equivalent matrix problem obtained by discretizing the original BVP.
The most commonly used method for numerically solving bvps in 1d is called the Finite Difference Method. This method in essence uses linear derivative approximations such as where is the distance between neighbouring x values on the discretized domain to obtain a system of linear equations that can be solved, this linear system can then be solved by standard matrix methods.
For instance, suppose the equation to be solved is:
Then the next step could be, to discretize the problem and use linear derivative approximations such as and solve the resulting system of linear equations.
This would lead to equations such as:
On first viewing this system of equations appears to have a difficulty associated with the fact that the equation involves no terms that are not multiplied by variables but in fact this is false. Because see, at i=1 and N-1 there's a term involving the boundary values and and since these two values are known one can simply substitute them into this equation and as a result have a non-homogenous linear system of equations that has non-trivial solutions
Read more about this topic: Numerical Methods For Ordinary Differential Equations
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