Finite-Difference Methods for Boundary-Value Problems
In this chapter, we approximate by means of finite-differences several prototype examples of boundary-value problems in both ordinary and partial differential equations. For each individual problem, we develop one or more finite-difference schemes and state some results on the solvability of the associated (linear or non-linear) systems of equations. Then we formulate the original problem and its approximations as operator equations in suitable function spaces. In such a setting, we are then able to investigate accuracy properties of the finite-difference approximations themselves by analyzing the behavior of the truncation errors.
KeywordsTruncation Error Mesh Point Difference Quotient Order Ordinary Differential Equation Approximate Boundary Condition
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