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Solving the Equations - uxx - euyy = f(x,y,u) by an O(h^4) Finite Difference Method
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| Numerical Methods for Partial Differential Equations |
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The semi-linear equation - uxx - euyy = f(x,y,u) with Dirichlet boundary conditions is solved by an O(h^4) finite different method, which has local truncation error O(h^2) at the mesh points neighboring the boundary and O(h^4) at most interior mesh points. It is proved that the finite difference method is O(h^4) uniformly convergent as h -> 0. The method is considered in the form of a system of algebraic equations with a nine diagonal sparse matrix. The system of algebraic equations is solved by an implicit iterative method combined with Gauss elimination. A Mathematica module is designed for the purpose of testing and using ht method. To illustrate the method, the equation of twisting a springy rod is solved.
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