边界值的有限差分法模块
Module For The FiniteDifference Method for Boundary Value Problems
Background The notation Corollary (Linear Boundary Value Problem). Assume that Finite-Difference Method Use the notation Proof Finite Difference Method for ODE's Finite Difference Method for ODE's Footnote. The significance of the theory. Computer Programs Finite Difference Method for ODE's Finite Difference Method for ODE's Program (Finite-Difference Method). To approximate the solution of the boundary value problem Example 1. Solve Example 2. Solve Example 3. Solve Example 4. Determine how much the solutions in Example 2 and 3 differ. Example 5. Use Richardson's extrapolation and the results of Example 2 and 3 to construct a more accurate solution for 25 subintervals. Example 6. How good did it get? Research Experience for Undergraduates Finite Difference Method for O.D.E.'s Finite Difference Method for O.D.E.'s Internet hyperlinks to web sites and a bibliography of articles. Download this Mathematica Notebook The Finite Difference Method for Boundary Value Problems
Theorem (Boundary Value Problem). Assume that
and
then the boundary value problem
has a unique solution
and
then the linear boundary value problem
has a unique solution
Methods involving difference quotient approximations for derivatives can be used for solving certain second-order boundary value problems. Consider the linear equation
(1)
over [a,b] with
(2)
and
(3)
which is used to compute numerical approximations to the differential equation (1). This is carried out by multiplying each side by
for
We are all familiar with the differential equation
Procedures.
(i) Construct the tri-diagonal matrix and vector.
(ii) Solve the system in (i).
(iii) Join the mesh points and vector from (ii) to form the solution points.
Use the finite difference method with 25 subintervals (total of 26 points).
Solution 1.
Use the finite difference method with 25 subintervals (total of 26 points).
Just use the subroutine and skip all the details.
Solution 2.
Use the finite difference method with 50 subintervals (total of 51 points).
Just use the subroutine and skip all the details.
Solution 3.
Solution 4.
Solution 5.
Solution 6.