Checking Finite Difference Errors
Checking Finite Difference Errors
The exact solution of the boundary value problem f(x)+mf(x)=0, =0 at , and at with parameter is .
2
∂
∂
2
x
∂f(x)
∂x
x=0
f(x)=1
x=1
m
f(x)cos(
m
x)sec(m
)The second-order discretization for the discrete variable at points with step size gives the recurrence equation , where , , which is solved exactly using Mathematica's RSolve function.
F(j)=f(x)
j=1,…,n
h
F(j-1)+mF(j)+F(j+1)=2F(j)
2
h
3F(1)+F(3)=4F(2)
F(n)=1
The difference between the finite difference and exact solutions in the region is dynamically plotted in the solution region for 3 to 300 discretization points and parameter between and .
0≤x≤1
0≤x≤1
m
-10
10
By using DSolve and RSolve, a numerical solution is completely avoided.