Math Problem Statement
Consider the differential equation:
dx
2
d
2
u
+u=x
2
for 0<x<1 under two sets of boundary conditions: Set 1:u(0)=0,u(1)=0 Set 2: u(0)=0,
dx
du
∣
∣
x=1
=1 Determine and numerically evaluate the three-parameter Ritz solutions (i.e., n=3 ) using algebraic polynomial approximation functions for both sets of boundary conditions. Plot and compare the results with the analytical solutions. (Hint: The analytical solution for Set 1 boundary condition is u(x)=
sin1
sinx+2sin(1−x)
+x
2
−2. The analytical solution for Set 2 boundary condition is u(x)=
cos1
2cos(1−x)−sinx
+x
2
−2.)
I want complete solution of set 2 only
Solution
The optimized values for the Ritz method approximation are:
- .
The plot compares the Ritz approximate solution (red curve) and the analytical solution (blue dashed curve) for the given boundary conditions. The Ritz method with provides a fairly good approximation, especially within the interior of the interval .
If you have any questions or would like more details, feel free to ask!
Here are some related questions:
- How does the Ritz method differ from other numerical approximation methods?
- Why do we use algebraic polynomials in the Ritz method?
- What are the limitations of using the Ritz method with a small number of terms (e.g., )?
- How does the analytical solution account for the boundary conditions differently than the approximate solution?
- How could we improve the accuracy of the Ritz solution?
Tip: Increasing the number of terms in the Ritz approximation (higher ) typically leads to a more accurate solution.
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Math Problem Analysis
Mathematical Concepts
Differential Equations
Boundary Value Problems
Ritz Method
Numerical Approximation
Formulas
Analytical solution for Set 2 boundary condition: u(x) = (cos(1)/2)cos(1-x) - sin(x) + x^2 - 2
Theorems
-
Suitable Grade Level
Graduate Level
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