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:

  • a1=1.232a_1 = 1.232
  • a2=0.116a_2 = -0.116.

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 n=3n = 3 provides a fairly good approximation, especially within the interior of the interval 0x10 \leq x \leq 1.

If you have any questions or would like more details, feel free to ask!

Here are some related questions:

  1. How does the Ritz method differ from other numerical approximation methods?
  2. Why do we use algebraic polynomials in the Ritz method?
  3. What are the limitations of using the Ritz method with a small number of terms (e.g., n=3n = 3)?
  4. How does the analytical solution account for the boundary conditions differently than the approximate solution?
  5. How could we improve the accuracy of the Ritz solution?

Tip: Increasing the number of terms in the Ritz approximation (higher nn) 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

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Suitable Grade Level

Graduate Level