Question

Find a two-parameter Galerkin approximation of the following problem which follows the second order differential equation. Use weightediresudual method. Use trigonometric approximation functions, Di = cos i7x L and Φ0 = 0. - d dx [ du(x) dx] = f₀ cos x L for 0 < x

Answer 1

To find the Galerkin approximation of the given problem, we will use the weighted residual method with trigonometric approximation functions. The equation we have is -d/dx [du(x)/dx] = f₀ cos(x/L) for 0 < x. First, we determine the weight function w(x) and set up the weighted residual equation. Next, we simplify the equation using a trigonometric identity, and finally, we find the Galerkin approximation solution by substituting the trigonometric approximation functions into the equation.

Step-by-step explanation:

To find the Galerkin approximation of the given problem, we will use the weighted residual method with trigonometric approximation functions. The equation we have is:

- d/dx [ du(x) dx] = f₀ cos(x/L) for 0 < x

First, we need to determine the weight function w(x). Since we are using the trigonometric approximation functions, we can choose w(x) to be equal to cos(iπx/L), where i is an integer. Now, we set up the weighted residual equation:

- d/dx [ du(x) dx] - f₀ cos(x/L) cos(iπx/L) = 0

Next, we use the trigonometric identity cos(a) cos(b) = (1/2)(cos(a+b) + cos(a-b)) to simplify the equation:

- d/dx [ du(x) dx] - (f₀/2) (cos((i+1)πx/L) + cos((i-1)πx/L)) = 0

We can now proceed to find the Galerkin approximation solution by substituting the trigonometric approximation functions u(x) = Σ Di cos(iπx/L), where i is an integer, into the equation.