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(Solved): Book problem 4.6(The book is Introduction to Partial Differential Equations A Computational Approach ...



Book problem 4.6(The book is Introduction to Partial Differential Equations A Computational Approach by Aslak Tveito Ragnar Winther) except you will not run the code. Your task is to develop the discrete equations, especially the new equations associated with the derivative boundary conditions. Then you will explicitly write down the matrix A, the right hand side vector, B. The right hand side vector is that associated with known data and the initial value function, f(x). As indicated in the book problem, use the function of Example 3.5, but just to build the explicit data vector of initial values.EXAMPLE 3.5 We want to solve (3.37) with the initial data
\[
f(x)=9+3 \cos (\pi x)+5 \cos (4 \pi x)
\]
Since this function is

EXAMPLE 3.5 We want to solve (3.37) with the initial data Since this function is written in the form (3.53), the Fourier coefficients are easily found, and the solution of is given by This solution is graphed, as a function of , in Fig. 3.6 for . You can observe from the figure that the Neumann-type boundary conditions are satisfied. EXERCISE 4.6 Derive an explicit scheme for the following Neumann problem: Use the analytical solution given in Example 3.5 on page 101 to check the quality of your approximations.


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