Chebyshev Differentiation Matrix

In summary, the conversation discusses the use of Chebyshev Polynomials to solve differential equations and how they can also be used to expand derivatives. The relation between the coefficients is given and the Chebyshev differentiation matrix is introduced. The conversation then moves on to discussing the possibility of multiplying two Chebyshev expansions, with the solution involving the use of Gaussian integration to solve inner product integrals.
  • #1
Leonardo Machado
57
2
TL;DR Summary
It is a question about the method to obtain the Chebyshev coefficients for differential operators.
Hi everyone.

I am studying Chebyshev Polynomials to solve some differential equations. I found in the literature that if you have a function being expanded in Chebyshev polynomials such as
$$
u(x)=\sum_n a_n T_n(x),
$$

then you can also expand its derivatives as
$$
\frac{d^q u}{dx^q}=\sum_n a^{(q)}_n T_n(x),
$$

with the following relation
$$
a^{(q)}_{k-1}= \frac{1}{c_{k-1}} ( 2 k a^{(q-1)}_k+ a^{(q)}_{k+1}),
$$

being $c_k=2$ for k=0 and 1 if k>0.

It all together defines the Chebyshev differentiation matrix, which is $D$ in
$$
a^{(1)}_i=D_{ij} a^{(0)}_j.
$$

Now I would like to know if there is any way of doing

$$
x^l \frac{du}{dx}=\sum_n a^{(x)}_n T_n(x),
$$

I am looking for it everywhere in the literature but I can't find a way of dealing with this kind of operator that appears in the Laplacian. I can't describe every linear operator without it
 
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  • #2
I haven't worked with Chebyshev polynomials since last millennia, but to me it looks like you are asking for the multiplication of two Chebyshev expansions. If so, that should be fairly straight forward.
 
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Likes Leonardo Machado
  • #3
I have solved it today. As a computational problem i used a Gaussian integration to take inner products from both sides, as

$$
\sum_k a^{(0)}_k (T_n(x),x^l \frac{du}{dx})=a^{(x)}_n (T_n(x),T_n(x)).
$$

Using Chebyshev collocation points to solve the inner product integral.
 

1. What is a Chebyshev Differentiation Matrix?

A Chebyshev Differentiation Matrix is a matrix used in numerical analysis to approximate the derivative of a function at a set of points. It is based on the Chebyshev polynomials, which are a set of orthogonal polynomials that have useful properties for numerical computations.

2. How is a Chebyshev Differentiation Matrix constructed?

A Chebyshev Differentiation Matrix is constructed by first choosing a set of points, typically Chebyshev points, at which the derivative of the function will be approximated. Then, the Chebyshev polynomials are evaluated at these points and used to construct a matrix that represents the derivative approximation.

3. What are the advantages of using a Chebyshev Differentiation Matrix?

One advantage of using a Chebyshev Differentiation Matrix is that it can provide a more accurate approximation of the derivative compared to other methods, such as finite differences. It also has the ability to handle functions with discontinuities or singularities, which can be challenging for other numerical methods.

4. How is a Chebyshev Differentiation Matrix used in practice?

A Chebyshev Differentiation Matrix is often used as a tool in scientific computing and engineering to approximate derivatives in numerical simulations. It can also be used in solving differential equations and optimization problems.

5. Are there any limitations to using a Chebyshev Differentiation Matrix?

While a Chebyshev Differentiation Matrix has many advantages, it does have limitations. It is most effective for smooth functions and may not perform well for functions with high-frequency oscillations. It also requires careful selection of the points at which the derivative will be approximated, which can be time-consuming for complex functions.

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