Question

The Chebyshev Equation. The Chebyshev? differential equation is $$ \left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}+\alpha^{2} y=0 $$ where \(\alpha\) is a constant. (a) Determine two linearly independent solutions in powers of \(x\) for \(|x|<1 .\) (b) Show that if \(\alpha\) is a nonnegative integer \(n\), then there is a polynomial solution of degree \(n\). These polynomials, when properly normalized, are called the Chebyshev polynomials. They are very useful in problems requiring a polynomial approximation to a function defined on \(-1 \leq x \leq 1\). (c) Find a polynomial solution for each of the cases \(\alpha=n=0,1,2,\) and \(3 .\)

Short answer

Question: Determine two linearly independent solutions for the Chebyshev differential equation within the interval |x|<1 and show that if α is a nonnegative integer n, there is a polynomial solution of degree n. Answer: The two linearly independent solutions for the Chebyshev differential equation within the interval |x|<1 are: $$ y_1(x) = \sum_{n=0}^{\infty} a_n x^{n+\alpha} $$ $$ y_2(x) = \sum_{n=0}^{\infty} a_n x^{n-\alpha} $$ If α is a nonnegative integer n, the recurrence relation becomes: $$ a_{m+1} = \frac{n^2 - (m+1+r)}{m+2+r}a_m $$ When the value of m becomes equal to n, the numerator of the fraction becomes zero, causing the series to terminate and yielding a polynomial solution of degree n.

Step-by-step solution

Part (a): Identify the Chebyshev differential equation and determine two linearly independent solutions for |x|

The given Chebyshev differential equation is: $$ (1-x^2) y^{\prime \prime} - x y^{\prime} + \alpha^{2} y = 0 $$ To find two linearly independent solutions in powers of x, we will use the Frobenius method. We can assume a solution of the form: $$ y(x) = \sum_{n=0}^{\infty} a_n x^{n+r} $$ Then, by calculating the first and second derivatives, we can plug them back into the original equation. _DERIVATIVES_ $$ y'(x) = \sum_{n=0}^{\infty} (n+r) a_n x^{n+r-1} $$ $$ y''(x) = \sum_{n=0}^{\infty} (n+r)(n+r-1) a_n x^{n+r-2} $$ _SUBSTITUTION_ Substituting y(x), y'(x), and y''(x) back into the original equation: $$ (1-x^2)\left(\sum_{n=0}^{\infty} (n+r)(n+r-1) a_n x^{n+r-2}\right) - x\left(\sum_{n=0}^{\infty} (n+r)a_n x^{n+r-1}\right) + \alpha^{2}\left(\sum_{n=0}^{\infty} a_n x^{n+r}\right) = 0 $$ In order to ensure the series converges, we must equate the coefficients to zero. Let \(n = m + 2\) to have a uniform power for x: $$ \sum_{m=0}^{\infty}\left[((m+2+r)(m+1+r) a_m x^{m+r} - (m+2+r)a_{m+1} x^{m+r+1} + \alpha^{2} a_m x^{m+r})\right]=0 $$ By comparing coefficients, we will have the following recurrence relationship: $$ (m+2+r) a_m - (m+2+r) a_{m+1} + \alpha^2 a_m = 0 $$ $$ a_{m+1} = \frac{\alpha^2 - (m+1+r)}{m+2+r}a_m $$ Using this recurrence relation, we can find two different cases of r: $$ r = \pm \alpha $$ Thus, the two linearly independent solutions are given by: $$ y_1(x) = \sum_{n=0}^{\infty} a_n x^{n+\alpha} $$ $$ y_2(x) = \sum_{n=0}^{\infty} a_n x^{n-\alpha} $$ These are the linearly independent solutions for the Chebyshev differential equation.

Key concepts

Linearly Independent Solutions

Linearly independent solutions are essential in solving differential equations, as they form the basis for the solution space. In the context of the Chebyshev differential equation \((1-x^{2}) y''-x y'+\alpha^{2} y=0\), we aim to find two solutions that are independent of each other to adequately describe the general solution of the equation within the domain \(|x| < 1\). The solutions are independent if no linear combination of them yields the trivial function.To discover these solutions, we employ the Frobenius method. This method involves expressing each solution as a power series in the form \(y(x) = \sum_{n=0}^{\infty} a_n x^{n+r}\). This series considers all potential values of \(r\) from which valid values will determine different series and solutions. We find such solutions are given by different choices of \(r\): \(r=\pm \alpha\). Thus, the solutions are:

• \(y_1(x) = \sum_{n=0}^{\infty} a_n x^{n+\alpha}\)
• \(y_2(x) = \sum_{n=0}^{\infty} b_n x^{n-\alpha}\)

These solutions satisfy the Chebyshev differential equation and do not express one solution as a multiple of the other, confirming their linear independence.

Polynomial Approximation

Polynomial approximation is a mathematical strategy used to safely and effectively estimate functions. The approximation process involves creating polynomials that mirror the behavior of a more complex function within a specific range. In many real-world applications and problems, especially those arising in numerical analysis and computer science, polynomial approximations are immensely valuable.For the Chebyshev differential equation, specifically, when \(\alpha\) is a nonnegative integer \(n\), the resulting polynomial solutions are of degree \(n\). These solutions are particularly significant because they serve as approximations to complex functions, efficiently captured within the interval \(-1 \leq x \leq 1\). In practice, such polynomials allow us to closely match the behavior of a function within this range, minimizing error in approximation. Chebyshev polynomials are commonly used for this purpose due to their advantageous mathematical properties, such as minimizing the maximum error in approximation across these intervals. This optimization makes them ideal for applications like signal processing, numerical integration, and solving differential equations, where precision is paramount.

Chebyshev Polynomials

Chebyshev polynomials, denoted \(T_n(x)\), are a sequence of orthogonal polynomials that arise in problems involving polynomial approximation. They are named after the mathematician Pafnuty Chebyshev and are derived from solving the Chebyshev differential equation specifically for values of \(\alpha\) that are nonnegative integers.When properly normalized, these polynomials serve important purposes, for example, in minimizing the error of polynomial approximations over the interval \([-1, 1]\). The most notable feature of Chebyshev polynomials is that they exhibit an equiripple property, meaning the extrema of the polynomial between \(-1\) and \(1\) are equally distant from each other, which helps to distribute the run-time error evenly.For instance, the first few Chebyshev polynomials are:

• \(T_0(x) = 1\)
• \(T_1(x) = x\)
• \(T_2(x) = 2x^2 - 1\)
• \(T_3(x) = 4x^3 - 3x\)

These polynomials are widely used in numerical solutions and are instrumental in areas requiring high degrees of accuracy, such as computational graphics, finite element analysis, and control systems in engineering.

Frobenius Method

The Frobenius method is a powerful technique to find solutions to differential equations, especially when they have regular singular points. It expands a differential equation's solution as a power series about some point, typically \(x=0\). This method is particularly useful when standard power series solutions do not suffice.For the Chebyshev differential equation \((1-x^2) y'' - x y' + \alpha^2 y = 0\), the method commences with an assumed solution: \(y(x) = \sum_{n=0}^{\infty} a_n x^{n+r}\). The next step involves calculating derivatives and substituting these back into the equation. By equating series coefficients to zero, we derive a recurring relationship that helps in determining the coefficients \(a_n\) of the series.A crucial part is determining the possible values for \(r\), known as the roots of the indicial equation. These roots offer us multiple series solutions, and for Chebyshev’s equation, yield two linearly independent solutions. In the case of Frobenius method applied to Chebyshev's equation, the solutions correspond to different \(r\), specifically \(\pm\alpha\). The entire process involves careful algebraic manipulation but results in precise and robust solutions for the types of differential equations often encountered in mathematical physics and engineering.