Question

Use Theorem 11.3.5(c) or, where applicable, Exercise 11.1.42(b), to find the mixed Fourier cosine series of \(f\) on \([0, L]\). $$ f(x)=2 x^{3}+3 L x^{2}-5 L^{3} $$

Short answer

Based on the step-by-step solution, provide a short answer to the problem: The mixed Fourier cosine series of the given function \(f(x) = 2x^3 + 3Lx^2 - 5L^3\) on the interval \([0, L]\) is: $$ f(x) = -2L^3 +\sum_{n=1}^{\infty} \left[\frac{4L^3}{n^4\pi^4}[-n^2\pi^2\cos(n\pi) + 2n\pi\sin(n\pi) + 2 (-1)^n - 2] + \frac{12}{n^2 \pi^2}(1 - (-1)^n)L^2 \right] \cos(\frac{n\pi x}{L}) $$

Step-by-step solution

Compute the Fourier Cosine Coefficients

To find the Fourier cosine coefficients, we'll need to calculate the following integrals: $$ A_n = \frac{2}{L} \int_{0}^{L} f(x) \cos(\frac{n\pi x}{L})dx $$ The coefficient \(A_0\) is given by: $$ A_0 = \frac{2}{L} \int_{0}^{L} (2x^3+3Lx^2-5L^3) dx $$ For other coefficients \(A_n\) with \(n > 0\): $$ A_n = \frac{2}{L} \int_{0}^{L} (2x^3+3Lx^2-5L^3) \cos(\frac{n\pi x}{L}) dx $$ Now we will compute \(A_0\) and \(A_n\).

Calculate \(A_0\)

We'll start by calculating \(A_0\): $$ A_0 = \frac{2}{L} \int_{0}^{L} (2x^3+3Lx^2-5L^3) dx \\ A_0 = \frac{2}{L} [\frac{1}{2}x^4 + Lx^3 - 5L^3x]_0^L \\ A_0 = -4L^3 $$

Calculate \(A_n\) for \(n > 0\)

Now, we will compute \(A_n\) for \(n > 0\): $$ A_n = \frac{2}{L} \int_{0}^{L} (2x^3+3Lx^2-5L^3) \cos(\frac{n\pi x}{L}) dx $$ We can solve this integral term by term: $$ A_n = \frac{4}{L} \int_{0}^{L} x^3 \cos(\frac{n\pi x}{L}) dx + \frac{6}{L} \int_{0}^{L} Lx^2 \cos(\frac{n\pi x}{L}) dx - \frac{10}{L}\int_{0}^{L} L^3 \cos(\frac{n\pi x}{L}) dx $$ Let's solve these three integrals, then substitute the values back in the above equation for \(A_n\): 1) \(\int_{0}^{L} x^3 \cos(\frac{n\pi x}{L}) dx = \frac{L^4}{n^4\pi^4}[-n^2\pi^2\cos(n\pi) + 2n\pi\sin(n\pi) + 2 (-1)^n - 2]\) 2) \(\int_{0}^{L} Lx^2 \cos(\frac{n\pi x}{L}) dx = \frac{2}{n^2 \pi^2}(1 - (-1)^n)L^3\) 3) \(\int_{0}^{L} L^3 \cos(\frac{n\pi x}{L}) dx = \frac{2 (-1)^n \sin(n\pi) L^3}{n\pi}=0\) (since \(\sin(n\pi)=0\)) Now substitute the values of the integrals back into \(A_n\): $$ A_n = \frac{4}{L}(\frac{L^4}{n^4\pi^4}[-n^2\pi^2\cos(n\pi) + 2n\pi\sin(n\pi) + 2 (-1)^n - 2] + \frac{6}{L} \frac{2}{n^2 \pi^2}(1 - (-1)^n)L^3) $$ Now, simplify the expression for \(A_n\): $$ A_n = \frac{4L^3}{n^4\pi^4}[-n^2\pi^2\cos(n\pi) + 2n\pi\sin(n\pi) + 2 (-1)^n - 2] + \frac{12}{n^2 \pi^2}(1 - (-1)^n)L^2 $$

Write the Fourier Cosine Series

With the computed coefficients \(A_0\) and \(A_n\), we can now write the mixed Fourier cosine series for \(f(x)\): $$ f(x) = \frac{A_0}{2} + \sum_{n=1}^{\infty} A_n \cos(\frac{n\pi x}{L}) \\ f(x) = -2L^3 +\sum_{n=1}^{\infty} \left[\frac{4L^3}{n^4\pi^4}[-n^2\pi^2\cos(n\pi) + 2n\pi\sin(n\pi) + 2 (-1)^n - 2] + \frac{12}{n^2 \pi^2}(1 - (-1)^n)L^2 \right] \cos(\frac{n\pi x}{L}) $$ This is the mixed Fourier cosine series of the given function \(f(x) = 2x^3 + 3Lx^2 - 5L^3\) on the interval \([0, L]\).

Key concepts

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. In simpler terms, these are equations involving rates of change. They commonly arise naturally in numerous scientific fields including physics, engineering, and biology.

One reason they are important is that they help describe a wide range of phenomena, from the motion of energy waves to the growth rate of populations. They can often be classified into ordinary differential equations (ODEs), which involve functions of one variable and their derivatives, or partial differential equations (PDEs), which involve multiple variables. Understanding and solving differential equations is crucial because they model the behavior of dynamic systems.

• Ordinary Differential Equations (ODEs): These involve functions of single variables and their derivatives.
• Partial Differential Equations (PDEs): These engage multiple variable functions integrating various derivatives.

Each type of differential equation requires different methods and solutions, often including the use of series, such as Fourier series, to express solutions in a more comprehensive manner.

Fourier Coefficients

Fourier coefficients are key components when expressing a function as a Fourier series, which is a method to represent functions over a certain interval as an infinite sum of sine and cosine terms. The Fourier cosine series is particularly valuable when dealing with functions defined on a symmetrical interval such as \(0, L\).

These coefficients, denoted by \(A_n\) in this context, inform us how much of each cosine component is needed in the series to accurately reconstruct the function. Calculating these requires using an integral formula that considers both the function itself and the specific cosine basis function:

• The formula: \[ A_n = \frac{2}{L} \int_{0}^{L} f(x) \cos(\frac{n\pi x}{L})dx \]
• Accurate calculation of these integrals is crucial to finding the correct coefficients.
• The coefficients help in constructing a series that shows periodic behavior even in systems that are not inherently periodic.

Understanding this concept is important in fields that require signal analysis or solving differential equations, as it helps break complex functions into simpler periodic components.

Mathematical Analysis

Mathematical analysis is a vital branch of mathematics focusing on limits, functions, and integrals. It often underlines the theories and methods employed to investigate mathematical concepts at a foundational level.

In the context of Fourier series, mathematical analysis steps in to help evaluate and manipulate the series. It gives precise meaning to the way we observe the convergence of the series and ensures the series approximates the function accurately over a defined interval.

To understand how a series behaves or the implications of summing an infinite number of terms, mathematical analysis provides essential tools such as:

• Understanding of limits and convergence, ensuring the series approximations approach the actual function.
• Enhanced methods for evaluating integrals, crucial for computing the necessary coefficients accurately.
• In-depth understanding of continuity and differentiability, ensuring series and functions can be analyzed robustly.

Grasping mathematical analysis ensures a deep understanding of how series expansions like Fourier cosine series work and enables solving complex mathematical problems with greater efficiency.

Boundary Value Problems

Boundary Value Problems (BVPs) pertain to differential equations that specify values or behaviors of the solutions at the boundaries of the defined domain. These conditions are crucial because they deeply affect the behavior and solutions of the differential equations involved.

In the case of Fourier series, particularly Fourier cosine series, BVPs are often used to solve problems with specified conditions on boundaries, such as at \(x = 0\) and \(x = L\). These problems are paramount in fields like engineering and physics where determining the behavior of phenomena over particular regions is required.

• They ensure that the solutions obtained match real-world conditions and constraints.
• Boundary conditions could be in terms of the function’s value or its derivatives at the borders.
• Such conditions can represent physical constraints, ensuring the function behaves realistically over the domain.

Understanding and solving BVPs with Fourier series is crucial as it allows us to model physical systems accurately and predict their behavior under specified boundary conditions.