Question

(a) As in Example 1, use (13) and (15) to solve the initial value problem $$ \begin{aligned} &u_{t}(x, t)=u_{x x}(x, t), \quad 0

Short answer

Question: Evaluate the solution of the given heat equation at (x, t) = (l/2, 1). Answer: The solution at (x, t) = (l/2, 1) is given by: $$u(\frac{l}{2},1) = \sum_{n\, \text{even}}^\infty \frac{2l}{(n - 2)(n + 2) \pi} \sin \frac{n \pi}{2} e^{-\frac{n^2 \pi^2}{l^2}}$$

Step-by-step solution

Applying eigenfunction expansion method

The solution to the heat equation will be in the form: $$u(x, t) = \sum_{n=1}^\infty b_n \sin \frac{n \pi x}{l} e^{-\frac{n^2 \pi^2}{l^2} t}$$ where \(b_n\) are coefficients to be determined.

Evaluate the coefficients

To find the \(b_n\) coefficients, we use orthogonality and the provided initial condition, \(u(x,0)=f(x)\): $$b_n = \frac{2}{l} \int_{0}^{l} f(x) \sin\frac{n \pi x}{l} dx$$ We are given \(f(x)=\sin\frac{2\pi x}{l}\). Therefore, $$b_n = \frac{2}{l} \int_{0}^{l} \sin\frac{2\pi x}{l} \sin\frac{n \pi x}{l} dx$$

Solve for b_n using trigonometric manipulations

To simplify the integral, we use the identity: $$2\sin A\sin B = \cos(A - B) - \cos(A + B)$$ So, the integral becomes: $$b_n = \frac{1}{l}\int_{0}^{l}\left[\cos\left(\frac{(n-2)\pi x}{l}\right) - \cos\left(\frac{(n+2)\pi x}{l} \right)\right] dx$$ Now, integrating and using the boundary values, we get $$b_n = \frac{l}{(n-2)(n+2) \pi}[(1 - (-1)^{n-2}) - (1 - (-1)^{n+2})]$$ However, we can determine that \(b_n = 0\) when 'n' is odd (due to orthogonality). Therefore, for even 'n', the \(b_n\) coefficient is:$$b_n=\frac{2l}{(n - 2)(n + 2) \pi}$$

Find the solution u(x,t)

The solution to the heat equation is given by: $$u(x, t) = \sum_{n\, \text{even}}^\infty \frac{2l}{(n - 2)(n + 2) \pi} \sin \frac{n\pi x}{l} e^{-\frac{n^2 \pi^2}{l^2} t}$$

Evaluate the solution at (x,t)=(l/2,1)

Using the solution we found and the given point \((\frac{l}{2},1)\), we get: $$u(\frac{l}{2},1) = \sum_{n\, \text{even}}^\infty \frac{2l}{(n - 2)(n + 2) \pi} \sin \frac{n \pi}{2} e^{-\frac{n^2 \pi^2}{l^2}}$$ This is the value of the solution at \((x,t)=(\frac{l}{2},1)\).

Key concepts

Eigenfunction Expansion Method

The eigenfunction expansion method is a powerful tool used to solve partial differential equations (PDEs) with boundary conditions. It involves representing the solution of a PDE as an infinite series of eigenfunctions, each associated with a corresponding eigenvalue.

These eigenfunctions are determined by the type of PDE and the boundary conditions, and they form a basis for the space of possible solutions. When solving a PDE such as the heat equation, the initial condition is projected onto this basis set, which allows us to find the series coefficients through orthogonality relationships.

The eigenfunction expansion method is particularly useful for problems with simple geometric domains where explicit eigenfunctions can be found. The emergence of series solutions provides a comprehensive understanding of the behavior of the PDE over time.

Heat Equation

The heat equation, \(u_t = u_{xx}\), is a second-order linear partial differential equation that describes the distribution of heat (or temperature variation) in a given region over time. It is based on Fourier's law of heat conduction, which essentially states that heat flows from regions of higher temperature to regions of lower temperature.

In solving the heat equation, boundary conditions such as fixed or insulated ends are specified to give the problem a physical context. Initial conditions provide the temperature distribution at time zero, setting the stage for how the heat will evolve. The eigenfunction expansion method, often utilized to solve such problems, gives a clear depiction of how the initial heat distribution evolves as time progresses.

Orthogonality of Functions

Orthogonality is a crucial concept in mathematics, particularly in the realm of function spaces. Functions are said to be orthogonal if their inner product (integral of the product of the functions over a given interval) is zero. This property is analogous to perpendicular vectors in Euclidean space.

When dealing with eigenfunction expansion methods, the orthogonality of the eigenfunctions permits us to solve for the coefficients in the series solution. For example, in the provided solution, the sine functions are orthogonal on the interval \([0, l]\). It simplifies the process of finding the coefficients \(b_n\) because the integral of the product of two different orthogonal eigenfunctions over the interval will be zero. The coefficients can then be reduced to contributions only from terms that are not orthogonal to the initial condition.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. These identities are invaluable tools for simplifying and transforming expressions in mathematical, engineering, and physical sciences.

In solving differential equations using the eigenfunction expansion method, trigonometric identities enable us to manipulate terms to evaluate integrals or simplify expressions. For instance, in the exercise provided, the product-to-sum identities are used to express the product of sine functions as a difference of cosine functions, which significantly simplifies the integration process. Learning to leverage these identities can allow students to streamline the computation of Fourier coefficients, which is a fundamental step in solving PDEs with periodic or symmetrical boundary conditions.