Question

Consider the initial boundary value problem for the heat equation: $$ \begin{array}{rr} u_{t}=2 u_{x x}, & 00 \\ u(1, t)=0, & t>0 \end{array} $$ Use the finite transform method to solve this problem. Namely, assume that the solution takes the form \(u(x, t)=\sum_{n=1}^{\infty} b_{n}(t) \sin n \pi x\) and obtain an ordinary differential equation for \(b_{n}\) and solve for the \(b_{n}\) 's for each \(n\).

Short answer

The series solution for the heat equation is \(u(x, t)=\sum_{n=1}^{\infty} \left[ \int_0^1 x(1-x) \sin (n \pi x) dx \cdot e^{-2n^2 \pi^2 t}\right] \sin n \pi x\). Each \(b_{n}(t)\) is given by \(b_{n}(t) = \int_0^1 x(1-x) \sin (n \pi x) dx \cdot e^{-2n^2 \pi^2 t}\).

Step-by-step solution

Substitute the assumed form of the solution

Substitute \(u(x, t)=\sum_{n=1}^{\infty} b_{n}(t) \sin n \pi x\) into the heat equation, \(u_{t}=2 u_{x x}\). This gives \(\sum_{n=1}^{\infty} \dot{b}_{n}(t) \sin n \pi x=2 \sum_{n=1}^{\infty} b_{n}(t) (-n^2 \pi^2) \sin n \pi x\). Here, the dot above \(b_{n}\) indicates a derivative with respect to time.

Set up the ordinary differential equation (ODE)

We can now state that the respective terms in the infinite sum must be equal. This allows us to write down the ordinary differential equation (ODE) for the function \(b_{n}(t)\): \(\dot{b}_{n}(t) = -2n^2 \pi^2 b_{n}(t)\). This is a first order homogeneous linear ODE for \(b_{n}(t)\).

Solve the ODE

Since we have a first order linear ODE, we can make use of standard methods to solve it. The solution will have the form: \(b_{n}(t) = C_n e^{-2n^2 \pi^2 t}\). The constant \(C_n\) will have to be determined for each \(n\).

Use the initial condition

Insert \(t=0\) into the found expresssion for \(b_{n}(t)\) and compare it with the given initial condition \(u(x, 0)=x(1-x)\). This results in \(b_{n}(0) = C_n = \int_0^1 x(1-x) \sin n \pi x dx\). Calculate this integral for \(C_n\).

Insert \(C_n\) back into the solution

After calculating \(C_n\), insert it into the general solution \(b_{n}(t)\). This gives a final answer for \(b_{n}(t)\) of \(b_{n}(t) = \int_0^1 x(1-x) \sin (n \pi x) dx \cdot e^{-2n^2 \pi^2 t}\).

The solution \(u(x, t)\)

Finally, substitute \(b_{n}(t)\) back into the assumed form of the solution \(u(x, t)=\sum_{n=1}^{\infty} b_{n}(t) \sin n \pi x\). This gives the final solution for \(u(x, t)\) in the form of a Fourier series.

Key concepts

Finite Transform Method

The finite transform method is a powerful technique used to solve partial differential equations (PDEs), especially those that model physical phenomena such as heat transfer, waves, and quantum mechanics. This method simplifies the problem by transforming it from the spatial domain to the frequency domain, where the solution can often be more readily obtained. The method takes advantage of orthogonality properties of the transform functions.

To illustrate, in the provided exercise, we apply this method to a heat equation by assuming the solution forms a series expansion in terms of sinusoidal functions. This series is reminiscent of a Fourier series, which we will discuss later. The next step is to substitute this assumed series form into the heat equation and use the orthogonality of sine functions to isolate equations for the coefficients, leading to an ordinary differential equation for each mode.

Ordinary Differential Equation

An ordinary differential equation (ODE) is an equation involving functions and their derivatives. ODEs provide a description of the dynamics of systems and are critical in modeling the behavior of physical systems, from motion to growth to the change in investment returns over time.

In our exercise, after employing the finite transform method, we obtain an ODE for each coefficient function, referred to as \(b_n(t)\). This ODE is a typical first-order homogeneous linear ODE. To solve it, we use standard methods such as separation of variables, integrating factors, or by recognizing it as an exponential growth/decay type equation. Once the general form of the solution is found, constants are determined using initial or boundary conditions, which are given as part of the exercise.

Fourier Series

A Fourier series is an expansion of a periodic function in terms of an infinite sum of sine and cosine functions. It's a crucial tool for analyzing periodic functions and is often used to solve PDEs, especially within the realm of heat, wave, and quantum equations. Fourier series allow us to decompose complex waveforms into simpler trigonometric functions, making them easier to work with.

In the context of our heat equation problem, once the ODE solution for \(b_n(t)\) is determined, it is plugged back into the assumed series solution to express the temperature distribution \(u(x, t)\) as a Fourier series. This series effectively captures the heat distribution over time and position. The initial condition provided in the exercise helps to calculate the coefficients of the series, finalizing the solution.

Initial Boundary Value Problem

An initial boundary value problem (IBVP) is a type of differential equation problem that includes initial conditions and boundary values to specify the solution uniquely. In physical problems, the initial conditions represent the state of the system at the beginning of observation, while boundary conditions represent constraints that the system must adhere to over its domain.

The heat equation exercise constitutes an IBVP as it provides the temperature distribution at time zero (initial condition) and temperature values at the boundaries of the spatial domain (boundary conditions). These are exceptionally important since they allow us to determine the unique coefficients for the Fourier series that solve the IBVP. Without these conditions, the problem would have infinitely many solutions, but the IBVP context bounds it to a single, physically meaningful solution.