Question

Solve the nonhomogeneous initial-boundary value problem. $$ \begin{array}{l} u_{t}=u_{x x}-2, \quad 00 \\ u(0, t)=1, \quad u(1, t)=3, \quad t>0 \\ u(x, 0)=2 x^{2}+1, \quad 0 \leq x \leq 1 \end{array} $$

Short answer

Question: Find the solution of the nonhomogeneous initial-boundary value problem $$ \begin{array}{l} u_t = u_{xx} - 2, \quad 0 < x < 1, \quad t > 0 \\ u(0,t) = 1, \quad u(1,t) = 3, \quad t > 0 \\ u(x,0) = 2x^2 + 1, \quad 0 \le x \le 1 \end{array} $$ Answer: The solution of the nonhomogeneous initial-boundary value problem is $$ u(x, t)=\sum_{n=1}^{\infty}c_n \Big(\frac{4}{e - 1} (e^{\sqrt{-\lambda_n}x} - 1)\Big)e^{n^2\pi^2 t} - 2x + 3 $$ where \(\lambda_n = -n^2\pi^2\) and the Fourier coefficients \(c_n\) are given by: $$ c_n = \int_0^1 (2x^2+1)\Big(\frac{4}{e - 1} (e^{\sqrt{-\lambda_n}x} -1)\Big)dx $$

Step-by-step solution

Solving the homogeneous heat equation

First, let's write down the homogeneous heat equation with the given boundary conditions: $$ \begin{array}{l} v_{t}=v_{x x}, \quad 00 \\ v(0, t)=1, \quad v(1, t)=3, \quad t>0 \end{array} $$ where \(v(x,t)\) denotes the solution of the homogeneous heat equation. Now, we can solve this homogeneous heat equation using the method of separation of variables. Assume that a solution can be written as \(v(x, t) = X(x)T(t)\). Plugging this into the PDE, we get $$ X(x)T'(t) = X''(x)T(t) $$ Now, divide both sides by \(XT\): $$ \frac{T'(t)}{T(t)} = \frac{X''(x)}{X(x)} = -\lambda $$ Here, the separation constant \(-\lambda\) is chosen since it leads to non-trivial solutions. This results in two ordinary differential equations: 1. \(T'(t) + \lambda T(t) = 0\) 2. \(X''(x) + \lambda X(x) = 0\) Solving the first equation, we get \(T(t) = Ce^{-\lambda t}\) where \(C\) is an integration constant. Solving the second equation (eigenvalue problem) with the boundary conditions \(X(0)=1\) and \(X(1)=3\), we get: $$ X_n(x) = \frac{4}{e - 1}(e^{\sqrt{-\lambda_n}x} - 1) $$ where \(\lambda_n = -n^2\pi^2\) are the eigenvalues with \(n = 1, 2, 3, ...\). Hence, the solution to the homogeneous equation is given by: $$ v(x, t) = \sum_{n=1}^{\infty}c_n \Big(\frac{4}{e - 1} (e^{\sqrt{-\lambda_n}x} - 1)\Big)e^{n^2\pi^2 t} $$

Finding the particular solution

Now we need to find a particular solution to the nonhomogeneous heat equation. We look for a solution in the form \(u_p(x, t) = Ax + B\). Plugging this into the nonhomogeneous PDE, we get $$ A = -2 $$ Using the boundary conditions, we can solve for \(A\) and \(B\): $$ \begin{array}{l} u_p(0, t) = 1,\quad u_p(1, t) = 3 \\ A+B=1, \quad A-B=-1 \end{array} $$ Solving these equations, we get \(A=-2\) and \(B=3\). So, \(u_p(x, t)=-2x+3\).

Combining homogeneous and particular solutions

Now we can use the principle of superposition to obtain the complete solution: $$ u(x, t) = v(x, t) + u_p(x, t) $$

Applying the initial condition

Finally, we can use the initial condition \(u(x, 0) = 2x^2 + 1\) to find the integration constants \(c_n\). $$ 2x^2+1 = \sum_{n=1}^{\infty}c_n \Big( \frac{4}{e - 1} (e^{\sqrt{-\lambda_n}x}-1) \Big) - 2x + 3 $$ To find \(c_n\), we'll use orthogonality properties of eigenfunctions and Fourier series: $$ c_n = \int_0^1 (2x^2+1)\Big(\frac{4}{e - 1} (e^{\sqrt{-\lambda_n}x}-1)\Big)dx $$ Calculating the Fourier coefficients \(c_n\), we finally obtain the complete solution \(u(x,t)\) as $$ u(x, t)=\sum_{n=1}^{\infty}c_n \Big(\frac{4}{e - 1} (e^{\sqrt{-\lambda_n}x} - 1)\Big)e^{n^2\pi^2 t} - 2x + 3 $$

Key concepts

Homogeneous Heat Equation

The homogeneous heat equation plays a pivotal role in understanding how heat diffuses through a medium over time. It is given by the partial differential equation (PDE) \( v_{t} = v_{xx} \). This equation assumes that no external heat sources or sinks are present - meaning that the heat spreads solely due to temperature gradients within the medium.

In this context, the terms relate to:

• \( v_{t} \): the rate of change of temperature with respect to time.
• \( v_{xx} \): the second derivative of temperature with respect to space, indicating how temperature varies spatially.

The boundary conditions \( v(0, t) = 1 \) and \( v(1, t) = 3 \) specify the fixed temperatures at the boundaries \( x = 0 \) and \( x = 1 \).

By solving this equation, especially using techniques like the separation of variables, we find how heat distribution changes over time with these specific boundary conditions. This forms the basis to approach more complex nonhomogeneous scenarios as in the given exercise.

Nonhomogeneous Initial-Boundary Value Problem

A nonhomogeneous initial-boundary value problem modifies the homogeneous PDE by introducing additional terms, typically representing an external influence or source of heat. In the exercise, this is seen as the adjusted heat equation:\[ u_{t} = u_{xx} - 2 \] Here, the \,\(-2\)\, term acts as a persistent heat sink throughout the domain, implying a uniform and constant removal of heat.

Unlike the homogeneous scenario, boundary conditions \( u(0, t) = 1 \) and \( u(1, t) = 3 \) describe the fixed temperatures at the edges while the initial condition \( u(x, 0) = 2x^2 + 1 \) provides the starting temperature distribution. Solving such problems requires finding both general solutions to the homogeneous part and particular solutions to the nonhomogeneous part.

Understanding these problems is essential, as they model real-life situations where a medium's heat distribution is affected by factors like ambient temperature changes or local heating mechanisms. Thus, techniques such as superposition and separation of variables become critical tools.

Method of Separation of Variables

The method of separation of variables is a powerful technique used to solve linear partial differential equations (PDEs). This method involves assuming that the solution can be written as a product of functions, each dependent on a single coordinate.

In the context of the homogeneous heat equation \( v(x, t) = X(x)T(t) \), we express the solution as a product of a spatial part \( X(x) \) and a temporal part \( T(t) \).

This assumption transforms the PDE into two separate ordinary differential equations (ODEs):

• \( T'(t) + \lambda T(t) = 0 \)
• \( X''(x) + \lambda X(x) = 0 \)

The separation constant \( \lambda \) allows the two functions to simultaneously satisfy the original PDE.

By solving these equations separately, we find pairs of functions that describe the evolution of temperature over time and space. The essence of this method lies in reducing a complex PDE problem into simpler, manageable ODEs for which solutions are more straightforward.

Superposition Principle

The superposition principle is a key aspect of solving linear systems. It states that if you have multiple solutions to a linear equation, their sum (or linear combination) is also a solution.

In solving our PDE, we find both the solution to the homogeneous equation \( v(x, t) \) and a particular solution \( u_p(x, t) \) to the nonhomogeneous part. By the superposition principle, the complete solution \( u(x, t) \) is found by adding these two solutions:

• \( u(x, t) = v(x, t) + u_p(x, t) \)

This principle greatly simplifies the process of solving nonhomogeneous boundary value problems, as it allows us to combine known solutions in a coherent manner. Understanding and applying superposition enable us to approach complex scenarios involving various initial and boundary conditions with more consistency and clarity.