Problem 19 - Advanced Engineering Mathematics

Question

Solve the boundary-value problem $$ \begin{aligned} &\frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t^{\prime}} \quad-\infty0 \\ &\left.\frac{\partial u}{\partial x}\right|_{x=1}=100-u(1, t), \quad \lim _{x \rightarrow-\infty} u(x, t)=0, t>0 \\ &u(x, 0)=0, \quad-\infty

Step-by-Step Solution

Answer

The solution involves separation of variables and using boundary conditions to determine the form of \( u(x,t) \). This involves finding \( u(x,t) \) as a series solution dependent on constants from boundary/initial values.

Step 1: Explore the Differential Equation

We identify that the given differential equation is a one-dimensional heat equation \( \frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t} \) where the dependent variable \( u(x,t) \) represents the temperature. It describes how the temperature evolves over the spatial domain \( -\infty < x < 1 \) over time \( t > 0 \).

Step 2: Boundary and Initial Conditions

The differential equation is subject to three conditions:1. A boundary condition at \( x = 1 \) is given by \( \left.\frac{\partial u}{\partial x}\right|_{x=1} = 100 - u(1, t) \).2. A boundary condition at \( x = -\infty \) is \( \lim_{x \to -\infty} u(x, t) = 0 \).3. An initial condition is \( u(x, 0) = 0 \) for \( -\infty < x < 1 \).

Step 3: Apply Ansatz Method

Assume a solution of the form \( u(x, t) = h(x) g(t) \). Substitute this into the differential equation. This leads to \( h''(x)g(t) = h(x)g'(t) \). Simplifying gives \( \frac{h''(x)}{h(x)} = \frac{g'(t)}{g(t)} = \lambda \), where \( \lambda \) is a separation constant.

Step 4: Solve Temporal Part

For the temporal part \( g'(t) = \lambda g(t) \), solve \( g(t) = g_0 e^{\lambda t} \), where \( g_0 \) is a constant determined by initial/boundary conditions.

Step 5: Solve Spatial Part

For the spatial part \( h''(x) = \lambda h(x) \), solve \( h(x) = A e^{\sqrt{\lambda} x} + B e^{-\sqrt{\lambda} x} \). Constants \( A \) and \( B \) will be determined using boundary conditions.

Step 6: Impose Boundary Conditions

Apply the boundary condition at \( x = -\infty \) to simplify the solution to \( h(x) = B e^{-\sqrt{\lambda} x} \). Applying \( \lim_{x \to -\infty} u(x, t) = 0 \) implies \( A = 0 \).

Step 7: Boundary Condition at x=1

Substitute \( x = 1 \) into \( u(x,t) \) and utilize \( \frac{\partial u}{\partial x} |_{x=1} = 100 - u(1, t) \) to solve for \( \lambda \) or constants \( B \) and \( g_0 \). This step involves differentiation and solving resulting equations.

Step 8: Complete Solution

The complete solution is then expressed as a series representation involving the solutions for temporal and spatial parts: \( u(x,t) = \sum B_n e^{-\sqrt{\lambda_n} x} g_0 e^{\lambda_n t} \) where \( B_n \) are coefficients determined from boundary/initial conditions.

Key Concepts

Boundary-Value Problems

Boundary-value problems are mathematical problems where the solution to a differential equation must meet specific conditions, known as boundary conditions. These conditions are defined at the boundaries of the domain in which the equation is applied.

• In our problem, the domain is defined over the spatial range from negative infinity to 1.
• Boundary conditions can be essential to ensuring the uniqueness and existence of solutions.

Boundary-value problems are common in various fields such as physics and engineering, as they can describe phenomena like the temperature distribution along a rod or the electric potential in a field. By understanding how to apply these conditions correctly, one can effectively solve real-world problems modeled by differential equations.

Heat Equation

The heat equation is a partial differential equation that represents the distribution of heat (or temperature) in a given region over time. It is a foundational concept in the study of heat transfer and thermodynamics. The standard form of the heat equation in one dimension is:\[\frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t}\] Where:

• \(u(x, t)\) denotes the temperature at position \(x\) and time \(t\).
• This equation illustrates how temperature changes with respect to space and time.

In the context of our exercise, the heat equation describes the evolution of temperature in an infinite domain extending to the left and bounded on the right at \(x = 1\). Understanding the heat equation helps conceptualize how thermal energy distributes over time, driven by the second spatial derivative, indicating the flow of heat due to temperature gradients and resulting changes over time.

Separation of Variables

Separation of variables is a method used to solve partial differential equations by breaking them down into simpler, separate equations. This technique assumes a solution that can be expressed as a product of functions, each depending on a single coordinate or variable.
For the heat equation:

• We assume a solution of the form \(u(x, t) = h(x) g(t)\).
• When substituted into the heat equation, this form lets us separate the equation into one for the spatial part and another for the temporal part.

This separation helps us solve the complex problem by addressing each part individually with their respective conditions. It results in a simpler set of ordinary differential equations (ODEs), each of which is easier to solve independently compared to the original partial differential equation.

Initial and Boundary Conditions

Initial and boundary conditions are vital in solving differential equations, as they specify the solution at initial moments or at spatial boundaries. For our boundary-value problem, they guide how the temperature field evolves and behaves.

• The initial condition \(u(x, 0) = 0\) suggests that at time \(t = 0\), the entire domain starts at a uniform baseline temperature.
• Boundary conditions like \(\left.\frac{\partial u}{\partial x}\right|_{x=1} = 100 - u(1, t)\) dictate the behavior at the spatial limit \(x = 1\).
• The condition \(\lim_{x \to -\infty} u(x, t) = 0\) denotes how temperature tends to a steady state towards negative infinity.

These conditions ensure that the differential equation has a well-defined solution, allowing for meaningful interpretation of phenomena such as heat distribution in the context provided by the problem constraints.