Problem 11 - Advanced Engineering Mathematics

Question

(a) Solve the boundary-value problem $$ \begin{aligned} &\frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t}, 00 \\ &u(0, t)=0, \quad u(\pi, t)=0, t>0 \\ &u(x, 0)=\sin x, 0

Step-by-Step Solution

Answer

(a) Solution: \(u(x,t) = \sin(x) e^{-t}\). (b) Solution: \(u(x,t) = 100\sin(3x)e^{-9t} - 30\sin(5x)e^{-25t}\).

Step 1: Understanding the Boundary-Value Problem

The given problem is a boundary-value problem involving the heat equation \( \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t} \) in the interval \(0

Step 2: Solving the Homogeneous Problem

Assume a solution of the form \(u(x,t) = X(x)T(t)\). Substitute into the PDE to obtain \(X''(x)T(t) = X(x)T'(t)\). Divide by \(X(x)T(t)\) to separate variables, giving \(\frac{X''(x)}{X(x)} = \frac{T'(t)}{T(t)} = -\lambda\). Solve the spatial part \(X''(x) = -\lambda X(x)\) with boundary conditions \(X(0) = 0\) and \(X(\pi) = 0\). Solutions are of the form \(X_n(x) = \sin(nx)\) where \(n\) is a positive integer.

Step 3: Solving the Temporal Problem

Given \(T'(t) = -\lambda T(t)\), solve for \(T(t)\) when the eigenvalue \(\lambda = n^2\). The general solution is \(T(t) = A_n e^{-n^2 t}\). The complete solution to the homogeneous problem is \(u_n(x,t) = A_n \sin(nx) e^{-n^2 t}\), where \(A_n\) are constants determined by initial conditions.

Step 4: Applying Initial Conditions

The initial condition \(u(x,0) = \sin x\) suggests an expansion: \(u(x,0) = \sum A_n \sin(nx)\). Coefficients \(A_n\) are determined by the orthogonality of sine functions: \(A_1 = 1\) (since \(A_1 \sin(x)\) satisfies the initial condition) and \(A_n = 0\) for all other \(n\). Thus, the solution is \(u(x,t) = \sin(x) e^{-t}\).

Step 5: Solving Part (b) with a New Initial Condition

Given a different initial condition \(u(x,0) = 100 \sin(3x) - 30 \sin(5x)\), the solution must be a combination of modes: \(u(x,t)=100 \sin(3x) e^{-9t} - 30 \sin(5x) e^{-25t}\). Each sine term decays exponentially with constants specifically chosen to satisfy new initial conditions.

Key Concepts

Heat Equation

The heat equation is a fundamental partial differential equation (PDE) pivotal in modeling the distribution of heat (or variation in temperature) in a given region over time. The general form of the heat equation in one spatial dimension is given by: \[ \frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t} \] In this equation, \( u(x,t) \) represents the temperature at a point \( x \) and at time \( t \). The term \( \frac{\partial^2 u}{\partial x^2} \) considers the spatial derivatives, capturing how temperature changes with position, while \( \frac{\partial u}{\partial t} \) describes the change in temperature over time. Heat equations are typically solved to determine how a given initial temperature distribution evolves as time progresses, and solutions often need to satisfy certain initial and boundary conditions.

Separation of Variables

Separation of variables is a powerful technique for solving partial differential equations such as the heat equation. The idea is to assume that the solution can be written as a product of functions, each depending on a single variable. For the heat equation, this means postulating that: \[ u(x, t) = X(x)T(t) \] This assumption allows us to express the PDE in a form where each side depends on a different variable, leading to separate ordinary differential equations (ODEs).

• The spatial component becomes: \( X''(x) + \lambda X(x) = 0 \)
• The temporal component is: \( T'(t) + \lambda T(t) = 0 \)

Here, \( \lambda \) is a separation constant, and each ODE can be solved independently. The superposition of these solutions provides a series that solves the original PDE.

Eigenvalues and Eigenfunctions

Eigenvalues and eigenfunctions arise naturally when using separation of variables to solve the heat equation. These terms originate from solving ODEs derived from the spatial component of the PDE. The problem of finding \( X(x) \) such that \( X''(x) + \lambda X(x) = 0 \) with boundary conditions \( X(0)=0 \) and \( X(\pi)=0 \) leads to eigenvalues \( \lambda_n = n^2 \). The corresponding eigenfunctions are \( X_n(x) = \sin(nx) \), particularly found by ensuring the boundary conditions make the only non-trivial solutions multiples of sine functions at certain harmonic frequencies. These eigenvalues provide the exponential decay rates in the temporal component, while eigenfunctions form a basis for representing any initial temperature distribution as a series.

Initial and Boundary Conditions

For solving PDEs like the heat equation, initial and boundary conditions are crucial as they dictate the specifics of the solution.

• Initial Conditions: These are the values of the solution at \( t=0 \). In this problem, the initial condition \( u(x,0) = \sin x \) specifies an initial temperature distribution across the domain.
• Boundary Conditions: These conditions describe how the solution behaves at the boundaries of the spatial domain. For this problem, \( u(0,t)=0 \) and \( u(\pi,t)=0 \) mean the boundaries are held at zero temperature.

Incorporating both sets of conditions determines the particular form of the solution, ensuring it evolves appropriately over time. The initial condition is often expanded in terms of the eigenfunctions, and the solution evolves as a series where each term decays over time based on the corresponding eigenvalue.

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Problem 11

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Problem 12

Other exercises in this chapter

Problem 11

Set up the boundary-value problem for the steady-state temperature \(u(x, y)\). A thin rectangular plate coincides with the region in the \(x y\) -plane defined

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Problem 11

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Problem 12

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Problem 12

Set up the boundary-value problem for the steady-state temperature \(u(x, y)\). A semi-infinite plate coincides with the region defined by \(0 \leq x \leq \pi,

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Key Concepts

Heat Equation

The heat equation is a fundamental partial differential equation (PDE) pivotal in modeling the distribution of heat (or variation in temperature) in a given region over time. The general form of the heat equation in one spatial dimension is given by: \[ \frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t} \] In this equation, \( u(x,t) \) represents the temperature at a point \( x \) and at time \( t \). The term \( \frac{\partial^2 u}{\partial x^2} \) considers the spatial derivatives, capturing how temperature changes with position, while \( \frac{\partial u}{\partial t} \) describes the change in temperature over time. Heat equations are typically solved to determine how a given initial temperature distribution evolves as time progresses, and solutions often need to satisfy certain initial and boundary conditions.

Separation of Variables

Separation of variables is a powerful technique for solving partial differential equations such as the heat equation. The idea is to assume that the solution can be written as a product of functions, each depending on a single variable. For the heat equation, this means postulating that: \[ u(x, t) = X(x)T(t) \] This assumption allows us to express the PDE in a form where each side depends on a different variable, leading to separate ordinary differential equations (ODEs).

• The spatial component becomes: \( X''(x) + \lambda X(x) = 0 \)
• The temporal component is: \( T'(t) + \lambda T(t) = 0 \)

Here, \( \lambda \) is a separation constant, and each ODE can be solved independently. The superposition of these solutions provides a series that solves the original PDE.

Eigenvalues and Eigenfunctions

Eigenvalues and eigenfunctions arise naturally when using separation of variables to solve the heat equation. These terms originate from solving ODEs derived from the spatial component of the PDE. The problem of finding \( X(x) \) such that \( X''(x) + \lambda X(x) = 0 \) with boundary conditions \( X(0)=0 \) and \( X(\pi)=0 \) leads to eigenvalues \( \lambda_n = n^2 \). The corresponding eigenfunctions are \( X_n(x) = \sin(nx) \), particularly found by ensuring the boundary conditions make the only non-trivial solutions multiples of sine functions at certain harmonic frequencies. These eigenvalues provide the exponential decay rates in the temporal component, while eigenfunctions form a basis for representing any initial temperature distribution as a series.

Initial and Boundary Conditions

For solving PDEs like the heat equation, initial and boundary conditions are crucial as they dictate the specifics of the solution.

• Initial Conditions: These are the values of the solution at \( t=0 \). In this problem, the initial condition \( u(x,0) = \sin x \) specifies an initial temperature distribution across the domain.
• Boundary Conditions: These conditions describe how the solution behaves at the boundaries of the spatial domain. For this problem, \( u(0,t)=0 \) and \( u(\pi,t)=0 \) mean the boundaries are held at zero temperature.

Incorporating both sets of conditions determines the particular form of the solution, ensuring it evolves appropriately over time. The initial condition is often expanded in terms of the eigenfunctions, and the solution evolves as a series where each term decays over time based on the corresponding eigenvalue.