Book cover for Advanced Engineering Mathematics

Advanced Engineering Mathematics

Dennis G. Zill, Michael R. Cullen

ISBN #9780763740955

3rd Edition

4,310 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section demonstrates how boundary-value problems in different coordinate systems (polar, cylindrical, and spherical) can be addressed using separation of variables. The solutions frequently involve series expansions with Fourier coefficients, Bessel functions, and Legendre polynomials. Ensuring that the solution remains bounded—through appropriate choices of constants and enforcing compatibility conditions—is crucial. These techniques are instrumental in modeling physical phenomena, such as steady-state temperature distribution and electrostatic potentials.

Learning Objectives

1

Apply the method of separation of variables to solve Laplace’s equation in polar, cylindrical, and spherical coordinate systems.

2

Interpret and derive Fourier and Bessel series solutions for steady?state problems including temperature distributions.

3

Explain the role of boundary conditions and compatibility conditions in obtaining physically bounded solutions.

4

Utilize special functions such as Bessel functions and Legendre polynomials to represent solutions in different coordinate systems.

Key Concepts

CONCEPT

DEFINITION

Separation of Variables

A method used to reduce a partial differential equation (PDE) into a set of ordinary differential equations (ODEs) by assuming the solution is the product of functions, each depending on a single coordinate.

Laplace’s Equation

A second-order linear PDE given by ∆u = 0. In various coordinate systems it takes different forms, for example, with additional terms in polar, cylindrical, or spherical coordinates.

Fourier Series

An expansion of a periodic function in terms of sine and cosine functions; used to represent boundary data when solving PDEs.

Bessel Functions

Special functions that are solutions to Bessel’s differential equation, commonly arising in problems with cylindrical symmetry.

Legendre Polynomials

A sequence of orthogonal polynomials which appear as solutions to Legendre’s differential equation, frequently used in spherical coordinate problems.

Compatibility Condition

A condition imposed on the boundary data to ensure that a full Fourier series exists (usually that a certain integral vanishes), ensuring the proper matching of non-homogeneous boundary conditions.

Example Problems

Example 1

We have $$\begin{aligned} A_{0} &=\frac{1}{2 \pi} \int_{0}^{\pi} u_{0} d \theta=\frac{u_{0}}{2} \\ A_{n} &=\frac{1}{\pi} \int_{0}^{\pi} u_{0} \cos n \theta d \theta=0 \\ B_{n} &=\frac{1}{\pi} \int_{0}^{\pi} u_{0} \sin n \theta d \theta=\frac{u_{0}}{n \pi}\left[1-(-1)^{n}\right] \end{aligned}$$ and so $$u(r, \theta)=\frac{u_{0}}{2}+\frac{u_{0}}{\pi} \sum^{\infty} \frac{1-(-1)^{n}}{n} r^{n} \sin n \theta$$.

Example 2

We have $$\begin{array}{l} A_{0}=\frac{1}{2 \pi} \int_{0}^{\pi} \theta d \theta+\frac{1}{2 \pi} \int_{\pi}^{2 \pi}(\pi-\theta) d \theta=0 \\ A_{n}=\frac{1}{\pi} \int_{0}^{\pi} \theta \cos n \theta d \theta+\frac{1}{\pi} \int_{\pi}^{2 \pi}(\pi-\theta) \cos n \theta d \theta=\frac{2}{n^{2} \pi}\left[(-1)^{n}-1\right] \\ B_{n}=\frac{1}{\pi} \int_{0}^{\pi} \theta \sin n \theta d \theta+\frac{1}{\pi} \int_{\pi}^{2 \pi}(\pi-\theta) \sin n \theta d \theta=\frac{1}{n}\left[1-(-1)^{n}\right] \end{array}$$ and so $$u(r, \theta)=\sum_{n=1}^{\infty} r^{n}\left[\frac{(-1)^{n}-1}{n^{2} \pi} \cos n \theta+\frac{1-(-1)^{n}}{n} \sin n \theta\right]$$.

Example 3

We have $$\begin{array}{l} A_{0}=\frac{1}{2 \pi} \int_{0}^{2 \pi}\left(2 \pi \theta-\theta^{2}\right) d \theta=\frac{2 \pi^{2}}{3} \\ A_{n}=\frac{1}{\pi} \int_{0}^{2 \pi}\left(2 \pi \theta-\theta^{2}\right) \cos n \theta d \theta=-\frac{4}{n^{2}} \\ B_{n}=\frac{1}{\pi} \int_{0}^{2 \pi}\left(2 \pi \theta-\theta^{2}\right) \sin n \theta d \theta=0 \end{array}$$ and so $$u(r, \theta)=\frac{2 \pi^{2}}{3}-4 \sum_{n=1}^{\infty} \frac{r^{n}}{n^{2}} \cos n \theta$$.

Example 4

We have $$\begin{aligned} &A_{0}=\frac{1}{2 \pi} \int_{0}^{2 \pi} \theta d \theta=\pi\\ &\begin{array}{l} A_{n}=\frac{1}{\pi} \int_{0}^{2 \pi} \theta \cos n \theta d \theta=0 \\ B_{n}=\frac{1}{\pi} \int_{0}^{2 \pi} \theta \sin n \theta d \theta=-\frac{2}{n} \end{array} \end{aligned}$$ and so $$u(r, \theta)=\pi-2 \sum_{n=1}^{\infty} \frac{r^{n}}{n} \sin n \theta$$.

Example 5

As in Example 1 in the text we have $R(r)=c_{3} r^{n}+c_{4} r^{-n}$. In order that the solution be bounded as $r \rightarrow \infty$ we must define $c_{3}=0 .$ Hence $$u(r, \theta)=A_{0}+\sum_{n=1}^{\infty} r^{-n}\left(A_{n} \cos n \theta+B_{n} \sin n \theta\right)$$ where $$\begin{array}{l} A_{0}=\frac{1}{2 \pi} \int_{0}^{2 \pi} f(\theta) d \theta \\ A_{n}=\frac{c^{n}}{\pi} \int_{0}^{2 \pi} f(\theta) \cos n \theta d \theta \\ B_{n}=\frac{c^{n}}{\pi} \int_{0}^{2 \pi} f(\theta) \sin n \theta d \theta \end{array}$$.
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Step-by-Step Explanations

QUESTION

Solve Laplace’s equation in polar coordinates with a boundary condition given by u(1,θ)=f(θ) for 0 ≤ θ < 2π and a bounded solution as r→0.

STEP-BY-STEP ANSWER:

Step 1: Assume a solution of the form u(r,θ)= R(r) Θ(θ) and substitute into Laplace’s equation in polar coordinates: u_rr + (1/r) u_r + (1/r^2) u_θθ = 0.
Step 2: Separate the variables by introducing a separation constant λ. This leads to two ODEs: r^2 R''(r) + r R'(r) - λR(r) = 0 and Θ''(θ) + λΘ(θ) = 0.
Step 3: Solve the angular equation. For periodicity, choose λ = n^2 (n = 0, 1, 2, …) so that Θ(θ)= A_n cos nθ + B_n sin nθ.
Step 4: Solve the radial equation which becomes Euler’s equation with solutions R(r)= C_n r^n + D_n r^(-n). Boundedness as r→0 forces D_n = 0.
Step 5: Compose the full solution and match the boundary condition at r=1, expanding f(θ) into its Fourier series to determine coefficients.
Final Answer: u(r,θ)= A_0 + Σ_{n=1}^∞ (A_n cos nθ + B_n sin nθ) r^n, with Fourier coefficients determined by A_0= (1/2π)∫_0^(2π)f(θ)dθ, A_n= (1/π)∫_0^(2π)f(θ) cos nθ dθ, and B_n= (1/π)∫_0^(2π)f(θ) sin nθ dθ.

Polar Coordinate BVP Example

QUESTION

Obtain the solution for the initial-boundary value problem for the heat equation in a cylinder using the eigenfunction expansion involving Bessel functions.

STEP-BY-STEP ANSWER:

Step 1: Assume a separable solution u(r,t)= R(r)T(t). Substitute into the cylindrical heat equation: u_t = k [u_rr + (1/r) u_r].
Step 2: Separate variables to get (T'(t)/T(t)) = k (R''(r) + (1/r) R'(r))/R(r) = -λ and identify the radial equation: r^2 R'' + r R' + λr^2 R = 0.
Step 3: Recognize the radial ODE as Bessel’s differential equation of order zero, so the solution is R(r)= J_0(αr) with eigenvalues defined by J_0(α c)=0 if a boundary (r=c) with Dirichlet condition is imposed.
Step 4: Solve T(t): T(t)= e^{-λkt} with λ= α^2.
Step 5: Expand the initial condition f(r) in terms of the orthogonal Bessel functions to determine the series coefficients.
Final Answer: u(r,t)= Σ_{n=1}^∞ A_n J_0(α_n r)e^{-α_n^2 k t}, where coefficients A_n= (2/c^2 J_1^2(α_n c)) ∫_0^c r f(r) J_0(α_n r) dr.

Cylindrical Coordinate BVP Example

QUESTION

Solve Laplace’s equation in spherical coordinates for the potential u(r,θ) with u(b,θ)=0, and u(r,θ)→ -Er cosθ as r→∞.

STEP-BY-STEP ANSWER:

Step 1: Express Laplace’s equation in spherical coordinates assuming azimuthal symmetry, and use separation of variables u(r,θ)= R(r) P_n(cosθ).
Step 2: The angular part leads to Legendre’s differential equation with P_n as solutions and selection of n=1 due to cosθ dependence.
Step 3: The radial ODE has general solution R(r)= c1 r^n + c2 r^{-(n+1)}. Boundedness as r→∞ forces c1 = 0.
Step 4: Apply the boundary condition at r=b to determine c2, then combine with the asymptotic behavior as r→∞ to match −Er cosθ.
Final Answer: u(r,θ)= −Er cosθ + (E c^3/r^2) cosθ, where c is determined from the boundary condition u(b,θ)=0.

Spherical Coordinate BVP Example

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Common Mistakes

  • Failing to properly enforce boundary conditions such as boundedness at the origin, leading to non-physical solutions.
  • Forgetting to set incompatible Fourier constant terms to zero to satisfy the full series (compatibility condition).
  • Misidentifying the order of Bessel functions or applying incorrect orthogonality properties when computing coefficients.
  • Neglecting the correct separation constant which can lead to eigenvalue problems with incorrect eigenfunctions.
  • Overlooking the effect of nonhomogeneous boundary conditions, which might require additional substitutions.