Book cover for Advanced Engineering Mathematics

Advanced Engineering Mathematics

Dennis G. Zill, Michael R. Cullen

ISBN #9780763740955

3rd Edition

4,310 Questions

17,647 Students Helped

Homework Questions

Summary

Learning Objectives

Key Concepts

Example Problems

Explanations

Common Mistakes

Summary

This section emphasizes the use of finite difference methods to discretize and solve partial differential equations, including Laplace’s, heat, and wave equations. Key iterative techniques like Gauss–Seidel and Crank–Nicholson ensure convergence and stability, provided the proper conditions on parameters are met. In addition, the study of complex numbers further reinforces computational techniques in algebra, serving as a foundation for more advanced topics in complex analysis and its applications across mathematics and engineering.

Learning Objectives

1

Explain the finite difference methods used to approximate solutions for Laplace’s, heat, and wave equations.

2

Apply iterative techniques such as Gauss–Seidel and Crank–Nicholson methods to solve discretized partial differential equations (PDEs).

3

Determine stability conditions and parameter choices (e.g., grid spacing, time steps, and ?) for accurate numerical solutions.

4

Perform algebraic manipulations with complex numbers and compute operations as introduced in functions of a complex variable.

Key Concepts

CONCEPT

DEFINITION

Laplace’s Equation

A second‐order PDE of the form u_xx + u_yy = 0 representing steady state conditions. In numerical methods, its solution is approximated by finite difference equations with boundary values.

Heat Equation

A PDE given by u_t = k u_xx (or with additional factors) describing the diffusion of heat. Numerical schemes discretize both time and space, often using stability parameters like λ.

Wave Equation

A PDE such as u_tt = c^2 u_xx modeling vibrations or wave propagation, where finite difference methods approximate the time and spatial derivatives.

Finite Difference Method

A numerical method that replaces derivatives by approximations using differences between adjacent grid values to solve differential equations.

Gauss–Seidel Iteration

An iterative technique used to solve systems of algebraic equations by updating unknowns sequentially until convergence is achieved.

Crank–Nicholson Method

A time-stepping method for parabolic PDEs that uses a weighted average of explicit and implicit schemes to enhance stability and accuracy.

Complex Numbers

Numbers of the form a + bi where a and b are real numbers and i is the imaginary unit (i^2 = -1), fundamental in many areas of engineering and analysis.

Example Problems

Example 1

The figure shows the values of $u(x, y)$ along the boundary. We need to determine $u_{11}$ and $u_{21} .$ The system is \[ u_{21}+2+0+0-4 u_{11}=0 \quad-4 u_{11}+u_{21}=-2 \] \[ 1+2+u_{11}+0-4 u_{21}=0 \quad \text { or } \quad u_{11}-4 u_{21}=-3 \] Solving we obtain $u_{11}=11 / 15$ and $u_{21}=14 / 15.$

Example 2

The figure shows the values of $u(x, y)$ along the boundary. We need to determine $u_{11}$ $u_{21},$ and $u_{31} .$ By symmetry $u_{11}=u_{31}$ and the system is \[ \begin{aligned} u_{21}+0+0+100-4 u_{11} &=0 \\ u_{31}+0+u_{11}+100-4 u_{21} &=0 \quad \text { or } \end{aligned} \quad-4 u_{11}+u_{21}=-100 \] \[ 0+0+u_{21}+100-4 u_{31}=0 \] Solving we obtain $u_{11}=u_{31}=250 / 7$ and $u_{21}=300 / 7.$

Example 3

The figure shows the values of $u(x, y)$ along the boundary. We need to determine $u_{11}, u_{21}$ $u_{12},$ and $u_{22} .$ By symmetry $u_{11}=u_{21}$ and $u_{12}=u_{22} .$ The system is \[ \begin{aligned} u_{21}+u_{12}+0+0-4 u_{11} &=0 \\ 0+u_{22}+u_{11}+0-4 u_{21} &=0 & & 3 u_{11}+u_{12}=0 \\ u_{22}+\sqrt{3} / 2+0+u_{11}-4 u_{12} &=0 & \text { or } & u_{11}-3 u_{12}=-\frac{\sqrt{3}}{2} \\ 0+\sqrt{3} / 2+u_{12}+u_{21}-4 u_{22} &=0 & & \end{aligned} \] Solving we obtain $u_{11}=u_{21}=\sqrt{3} / 16$ and $u_{12}=u_{22}=3 \sqrt{3} / 16.$

Example 4

The figure shows the values of $u(x, y)$ along the boundary. We need to determine $u_{11}, u_{21}$ $u_{12},$ and $u_{22} .$ The system is \[ \begin{aligned} u_{21}+u_{12}+8+0-4 u_{11} &=0 & &-4 u_{11}+u_{21}+u_{12} &=-8 \\ 0+u_{22}+u_{11}+0-4 u_{21} &=0 & & u_{11}-4 u_{21}+u_{22} &=0 \\ u_{22}+0+16+u_{11}-4 u_{12} &=0 & \text { or } & u_{11}-4 u_{12}+u_{22} &=-16 \\ 0+0+u_{12}+u_{21}-4 u_{22} &=0 & & u_{21}+u_{12}-4 u_{22} &=0 \end{aligned} \] Solving we obtain $u_{11}=11 / 3, u_{21}=4 / 3, u_{12}=16 / 3,$ and $u_{22}=5 / 3.$

Example 5

The figure shows the values of $u(x, y)$ along the boundary. For Gauss-Seidel the coefficients of the unknowns $u_{11}, u_{21}, u_{31}, u_{12}, u_{22}, u_{32}, u_{13}, u_{23}, u_{33}$ are shown in the matrix $$\left[\begin{array}{ccccccccc} 0 & .25 & 0 & .25 & 0 & 0 & 0 & 0 & 0 \\ 25 & 0 & .25 & 0 & .25 & 0 & 0 & 0 & 0 \\ 0 & .25 & 0 & 0 & 0 & .25 & 0 & 0 & 0 \\ 25 & 0 & 0 & 0 & .25 & 0 & .25 & 0 & 0 \\ 0 & .25 & 0 & .25 & 0 & .25 & 0 & .25 & 0 \\ 0 & 0 & .25 & 0 & .25 & 0 & 0 & 0 & .25 \\ 0 & 0 & 0 & .25 & 0 & 0 & 0 & .25 & 0 \\ 0 & 0 & 0 & 0 & .25 & 0 & .25 & 0 & .25 \\ 0 & 0 & 0 & 0 & 0 & .25 & 0 & .25 & 0 \end{array}\right]$$ The constant terms in the equations are $0,0,6.25,0,0,12.5,6.25,12.5,37.5 .$ We use 25 as the initial guess for each variable. Then $u_{11}=6.25, u_{21}=u_{12}=12.5, u_{31}=u_{13}=18.75, u_{22}=25, u_{32}=u_{23}=37.5,$ and $u_{33}=56.25.$

Step-by-Step Explanations

QUESTION

Given a discretized Laplace problem with boundary conditions, how do you form and solve the system using Gauss–Seidel iteration?

STEP-BY-STEP ANSWER:

Step 1: Discretize the domain into a grid and approximate the second derivatives using finite differences, e.g., u(i+1,j) + u(i−1,j) + u(i,j+1) + u(i,j−1) − 4u(i,j) = 0.
Step 2: Apply the boundary values provided by the problem to set up the system for interior nodes (like u11, u21, u31, etc.).
Step 3: Rearrange the finite difference equations to express each unknown as a combination of its neighbors.
Step 4: Use an iterative approach (Gauss–Seidel method): start with an initial guess, then update each unknown sequentially with the new values until convergence is achieved.
Final Answer: The converged values for u11, u21, etc., form the numerical approximation of the solution of Laplace’s equation within the domain.

Solving a Finite Difference System for Laplace’s Equation

QUESTION

How do you discretize the heat equation u_t = k u_xx and determine the stability parameter λ?

STEP-BY-STEP ANSWER:

Step 1: Divide the spatial domain into n segments with step size h and the time interval into m steps with step size k.
Step 2: Replace the time derivative with a forward difference and the spatial second derivative with a central difference, yielding u(i,j+1) = u(i,j) + λ (u(i+1,j) + u(i−1,j) − 2u(i,j)) where λ = k/h^2 (possibly multiplied by other constants).
Step 3: Incorporate boundary conditions to yield a complete system.
Step 4: Check the stability condition (e.g., for many explicit schemes, λ must be ≤ 0.5) to ensure the numerical solution remains stable.
Final Answer: With proper h, k, and λ values, the scheme approximates the steady and transient behavior of the heat equation.

Discretizing the Heat Equation

QUESTION

How do you simplify an expression involving complex numbers, such as i^11 or products/divisions of complex numbers?

STEP-BY-STEP ANSWER:

Step 1: Recognize that powers of i cycle with period 4 (i^2 = -1, i^3 = -i, i^4 = 1, etc.).
Step 2: Express the exponent modulo 4 (for i^11, 11 mod 4 = 3, so i^11 = i^3 = -i).
Step 3: For multiplication/division, express each complex number in the form a + bi and combine like terms, using conjugates for division.
Final Answer: The expression simplifies through systematic use of i’s periodicity and conjugate multiplication techniques.

Complex Number Operations

Common Mistakes

Failing to correctly apply boundary conditions when setting up the discretized equations.
Incorrect computation of the stability parameter (?), leading to unstable or divergent solutions.
Misordering iterative updates in the Gauss–Seidel method, which can slow down or prevent convergence.
Errors in handling complex arithmetic, such as forgetting that i^2 = -1 or misapplying the conjugate method in division.