Cover
Preface
I Introduction to PDE2D
1. I.1 The Collocation and Galerkin Finite Element Methods
2. I.2 The PDE2D User Interfaces
3. I.3 Accuracy
4. I.4 Computer Time and Memory
5. I.5 Programming Hints
1 The Damped Spring and Pendulum Problems
1. 1.1 Derivation of the Damped Spring and Pendulum Equations
2. 1.2 Damped Spring and Pendulum Examples
3. 1.3 Problems
2 Beam and Plate Bending
1. 2.1 Derivation of Beam Bending Equation
2. 2.2 Derivation of Plate Bending Equation
3. 2.3 Beam and Plate Examples
4. 2.4 Problems
3 Diffusion and Heat Conduction
1. 3.1 Derivation of Diffusion Equation
2. 3.2 Diffusion and Heat Conduction Examples
3. 3.3 Problems
4 Pricing Options
1. 4.1 Derivation of Black–Scholes Equation
2. 4.2 Option Pricing Examples
3. 4.3 Problems
5 Elasticity
1. 5.1 Derivation of Elasticity Equations
2. 5.2 Elasticity Examples
3. 5.3 Problems
6 Incompressible Fluid Flow
1. 6.1 Derivation of Navier–Stokes Equations
2. 6.2 Stream Function and Penalty Method Approaches
3. 6.3 Fluid Flow Examples
4. 6.4 Problems
7 The Schrödinger and Other Eigenvalue Equations
1. 7.1 The Schrödinger Equation
2. 7.2 Schrödinger and Maxwell Equations Examples
3. 7.3 Problems
8 Minimal Surface and Membrane Wave Equations
1. 8.1 Derivation of Minimal Surface Equation
2. 8.2 Derivation of Membrane Wave Equation
3. 8.3 Examples
4. 8.4 Problems
9 The KPI Wave Equation
1. 9.1 A Difficult Nonlinear Problem
2. 9.2 Numerical Results
Appendix A: Formulas from Multivariate Calculus
Appendix B: Algorithms Used by PDE2D
1. B.1 The Galerkin and Collocation Finite Element Methods
2. B.2 1D Steady‐state Collocation Problems
3. B.3 2D Steady‐state Galerkin Problems
4. B.4 3D Steady‐state Collocation Problems
5. B.5 Nonrectangular 3D Regions
6. B.6 Time‐dependent Problems
7. B.7 Eigenvalue Problems
8. B.8 The PDE2D Parallel Solvers
Appendix C: Equations Solved by PDE2D
1. C.1 0D Problems
2. C.2 1D Problems (Galerkin Method)
3. C.3 2D Problems (Galerkin Method)
4. C.4 1D Problems (Collocation Method)
5. C.5 2D Problems (Collocation Method)
6. C.6 3D Problems
Appendix D: Problem 5.7 Local Solvers
1. D.1 DTD3M,DTD3N,DCG (problem57.f)
2. D.2 DTD3M,DTD3N,PCG,PBAND (problem57d.f)
References
Index
End User License Agreement

List of Illustrations

Introduction
1. Figure I.1 Triangulation of Venezuela.
2. Figure I.2 3D grid, hollowed‐out half torus.
3. Figure I.3 Collocation grid for Problem 1, Chapter 8.
4. Figure I.4 Manually supplied initial triangulation (Galerkin method).
5. Figure I.5 Automatically generated initial triangulation (Galerkin method).
6. Figure I.6 (a and b) Two pages from GUI session.
7. Figure I.7 Portion of PDE2D interactive session.
8. Figure I.8 Portion of Fortran program created by interactive session.
9. Figure I.9 Region used for accuracy tests.
Chapter 01
1. Figure 1.1 Damped spring oscillation and decay, Example 1.1.
2. Figure 1.2 Plot of , pendulum Example 1.2. (a) NSTEPS=1000 and (b) NSTEPS=100 000.
3. Figure 1.3 Resonance when Problem 2.
4. Figure 1.4 Pendulum Problem 3, . (a) and (b) .
5. Figure 1.5 Boundary value Problem 6 solution.
Chapter 02
1. Figure 2.1 Example 2.2 solution.
2. Figure 2.2 Solution for beam Problems 1 and 2.
3. Figure 2.3 MATLAB plot of Problem 3a solution.
Chapter 03
1. Figure 3.1 Conduction only, Example 3.1. (a) Initial temperature and (b) temperature at .
2. Figure 3.2 Conduction and convection, Example 3.1.
3. Figure 3.3 Convection only, Example 3.1.
4. Figure 3.4 Diffusion Example 3.2. (a) Manually‐generated initial triangulation and (b) adaptively generated final triangulation.
5. Figure 3.5 Contour plot of density, Example 3.2.
6. Figure 3.6 Bullet‐shaped 3D region, Example 3.3.
7. Figure 3.7 Concentration at a cross section of bullet, Example 3.3.
8. Figure 3.8 Axisymmetric solution, Example 3.3.
9. Figure 3.9 Original picture, Example 3.4.
10. Figure 3.10 Image enhancement, Example 3.4. (a) Initial conditions, with noise added and (b) final picture, after nonlinear diffusion.
11. Figure 3.11 Cross sections of and at steady state, Problem 1. (a) density vs. X and (b) density vs. X.
12. Figure 3.12 Fluid velocity field, Problem 2.
13. Figure 3.13 Temperature at , Problem 2. (a) and (b) .
14. Figure 3.14 Shock wave formation, Problem 3.
15. Figure 3.15 p1= cross section, Problem 4.
16. Figure 3.16 MATLAB cross‐sectional plots, Problem 4. (a) and (b) .
17. Figure 3.17 Noise reduction using nonlinear diffusion, Problem 5a. (a) Initial picture, with noise added and (b) final picture, after diffusion.
18. Figure 3.18 Initial conditions with LSQFIT=.FALSE., Problem 5b.
19. Figure 3.19 Noise reduction (“surf” plots), Problem 5b. (a) Initial conditions with LSQFIT=.TRUE. and (b) after diffusion.
20. Figure 3.20 Noise reduction (“imagesc” plots), Problem 5b. (a) Initial picture, with noise added and (b) final picture, after diffusion.
Chapter 04
1. Figure 4.1 Black–Scholes solution, Example 4.1.
2. Figure 4.2 Two‐asset solution of Example 4.3.
3. Figure 4.3 Value of put option, Problem 2a.
4. Figure 4.4 Jump diffusion model, Problem 4. (a) and (b) without jumps ().
Chapter 05
1. Figure 5.1 Displacement field, Example 5.1.
2. Figure 5.2 Stress field, Example 5.2.
3. Figure 5.3 3D arch, Example 5.3.
4. Figure 5.4 Displacement field cross sections, Example 5.3. (a) X=5 and (b) X=10.
5. Figure 5.5 Displacement field, Problem 1.
6. Figure 5.6 L‐shaped block, Problem 2. (a) Graded final triangulation and (b) displacement field.
7. Figure 5.7 Stress field, axisymmetric Problem 3.
8. Figure 5.8 Half ellipsoid, Problem 5.
9. Figure 5.9 Displacement field cross sections, Problem 5. (a) Z = 0 and (b) Y = 0.
10. Figure 5.10 Vector eigenfunction for third eigenvalue, Problem 6.
Chapter 06
1. Figure 6.1 Initial triangulation, Example 6.1.
2. Figure 6.2 Streamlines (lower Reynold's number), Example 6.1.
3. Figure 6.3 Higher Reynold's number velocity field. (a) Using stream function, Example 6.1 and (b) using penalty method, Example 6.2.
4. Figure 6.4 Penalty method pressure field, Example 6.2.
5. Figure 6.5 Coupled fluid flow and heat convection, Example 6.3.
6. Figure 6.6 Torus of Example 6.4.
7. Figure 6.7 Velocity field cross sections, Example 6.4. (a) () with contour of , at and (b) () with contour of −, at .
8. Figure 6.8 Problem 2, after steady‐state reached. (a) Streamlines and (b) vorticity.
9. Figure 6.9 Penalty method formulation, Problems 3a and b. (a) Velocity field, using collocation method and (b) pressure, using Galerkin method.
10. Figure 6.10 Stream function formulation, Problem 3c. (a) Streamlines and (b) vorticity.
11. Figure 6.11 Problem 4. (a) Velocity field and (b) pressure.
12. Figure 6.12 Axisymmetric flow, Problem 5.
13. Figure 6.13 Zoom of back flow, Problem 5.
14. Figure 6.14 Automatically generated initial triangulation of Problem 6.
15. Figure 6.15 Irrotational flow, Problem 6.
16. Figure 6.16 X=0.5 cross section, Problem 7. (a) Fluid velocity and (b) MATLAB plot of pressure.
17. Figure 6.17 MATLAB streamline plot, Problem 7.
Chapter 07
1. Figure 7.1 for orbital in hydrogen atom.
2. Figure 7.2 for a nonspherically symmetric orbital.
3. Figure 7.3 First 3 eigenvalues, Kronig–Penney, Example 7.2.
4. Figure 7.4 A sharp peak indicates an eigenvalue, Example 7.3.
5. Figure 7.5 Eigenfunctions for Example 7.3. (a) and (b) .
6. Figure 7.6 Probability density for Problem 1a.
7. Figure 7.7 molecule eigenfunctions, Problem 2a. (a) and (b) .
8. Figure 7.8 Axisymmetric eigenfunctions, Problem 2b. (a) and (b) .
9. Figure 7.9 for third eigenfunction, with , Problem 3b.
10. Figure 7.10 Eigenfunction cross sections, Problem 4. (a) and (b)
11. Figure 7.11 Second axisymmetric eigenfunction, Problem 6.
12. Figure 7.12 Parallelepiped of Problem 7.
13. Figure 7.13 Peaks at two known eigenvalues, Problem 9.
Chapter 08
1. Figure 8.1 Minimal surface, Example 8.1. (a) Surface plot and (b) contour plot.
2. Figure 8.2 Time variation of at parallelogram midpoint, Example 8.2.
3. Figure 8.3 MATLAB plots of membrane, Example 8.2. (a) Contour plot at and (b) surface plot at .
4. Figure 8.4 Minimal surface in annulus, Problem 2. (a) Full region (Problem 2a) and (b) symmetry used to reduce region (Problem 2b).
5. Figure 8.5 Damped membrane, Problem 3. (a) PDE2D plot of membrane at and (b) total energy as function of .
6. Figure 8.6 String height as function of and , Problem 4.
Chapter 09
1. Figure 9.1 Oblique collision, . (a) and (b) triangulation.
2. Figure 9.2 Oblique collision, . (a) and (b) triangulation.
3. Figure 9.3 Oblique collision, . (a) and (b) triangulation.
4. Figure 9.4 Oblique collision exact solution, .
5. Figure 9.5 Direct collision, t = 0. (a) and (b) triangulation.
6. Figure 9.6 Direct collision, . (a) and (b) triangulation.
7. Figure 9.7 Direct collision, . (a) and (b) triangulation.
8. Figure 9.8 Direct collision, . (a) and (b) triangulation.
9. Figure 9.9 Direct collision exact solution, and . (a) and (b) .
10. Figure 9.10 Results with uniform grid. (a) Oblique collision, and (b) direct collision, .
Appendix B
1. Figure B.1 PDE2D's triangular elements: (a) linear, (b) quadratic, (c) cubic, and (d) quartic. O = nodes, X = integration points.
2. Figure B.2 GUI page defining PDEs, Example 5.3.
3. Figure B.3 GUI page defining boundary conditions at P1 = 0 and 10.
4. Figure B.4 Distributed band matrix. x = may be nonzero in original matrix and y = may become nonzero after pivoting.

This edition first published 2018
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Library of Congress Cataloging-in-Publication Data:

Names: Sewell, Granville, author.
Title: Solving partial differential equation applications with PDE2D / Granville Sewell.
Description: Hoboken, NJ : John Wiley & Sons, 2018. | Includes bibliographical references and index. |
Identifiers: LCCN 2018018589 (print) | LCCN 2018031767 (ebook) | ISBN 9781119507956 (Adobe PDF) | ISBN 9781119507963 (ePub) | ISBN 9781119507932 (hardcover)
Subjects: LCSH: Differential equations, Partial–Numerical solutions–Computer programs.
Classification: LCC QA377 (ebook) | LCC QA377 .S4643 2018 (print) | DDC 515/.353028553–dc23
LC record available at https://lccn.loc.gov/2018018589

Cover Design by Wiley
Cover image: © Courtesy of Granville Sewell

Preface

This book looks at a wide range of ordinary and partial differential equation (PDE) applications. Students using this text will actually solve many interesting science and engineering applications using PDE2D, an easy‐to‐use, general‐purpose PDE solver developed by the author over a 40‐year period, which is free with the purchase of this book. They will learn to derive and solve the ordinary or partial differential equations, with boundary and initial conditions, for many time‐dependent, steady‐state, and eigenvalue applications, including diffusion, heat conduction and convection, image processing, math finance, fluid flow, elasticity, and quantum mechanics, in one, two, and three space dimensions. That PDE2D can be used to solve a wide variety of applications is evidenced by the list of over 250 journal articles and books at www.pde2d.com, in which it has been used to generate some or all of the numerical results.

Much of the material in the book was developed for two graduate courses, “Seminar in Applied Mathematics” at Texas A&M University and “Advanced Scientific Computing” at the University of Texas El Paso, but the book could also be used as a supplementary text for a number of science or engineering courses in which PDE applications are studied. It could also be used as a reference by individual students or researchers interested in using PDE2D to solve their specific applications.

Some documentation on the mathematical algorithms used by PDE2D can be found in Appendix B. The focus in this book, however, is on use of PDE2D, not the mathematics behind it, and students are not primarily learning about numerical methods, though they will learn some things, but rather about modeling real‐world applications using differential equations.

The book starts with some simple ordinary differential equation problems in Chapter 1, which give the student a chance to become familiar with PDE2D before proceeding to more difficult problems, and ends with the solution in Chapter 9 of a very difficult nonlinear problem, which requires a moving adaptive grid because the solution has sharp, moving peaks.

The Windows/GFortran version of PDE2D is available at no cost with the book purchase: go to the “Free with Book” page at www.pde2d.com.

Linux and Mac OS X versions are available for a fee, also from www.pde2d.com. All three versions require the GNU GFortran compiler, but this can be downloaded from http://gcc.gnu.org/wiki/GFortranBinaries for free. Instructors using this book as a required text for a university class, and their students, can also obtain the Linux or Mac OS X version at no cost.

Each problem comes with some graphical or numerical output so the student can tell when his/her program is working correctly. The example programs can be downloaded from www.pde2d.com.