About the Book
Advanced Calculus by Example first covers linear ordinary differential equations (ODEs). First, you learn and use 1st order linear ODEs. Second, you'll get to learn and use 2nd order linear ODEs. Third, you 3rd or higher order ODEs. And, then, you learn about ODEs of various orders that have polynomial functions. Many of these ODEs typically model classical, analytic mechanics found in engineering or classical Newtonian physics.
The second part of this book turns its focus onto linear and nonlinear partial differential equations (PDEs). First, you learn and use 1st order linear PDEs, often used in applications like classical mechanics. The variable u is dependent upon the independent spatial variables x, y, and maybe z if available. Next, you'll learn second or higher order linear PDEs (elliptic). These usually model steady state conduction of electric charges or heat transfer along a sheet or plate, etc.
Finally, this book dives into specific types of popular linear and nonlinear PDEs with applications found all over engineering and/or science, especially physics:
Explore examples and solutions of the mostly nonlinear KdV or Korteweg-de Vries PDEs. These PDEs typically model the fluid dynamics of waves on/over shallow water surfaces or solitons found in optics.
Cover examples and solutions of the linear and then nonlinear Schrodinger PDEs which are very important in applications including wave mechanics, quantum mechanics, particle physics, optics and much more found in physics, electrical engineering, and industrial/applied mathematics
Discuss a popular type of hyperbolic PDEs, the Telegraph PDEs. Both linear and then nonlinear examples are covered. These PDEs typically model the transmission of electro-magnetic (EM) waves and/or the flow of particles along a wire or similar, related medium.
Conclude with mostly some nonlinear PDEs of fluid mechanics. First, you'll learn about some hyrodynamic and boundary layer models. Then, learn to solve various nonlinear PDEs including Boussenesq, Euler, Hopf (gas), Tricomi, Ostrovsky (Ocean Waves), BBM (in-dispersive long waves), anisotropic media and more. Also, the Thin Film equation for bubbles and the liquid film mass transfer equation are covered.
This advanced calculus book for coursework purposes is essentially a second course on Advanced Calculus in mathematics or applied mathematics for undergraduate college/university students. It is an applied, definitions, then examples-driven approach. This book can also be used as an Engineering Mathematics and/or Mathematical Physics course textbook as well.
After reading and using this textbook, you'll come away with the skills to solve ODEs or PDEs on your own and take the next steps in your learning or career journey in data science, science, engineering or industrial / applied mathematics.
CONTENTS
1. 1st Order Linear ODEs
2. 2nd Order Linear ODEs
3. Higher Order Linear ODEs
4. Linear ODEs w/Polynomial Functions
5. 1st Order Linear PDEs
6. 2nd & Higher Order Linear PDEs
7. Nonlinear KdV PDEs
8. Linear Schrodinger PDEs
9. Nonlinear Schrodinger PDEs
10. Linear Telegraph PDEs
11. Nonlinear Telegraph PDEs
12. Nonlinear Boundary Layer PDEs
13. Other Nonlinear PDEs of Fluid Dynamics
ABOUT AUTHOR Steve Anglin, MSc, PhD(hc) is an applied mathematician, and has been a lecturer of mathematics for Case Western Reserve University and Saint Leo University. Steve has authored several books and 20+ journal articles on differential equations. Lastly, he is the founder, editor and publisher of the open access Journal of Applied Differential Equations (JADEs).
