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This text will be useful for the four different audiences listed below. It is expected that all readers will have knowledge of basic calculus, linear algebra, and ordinary differential equations, and that the successful student will either already know elementary partial differential equations, or be concurrently learning that subject. The material is intended to be accessible to those without expertise in MATLAB, although a little prior experience with programming is probably required.
1. This text serves as a supplement for the student in an introductory partial differential equations course. A selection of the included exercises can be assigned as projects throughout the semester. Through the use of this text, the student will develop the skills to run simulations corresponding to the primarily theoretical course material covered by the instructor.
2. These notes work well as a standalone graduate course text in introductory scientific computing for partial differential equations. With prerequisite knowledge of ordinary and partial differential equations and elementary numerical analysis, most of the material can be covered and many of the exercises assigned in a one-semester course. Some of the harder exercises make substantial projects, and relate to topics from the other graduate mathematics courses graduate students typically take, e.g., differential equations and topics in nonlinear functional analysis.
3. Established researchers in theoretical partial differential equations may find these notes useful as well, particularly as an introductory guide for their research students. Those unfamiliar with MATLAB can use the included material as a reference in quickly developing their own applications in that language. A mathematician who is new to the practical implementation of methods for scientific computation in general can with relative ease, by working through a selection of exercises, learn how to implement and execute numerical simulations of differential equations in MATLAB. These notes can serve as a practical guide in independent study, undergraduate or graduate research experiences, or for help simulating solutions to specific thesis- or dissertation-related experiments. The author hopes that the ease and brevity with which the notes provide solutions to fairly significant problems will serve as inspiration.
4. The text can serve as a supplement for the instructor teaching any course in differential equations. Many of the examples can be easily implemented and the resulting simulation demonstrated by the instructor. If the course has a numerical component, a few exercises of suitable difficulty can be assigned as student projects. Practical assistance in implementing algorithms in MATLAB can be found in this text.
Scientist and engineers have valid motivations to become proficient at implementing numerical algorithms for solving PDE. The text’s emphasis on enforcing boundary conditions, eigenfunctions, and general regions will be useful as an introduction to their advanced applications. For the mathematician, accomplished or student, a more powerful benefit can be the tangible, visual realization of the objects of calculus, differential equations, and linear algebra. The high-level programs are developed by the reader from earlier programs and short fragments of relatable code. The resulting simulations are demonstrations of the properties of the underlying mathematical objects, where vectors represent functions, matrix operations represent differentiation and integration, and calculations such as solving linear systems or finding eigenvalues are easily accomplished with one line of code. Even without much prior knowledge of programming or MATLAB, by working through a selection of exercises in this text, the reader will be able to create working programs that simulate many of the classic problems from PDE, while gaining an understanding of the underlying fundamental mathematical principles.
The approach of the text is to first review MATLAB and a small selection of techniques from elementary numerical analysis, and then introduce difference matrices in the context of ordinary differential equations. We then apply these ideas to PDE, including topics from the heat, wave, and Laplace equations, eigenvalue problems, and semilinear boundary value problems. We enforce boundary conditions on regions including the interval, square, disk, and cube, and more general domains. We push the general notion that linear problems can be expressed as a single linear system, while many nonlinear problems can be solved via Newton’s method