Differential Equations

This book introduces Finite Difference Methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations.

This text provides a very simple, initial introduction to the complete scientific computing pipeline: models, discretization, algorithms, programming, verification, and visualization.

This textbook teaches Finite Element Methods from a computational point of view. It focuses on how to develop flexible computer programs with Python, a programming language in which a combination of symbolic and numerical tools is used to achieve an explicit and practical derivation of finite element algorithms.

The FEniCS Project is an open-source Finite Element library. It gives its users easy access to powerful and scalable numerical tools for solving a great number of problems based on partial differential equations (PDEs).

The theory of difference equations, the methods used in their solutions, and their wide applications have advanced beyond their adolescent stage to occupy a central position in applicable analysis. In fact, in the last ten years, the proliferation of the subject is witnessed by numerous research articles and several monographs, annual international conferences, and new journals devoted to the study of difference equations and their applications.

This easy-to-read book introduces the basics of solving partial differential equations by means of Finite Difference Methods. Unlike many of the traditional academic works on the topic, this book was written for practitioners. Accordingly, it especially addresses: the construction of finite difference schemes, formulation and implementation of algorithms, verification of implementations, analyses of physical behavior as implied by the numerical solutions, and how to apply the methods and software to solve problems in the fields of physics and biology.

This book offers a concise and gentle introduction to Finite Element programming in Python based on the popular FEniCS software library. Using a series of examples, including the Poisson equation, the equations of linear elasticity, the incompressible Navier–Stokes equations, and systems of nonlinear advection–diffusion–reaction equations, it guides readers through the essential steps to quickly solving a PDE in FEniCS, such as how to define a finite variational problem, how to set boundary conditions, how to solve linear and nonlinear systems, and how to visualize solutions and structure finite element Python programs.