Numerical Solutions to Partial Differential Equations with Finite Difference Methods

This book presents finite difference methods for three types of classical linear PDEs, three types of nonlinear PDEs and fractional PDEs. Specific topics cover two-point boundary value problems, elliptic equations, parabolic equations, hyperbolic equations, high-dimensional evolution equations, Schr\''{o}dinger equations, the Burgers' equation, the Korteweg-de Vries equation, and fractional diffusion-wave equations.

The book strives to achieve:

(a) Featured content. Thorough and dedicated presentations are provided for the finite difference methods. 

(b) Scattered difficulty. Starting from a simple two-point boundary value problem for an ODE, authors introduce core concepts and analytical techniques of the finite difference methods, then apply them to handle with various partial differential equations.

(c) Emphasis on practicability. For each algorithm, provided numerical examples enable students to learn how to apply it and verify theoretical results with numerical outcomes.

The book is suitable for advanced undergraduate and beginning graduate students in applied mathematics and engineering.

Zhi-Zhong Sun, Professor at the School of Mathematics, Southeast University. Sun was born in March 1963. He received his bachelor and master degree from Nanjing University in 1984 and 1987 respectively, and obtained his PhD from Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences in 1990. He engaged in the finite difference methods for the partial differential equations, especially the nonlinear evolutionary differential equations, fractional order differential equations.

Qifeng Zhang, Associate Professor at the Department of Mathematics, Zhejiang Sci-Tech University. Zhang was born in September 1987. He received his PhD from Huazhong University of Science and Technology in 2014. He visited Ecole Polytechnique Federale de Lausanne in 2020. His research interests are numerical analysis and simulation of nonlinear water wave equations and fractional order differential equations.

Guang-hua Gao, Associate Professor at the College of Science, Nanjing University of Posts and Telecommunications. Gao was born in November 1985. She obtained her PhD from Southeast University in 2012. She visited University of Macau in 2014. Her main research interest is numerical methods for partial differential equations.