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.