Wow! The below article is must read if you solve differential equations on a regular basis. Read it.
The quick summary table below is outstanding and directly copied from the article linked above. PDF link for full screen view.
Comparison Of Differential Equation Solver SoftwareTransmission Line Method (TLM) or Transmission Line Matrix method is an elegant technique used to solve lossless or lossy wave equations. Here I am gathering together resources on the topic since I have found it is not well studied in the US.
Christopoulos, Christos. “The transmission-line modeling (TLM) method in electromagnetics.” Synthesis Lectures on Computational Electromagnetics 1.1 (2005): 1-132. ISBN: 1598290509
This book is focused on the point of view of electrical engineering.
De Cogan, Donard. Transmission line matrix (TLM) techniques for diffusion applications. Gordon and Breach Science Publishers, 1998. ISBN: 978-9056991296
De Cogan, Donard, William J. O’Connor, and Susan Pulko. Transmission line matrix (TLM) in computational mechanics. CRC Press, 2005. ISBN: 978-0415327176
This is the book I found easiest to read from a Mechanical Engineering perspective. The book starts by looking at the wave equation in 1D. It uses the ideas of boundary conditions to explain the scattering effect that occurs in TLM. It is an easy read for an engineer who has studied the wave equation before.
http://dandadec.com/what-is-tlm/ – This page is Donard De Cogan’s page. Donard’s website has lots of information on TLM such as conferences and papers.
Often Ordinary Differential Equations (ODE’s) are studied in engineering, especially system dynamics and control theory. However, there are many cases where the system is described by Differential Algebraic Equations (DAE’s) rather than an ODE system.
Here are some resources related to the issue:
Computer methods for ordinary differential equations and differential-algebraic equations is the most famous book on DAE’s I have come across. When you read articles on DAE’s, the author usually references the following book.
Ascher, Uri M., and Linda R. Petzold. Computer methods for ordinary differential equations and differential-algebraic equations. Vol. 61. Siam, 1998. https://goo.gl/vmpq5l
I came across this book related to control theory and DAE’s
Berger, Thomas. On differential-algebraic control systems. Diss. Technische Universität Berlin, 2013. http://goo.gl/w4m5ms