Join a cohort of fellow mathematics enthusiasts to explore the fundamentals of partial differential equations (PDEs). This Daily Study Group offers a clear and comprehensive introduction to PDEs-their key properties, solution techniques and real-world applications-using the powerful built-in capabilities of Wolfram Language.
We’ll cover topics such as linear and nonlinear first-order PDEs, as well as important second-order equations including the heat, wave and Laplace equations, among others. Also, four famous modern PDEs - Black-Scholes, Schrodinger, Maxwell and Navier-Stokes - will be discussed.
The study group will be led by me and @Devendra Kapadia, and we’re excited to help you deepen your understanding of partial differential equations.
Some prior familiarity with Wolfram Language, multivariable calculus and differential equations is recommended.
Please feel free to use this thread to collaborate, ask questions and share ideas, materials and relevant resources with fellow learners.
Dates
January 12 - January 23, 2026, 11am - 12pm CT (4 - 5pm GMT)
Having a strange issue with the lessons. Starting with Lesson 5, when I scroll to example 5.4 (not evaluating each cell), there is a warning about dynamic content, and then Mathematica crashes. I tried it several times, and several Lessons (5, 6, 7, 21 & 23 for sure). Any thoughts on how to view the material and not crash the software? I am using version 14.0.0 Personal Edition, Windows 11 Pro with 24GB RAM. Thank you!
Are there any external resources for PDES please that you can recommend? In particular, I’m interested in the method of characteristics, the solutions method and the transformations of PDES to ODEs
It depends how you define the PDE. If it is defined as a*u_{xx}+b*u_{xy}+c*u_{yy}+...=0, then the discriminant will be b^2-4a*c. But if we write a*u_{xx}+2*b*u_{xy}+c*u_{yy}+...=0, where the factor 2 near second term is written for convenience, then the discriminant can be defined as b^2-a*c.
W. A. Strauss, Partial Differential Equations: An Introduction, Hoboken, NJ: John Wiley & Sons, 2008.
R. L. Herman, Introduction to Partial Differential Equations, 2015.
I. P. Stavroulakis and S. A. Tersian, Partial Differential Equations: An Introduction with Mathematica and MAPLE, River Edge, NJ: World Scientific, 2004.
P. Kythe, P. Puri and M. R. Schaferkotter, Partial Differential Equations and Mathematica, Boca Raton, FL: CRC Press, 1997.
E. Kreyszig, Advanced Engineering Mathematics, 10th ed., Hoboken, NJ: John Wiley & Sons, 2011.
I am sorry to hear about that. I will forward your message to our tech support team. Meanwhile you can try to disable Dynamic Updating from the Evaluation menue of your Wolfram App. I hope this will help.
Thanks for the question. The problem of PDEs with transparent boundary conditions is not covered in the course and study group. But you may find some useful information about PDEs with open boundaries using following links to Wolfram Documentations:
When Aram was discussing the Heat Equation in Lesson 8, I asked, “I assume that rho could be a function of position. Can it also be a function of time?” Juan Ortiz answered, “rho is the density of the material, it is mostly considered to be constant.” I replied, “Can standard solution techniques handle the case where rho varies with position (even if that is not the case in real-world problems)? If not, can they be extended to handle such a problem?” And Juan answered, “I do not think so. All the solutions here are for simpler examples but please post this question in the community forum so that Aram can discuss it better.” So here it is. Meanwhile, I asked Copilot, “The heat equation contains a term rho = the density of the medium. This is generally assumed to be constant, but are there real-world situations in which rho could be a function of position?” Then further, “The heat equation also contains a constant C = the heat capacity. Could that also be variable rather than constant?” I thought its responses might be of interest, so I will attach these.
